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Preface

This research monograph is a follow-up to the book Geometry of Cuts and Metrics by M.Deza and M.Laurent, published in 1997 by Springer-Verlag, Berlin (Russian translation was published in 2001 by MCNMO, Moscow). The main object of that book was the ℓ1 -metrics, i.e. those isometrically embeddable, up to a scale, into some hypercube Hm or, if infinite, into some cubic lattice Zm . During the last six years a lot of work was done on a special case of ℓ1 -metric: the graph distance of the skeleton of (finite or infinite) polytope. This monograph consists mainly of identifying such polytopes combinatorially ℓ1 -embeddable, within interesting lists of polytopal graphs, i.e. such that corresponding polytopes are either prominent mathematically (regular partitions, root lattices, uniform polytopes and so on), or applicable in Chemistry (fullerenes, polycycles etc.) The embeddability, if any, provides applications to chemical graphs and, in the first case, it gives new combinatorial perspective to ℓ2 -prominent affine polytopal objects. The lists of polytopal graphs in the book, come from broad areas of Geometry, Crystallography and Graph Theory; so, just to introduce them we need many definitions. The book concentrates on such concise and, as much as possible, independent definitions. The scale-isometric embeddability - the main unifying question, to which those lists will be subjected - will be presented with the minimum of technicalities. The main families of the considered graphs come from: various generalization of regular polytopes (or tilings), from (point) lattices and from applications in Chemistry. Some samples of results are: (i) All embeddable regular tilings and honeycombs of dimension d > 2, are, besides the hyper-simplices and hyper-octahedra, exactly those with v

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Scale Isometric Polytopal Graphs

bipartite skeleton: the hyper-cubes, cubic lattices and 11 special tilings of hyperbolic space. (ii) If P is an Archimedean polyhedron or a plane partition, other than 3-gonal prism, then exactly one of P and its dual P is embeddable. (iii) For the regular 4-polytope 24-cell, its usual and golden truncation (Gosset’s semi-regular 4-polytope) are embedded into H12 and half-H12 , respectively. (iv) The skeletons of Voronoi tiling for the lattice An and its dual lattice ∗ An are embedded into Zn+1 and Z(n+1) , respectively. 2 The book is organized as follows. Relatively long introduction (chapter 1) gives main notions, as well as methods of embedding. After reading it, any of the other chapters can be read independently. Chapters 14 and 15 consider, respectively, specifications and generalizations of the notion of embeddability. Each of chapters 2–13 is centered around embeddability for a particular list of graphs. We tried to give concise and, as much as possible, independent presentation of those lists; so that the readers of different backgrounds will be able to isolate “ready to use” chapters, which are of interest for them. Chapters 2, 4, 5, 6, 12, 13 treat various lists of 3-polytopes. Chapters 9, 10 and 11 consider infinite graphs coming from the tilings of R2 , R3 and from lattices. Chapters 3, 7, 11 consider graphs in Rn . Finally, chapters 2, 8 and 11 can be of interest for workers in Mathematical Chemistry and Crystallography. The authors are grateful to Marie Grindel, Jacques Beigbeder and, especially, to Mathieu Dutour for various help: in drawings, redaction and inspiration. 1991 Mathematics Subject Classification: primary 05C12; secondary 52C99

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Contents

Preface

v

1. Introduction: Graphs and their Scale-isometric Embedding

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Graphs . . . . . . . . . . . . . . . . . . . . . Embeddings of graphs . . . . . . . . . . . . Embedding of plane graphs . . . . . . . . . Types of regularity of polytopes and tilings Operations on polytopes . . . . . . . . . . . Voronoi and Delaunay partitions . . . . . . Infinite graphs . . . . . . . . . . . . . . . . .

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2. An Example: Embedding of Fullerenes 2.1 2.2 2.3

25

Embeddability of fullerenes and their duals . . . . . Infinite families of non-ℓ1 fullerenes . . . . . . . . . . Katsura model for vesicles cells versus embeddable fullerenes . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . dual . . .

3. Regular Tilings and Honeycombs 3.1 3.2 3.3 3.4

Regular tilings and honeycombs The planar case . . . . . . . . . Star-honeycombs . . . . . . . . The case of dimension d ≥ 3 .

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Semi-regular polyhedra . . . . . . . . . . . . . . . . . . . . Moscow, Globe and Web graphs . . . . . . . . . . . . . . . vii

26 30 30 35

4. Semi-regular Polyhedra and Relatives of Prisms and Antiprisms 4.1 4.2

1 3 10 14 17 18 19

35 36 40 40 43 43 45

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4.3 4.4

Stellated k-gons, cupolas and antiwebs . . . . . . . . . . . Capped antiprisms and columns of antiprisms . . . . . . .

5. Truncation, Capping and Chamfering 5.1 5.2 5.3

48 50 53

Truncations of regular partitions . . . . . . . . . . . . . . Partial truncations and cappings of Platonic solids . . . . Chamfering of Platonic solids . . . . . . . . . . . . . . . .

53 54 59

6. 92 Regular-faced (not Semi-regular) Polyhedra

63

7. Semi-regular and Regular-faced n-polytopes, n ≥ 4

71

7.1 7.2 7.3 7.4

Semi-regular (not regular) n-polytopes . . . . Regular-faced (not semi-regular) n-polytopes Archimedean 4-polytopes . . . . . . . . . . . The embedding of the snub 24-cell . . . . . .

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8. Polycycles and Other Chemically Relevant Graphs 8.1 8.2 8.3

75

(r,q)-polycycles . . . . . . . . . . . . . . . . . . . . . . . . Quasi-(r, 3)-polycycles . . . . . . . . . . . . . . . . . . . . Coordination polyhedra and metallopolyhedra . . . . . . .

9. Plane Tilings 9.1 9.2 9.3

71 72 72 73

75 77 80 83

58 embeddable mosaics . . . . . . . . . . . . . . . . . . . . Other special plane tilings . . . . . . . . . . . . . . . . . . Face-regular bifaced plane tilings . . . . . . . . . . . . . .

10. Uniform Partitions of 3-space and Relatives

83 87 89 99

10.1 28 uniform partitions . . . . . . . . . . . . . . . . . . . . . 100 10.2 Other special partitions . . . . . . . . . . . . . . . . . . . 103 11. Lattices, Bi-lattices and Tiles 11.1 11.2 11.3 11.4

Irreducible root lattices . . . . . . The case of dimension 3 . . . . . . Dicings . . . . . . . . . . . . . . . . Polytopal tiles of lattice partitions

12. Small Polyhedra

107 . . . .

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107 108 110 111 115

12.1 Polyhedra with at most seven faces . . . . . . . . . . . . . 115

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Contents

12.2 Simple polyhedra with at most eight faces . . . . . . . . . 116 13. Bifaced Polyhedra 13.1 13.2 13.3 13.4 13.5 13.6

119

Goldberg’s medial polyhedra . . . . Face-regular bifaced polyhedra . . Constructions of bifaced polyhedra Polyhedra 3n and 4n . . . . . . . . Polyhedra 5n (fullerenes) revisited Polyhedra ocn (octahedrites) . . .

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14. Special ℓ1 -graphs

120 123 125 126 129 129 137

14.1 Equicut ℓ1 -graphs . . . . . . . . . . . . . . . . . . . . . . . 137 14.2 Scale one embedding . . . . . . . . . . . . . . . . . . . . . 145 15. Some Generalization of ℓ1 -embedding 15.1 15.2 15.3 15.4

Quasi-embedding . . . Lipschitz embedding . Polytopal hypermetrics Simplicial n-manifolds

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153 . . . .

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153 157 157 160

Bibliography

163

Index

171

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Chapter 1

Introduction: Graphs and their Scale-isometric Embedding

In this chapter we introduce some basic definitions about graphs, embedding and polytopes. We present here only basic notions, further definitions will be introduced later. The reader can consult the following books for more detailed information: [Gr¨ un67], [Coxe73], [CoSl88], [DeLa97], [Crom97]. 1.1

Graphs

A simple graph G = (V, E) consists of a set V of vertices and a set E of edges. Every edge e ∈ E consists of two end-vertices u and v; we set e = (u, v). Those two vertices are called adjacent and each of these vertices u and v is incident to the edge e. The degree of a vertex v ∈ V is the number of edges, to which the vertex v is incident. If every two vertices of G are adjacent, then G is called complete. A complete graph on n vertices is denoted as Kn . If a graph has no edges, it is called empty. For U ⊆ V , let EU be the set of edges having both its end-vertices in U and the graph GU := (U, EU ) is called induced subgraph (by U ) of G. A graph G is called bipartite graph if its vertices can be partitioned into two non-void parts, V = V1 ∪ V2 , in such a way that the graphs induced by V1 and V2 are both empty. If G is bipartite with bipartition (V1 , V2 ) and if every vertex of V1 is adjacent to every vertex of V2 , then G is called a complete bipartite graph. Let Kn,m denote the complete bipartite graph with |V1 | = n and |V2 | = m. The complete bipartite graph K1,m (m ≥ 1) is called a star. A set E ′ of edges is called a matching if no two edges of E ′ have a common end-vertex. A matching is called perfect if every vertex is an end1

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vertex of the matching. (Only graphs with even number of vertices can have a perfect matching.) The following graphs will be frequently used in the book: • The multipartite graph Kn1 ,n2 ,...,nk with k parts of sizes n1 , n2 , . . . , nk is the graph, whose edges are all having end-vertices in different parts. An example is the complete bipartite graph Kn,m (k = 2). • The path Pn =P{v1 ,v2 ,...,vn } with vertex-set V = {v1 , v2 , . . . , vn }, whose edges are (vi , vi+1 ) for 1 ≤ i ≤ n − 1. • The circuit Cn =C{v1 ,v2 ,...,vn } (or n-gon) is obtained from the path P{v1 ,v2 ,...,vn } by adding the edge (v1 , vn ). • The hypercube graph Hn is the graph with vertex-set V = {0, 1}n, whose edges are the pairs of vectors (x1 , . . . , xn ) and (y1 , . . . , yn ) in {0, 1}n such that |{i : xi 6= yi }| = 1. • The half-cube graph 21 Hn is the graph with vertex-set V = Pn {(x1 , . . . , xn ) ∈ {0, 1}n : i=1 xi is even} (in even representation, Pn and i=1 xi is odd in odd representation), whose edges are pairs x, y ∈ {0, 1}n such that |{i : xi 6= yi }| = 2. • The cubic lattice graph Zn is an infinite graph with vertexset Zn = {(x1 , x2 , . . . , xn ) : xi ∈ Z}, whose edges are P ((x1 , . . . , xn ), (y1 , . . . , yn )) such that 1≤i≤n |xi − yi | = 1. • The half-cubic lattice graph 12 Zn is an infinite graph with vertex-set P V = {(x1 , . . . , xn ) ∈ Zn : ni=1 xi is even}, whose edges are pairs P of vectors x, y ∈ V such that 1≤i≤n |xi − yi | = 2. (The graph 1 2 Zn has the same set of vertices as the root lattice Dn ; see chapter 11). • The cocktail-party graph Kn×2 is obtained from the complete graph K2n by deleting the edges of a perfect matching. There are isomorphisms amongst above graphs: K2,2 = C4 = H2 = K2×2 , K2 = P2 =

1 1 1 H2 , H3 = K4 , H4 = K3×2 . 2 2 2

In Coxeter’s notation, αn , βn , γn and 21 γn denote, respectively, following n-dimensional convex polytopes: regular n-simplex, n-cross-polytope, ncube and half-n-cube; their respective 1-skeleton graphs (see chapter 1.4 below) are Kn+1 , Kn×2 , Hn and 21 Hn . A graph G is said to be connected if, for every two vertices u, v in G, there is a path in G joining u and v; a graph, which is not connected is said

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Introduction

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3

to be disconnected. Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs. Their Cartesian product G1 × G2 is the graph G := (V1 × V2 , E) with vertex-set V1 × V2 = {(v1 , v2 ) : v1 ∈ V1 and v2 ∈ V2 }, whose edges are the pairs ((u1 , u2 ), (v1 , v2 )), where u1 , v1 ∈ V1 and u2 , v2 ∈ V2 , such that either (u1 , v1 ) ∈ E1 and u2 = v2 , or (u2 , v2 ) ∈ E2 and u1 = v1 . For a graph G, its suspension ∇G is the graph obtained from G by adding a new vertex (called the apex of ∇G) and making it adjacent to all vertices of G. The graphs ∇Cn and Cn × K2 are called n-wheel and n-prism, respectively. The line graph L(G) of G is the graph, whose vertices are the edges of G with two vertices of L(G) adjacent in L(G) if the corresponding edges of G share a common vertex. We denote L(Kn ) by T (n). This graph is the special case k = 2 of the Johnson graph J(n, k). The Johnson graph J(n, k) has the family of all k-subsets of an n-set as its vertex-set. Two vertices of J(n, k) are adjacent if and only if the cardinality of the intersection of the corresponding k-sets equals to k − 1. In particular, J(n, 1) = Kn . 1.2

Embeddings of graphs

Before recalling some graph-related metric notions, we briefly discuss the relevance of such embedding. A graph G can be isometrically embedded into a half-cube 21 Hm if we can label or encode the vertices of G by a binary string of length m with an even number of ones such that the square of the Euclidean distance between the labels is twice the path distance of the vertices in the graph G. If, moreover, the embedding is ℓ1 -rigid, then we have an essentially unique encoding. Such labeling of vertices can be useful, for example, in Chemistry, for a nomenclature of fullerenes (see chapter 2) and for the calculation of molecular parameters depending only on the graphical distances such as the Wiener, J and other indices (see, for example, [BLKBSSR95]). A semimetric on a set V is a real symmetric function d(x, y) defined on all pairs of points x, y ∈ V satisfying d(x, x) = 0 and for each ordered triple (x, y, z) of points of V , the following triangle inequality: d(x, y) + d(y, z) − d(x, z) ≥ 0.

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Summing two inequalities for the triples (x, y, z) and (x, z, y), we obtain the inequality 2d(y, z) ≥ 0. Hence any semimetric d is non-negative, i.e. takes only non-negative values. A metric is a positive semimetric, i.e. such that d(x, y) = 0 if and only if x = y. Let G = (V, E) be a connected graph (finite or not), where V and E are the sets of its vertices and edges, respectively. The path-metric dG , associated with the graph G, is the integer valued metric on the vertices of G, which is defined by setting dG (v, u) equal to the length of the shortest path in G joining v and u. distances between the vertices of an n-vertex The sum of all n(n−1) 2 graph G is denoted by W (G). Chemists call it the Wiener number. A connected subgraph G1 of G is called an isometric subgraph of G if dG = dG1 on the vertices of G1 , i.e. if the distances of G are preserved in G1 . In this case we write G1 ≺ G. A geodesic is a (possibly, infinite in one or two directions) simple path P with the property that dP (x, y) = dG (x, y) for any x, y ∈ P . The metric dG allows us to introduce convex subsets on V . A subset S ⊆ V is called convex subset if for any vertices u, v ∈ S all vertices of every shortest (u, v)-path (i.e. geodesic), belong to S. The set Zn is naturally endowed with an ℓ1 -metric. Namely, for x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in Zn , the ℓ1 -distance between x and y is P d(x, y) = ni=1 |xi −yi |. Zn is the graph of Zn , whose path-metric coincides with the ℓ1 -metric d. Similarly, the hypercube graph Hn is the subgraph of Zn induced by {0, 1}n. (The path-metric dHn is also the square of the Euclidean l2 -metric.) An ℓ1 -graph is a graph G, whose path-metric dG is, up to a scale λ, isometrically embeddable into a hypercube Hm or, if G is infinite, into Zm . That is, for some λ, m ∈ IN , there is a mapping φ : V → Hm or Zm , such that: m X |φk (vi ) − φk (vj )| λ · dG (vi , vj ) = ||φ(vi ) − φ(vj )||ℓ1 = k=1

with vi ∈ V and φ(vi ) = (φ1 (vi ), . . . , φm (vi )) ∈ Hm or in Zm . The least integer λ is called the minimal scale of the embedding. For example, the half-cube graph 21 Hn and the half-cubic lattice graph 1 2 Zn are naturally embedded into Hn and Zn , respectively, with the scale λ = 2. Hence 12 Hn and 21 Zn are ℓ1 -graphs. Recall that the 2m vertices of the m-cube Hm can be labeled by all 2m subsets of the set {1, 2, . . . , m}, such that two vertices with labels A and

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Introduction

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5

B are adjacent if and only if |A△B| = 1, where A△B is the symmetric difference of the sets A and B. Hence the scale λ embedding φ : V → Hm is equivalent to a labeling of each vertex v ∈ V (G) by a set φ(v), such that vertices v and u are adjacent if and only if |φ(v)△φ(u)| = λ. On the other hand, this labeling implies a labeling of edges (v, u) by the sets φ(v)△φ(u). For i ∈ {1, 2, . . . , m}, call the set of edges, labels of which contain i, a i-zone or, simply, a zone. Both these labeling of vertices and edges of G are used in Figures of the chapter 2, for example. If G is embedded into a hypercube Hm , then any partition of Hm into two opposite facets induces a partition S ∪ S = V of the set of vertices of G. Any partition S ∪ S = V is called the cut {S, S}. The edge set E(S, S) of the cut {S, S} is the set of edges with one end in S and another one in S. Evidently, removing E(S, S) from G we obtain a graph with at least two connected components, i.e. E(S, S) is a cutset of edges. The edges of the set E(S, S) are cut by the cut {S, S}. The cut {S, S} defines the cut semimetric δ(S) on the set V : 0 if i, j ∈ S or i, j ∈ S δ(S)(i, j) = δ{S,S} (i, j) = 1 otherwise. Let us project the hypercube Hm with an embedded graph G along the edges connecting two opposite facets. Then we obtain an embedding of G into Hm−1 , such that some distances of G are diminished by one. In other words, we embed G endowed with the new semimetric dG − δ(S). In such a way we obtain a decomposition of the path-metric dG of an ℓ1 embeddable graph G (actually, of any ℓ1 -metric) into a non-negative linear combination of cut semimetrics. All ℓ1 -semimetrics on n vertices (that is, allℓ1 -semimetric spaces (Vn , d) n with |Vn | = n), considered as points of an 2 -dimensional space, form a n 2 -dimensional pointed cone called the cut cone. This cone is generated by the 2n−1 − 1 extreme rays. Each extreme ray is a non-zero cut semimetric δ(S) for some proper subset S of Vn = {1, 2, . . . , n}. In other words, a graph G (or any metric) is ℓ1 -embeddable graph if and only if the path-metric dG is a linear combination, with non-negative coefficients, of cut semimetrics: dG =

X

S⊂Vn

aS δ(S) with aS ≥ 0 for all S.

If G is embeddable into Hm with a scale λ, then the above decomposition

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can be rewritten as follows λdG =

X

S⊂Vn

aS δ(S) with integer aS ≥ 0 for all S.

(1.1)

An advantage of using (1.1) is that it allows to classify ℓ1 -embedding of G up to equivalence: different solutions to (1.1) with non-negative integers aS such that g.c.d.(λ, aS ) = 1 correspond to different embeddings. If such a solution is unique, the graph G is called ℓ1 -rigid graph (see [DeLa94]). Rigidity of any ℓ1 -metric is defined similarly. Restated in terms of the cut cone, a metric is ℓ1 -rigid if and only if it belongs to a face of the cut cone, which is a simplex subcone. If G is an ℓ1 -graph, then for every cut {S, S} occurring in the ℓ1 decomposition of dG (i.e. with aS > 0), both sets S and S are convex; we call such a cut convex cut. The following fact was established in [DeTu96]. Proposition 1.1 [DeTu96] A graph G is scale λ embeddable into a hypercube if and only if there exists a complete collection C(G) of (not necessary distinct) convex cuts of G, such that every edge of G is cut by exactly λ cuts from C(G).

For λ = 1 this is the well-known Djokovic characterization ([Djok73]) of graphs, which are isometrically embeddable into hypercubes. Lemma 1.1 rigid.

[CDG97] Any ℓ1 -graph, which does not contain K4 , is ℓ1 -

We say that a polyhedron P is ℓ1 -embeddable if its skeleton G(P ) (that is, the graph formed by its vertices and edges) is an ℓ1 -graph. A graph G is called polytopal graph if there is a polytope P (possibly, infinite) with G(P ) = G. For an embedding of a metric d into m-cube with scale λ, the ratio s(d) = m λ is called size of the embedding. For a graph G on n vertices there are the following lower and upper bounds on size s(dG ) (see [DeLa97], page 45): W (G) W (G)) ≤ s(dG ) ≤ , ⌊ n2 ⌋⌈ n2 ⌉ n−1 where, recall, W (G) =

X

i,j∈V

dG (i, j).

(1.2)

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7

Introduction

When the equality holds in the left-hand side of the inequality (1.2), we say that G is an equicut graph. This means that for such a graph, every S in the formula (1.1) satisfies aS 6= 0 if and only if S partitions V into parts of size ⌈ n2 ⌉ and ⌊ n2 ⌋, where n = |V |. On the other hand, the equality on the other side of (1.2) only happens in a very special case. Lemma 1.2

s(dG ) =

W (G) n−1

if and only if G is the star K1,n−1 .

As an example, consider embedding of an n-simplex αn having G(αn ) = Kn+1 . Any αn , n ≥ 3, is not ℓ1 -rigid, i.e. it admits at least two different embeddings. We give now three embeddings of αn into m-cubes Hm with scale λ, realizing, respectively, maximum, minimum and a middle (for n > 4) of size m λ . For the simplex αn the bounds (1.2) take the form

sn :=

n+1 n(n + 1) n(n + 1) W (Kn+1 ) < = . n+1 ≤ s(αn ) ≤ n+1 2 2(n − 1) n−1 2⌊ 2 ⌋⌈ 2 ⌉

For the left hand side, we have sn = 2 − 2(n+1) n+2

1 , ⌈ n+1 2 ⌉

i.e. sn =

2n n+1

for odd n,

for even n. and sn = The upper bound is realized by the embedding αn → 21 Hn+1 , where the P vertices of αn are mapped into vertices x of 12 Hn+1 with n+1 i=1 xi = 1 (in odd representation of 21 Hn+1 ). A middle (for n > 4) size is realized as follows. Take the set of all n(n+1) edges (ij), 1 ≤ i < j ≤ n + 1, of Kn+1 as the set of indices V of 2 Hm , i.e. m = n(n+1) . Map the vertex i of αn (i.e. of Kn+1 ) into the subset 2 Vi = {ij : 1 ≤ j ≤ n + 1, j 6= i} ⊂ V . Then |Vi ∆Vj | = 2(n − 1) is the scale 5 λ. In this case s(αn ) = n(n+1) 4(n−1) . For n = 4, this gives s(α4 ) = s4 = 3 , i.e. the lower bound for n = 4. Define λn be the minimal even positive number t such that tsn is an integer. Then there is an embedding of αn into λn sn -cube with scale λn realizing the lower bound sn . This is an equicut embedding (cf. the item (i) of Theorem 1.1 below). Any βn , n ≥ 4, is also not ℓ1 -rigid. All embedding of βn are into 2λ-cube with a such even scale λ that αn−1 is embeddable into m-cube, m ≤ 2λ, with scale λ. For minimal such scale, denote it µn , the following is known: n > µn ≥ 2⌈ n4 ⌉ with equality in the lower bound for any n ≤ 80 and, in the case of n divisible by four, if and only if there exists an Hadamard matrix of order n. In particular, β3 and β4 are embeddable into 21 H4 (in

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fact, G(β4 ) = 21 H4 , but there are two embeddings of β4 into 21 H4 ), β5 is embeddable only with scale four (into H8 ). The size s(dG ) has the following properties formulated in terms of n = |V |.

Theorem 1.1 [DePa01] Let G be an ℓ1 -graph. (i) s(dG ) = sn−1 = 2 − ⌈ n1 ⌉ if and only if G = Kn ; 2 (ii) s(dG ) = 2 if and only if G is a non-complete subgraph of a cocktailparty graph Kn×2 ; (iii) s(dG ) = n − 1 if and only if G is a tree; (iv) 2 < s(dG ) < n − 1, otherwise.

Clearly, the scale λ embeddability into Hm implies the scale 2λ embeddability into 21 H2m . Moreover, for any graph G, which is scale λ embeddable into a hypercube, we have: (i) λ = 1 ⇔ G is an isometric subgraph of a hypercube; we write: G → Hm . (ii) λ = 1 or 2 ⇔ G is an isometric subgraph of a half-m-cube; we write: G → 21 Hm . Similarly, for an infinite graph G we have: (i) λ = 1 ⇔ G is an isometric subgraph of a cubic lattice; we write: G → Zm . (ii) λ = 1 or 2 ⇔ G is an isometric subgraph of a half-cubic lattice graph; we write: G → 21 Zm . Lemma 1.3 two.

[CDG97] The scale λ of a planar ℓ1 -graph is either one or

Finally, we recall that a graph G is called hypermetric graph (see [DeGr93] for details) if its path-metric dG satisfies any hypermetric inequality defined by: X

1≤i 3; (4) a doubly infinite set of polytopes, which are direct products of two regular polygons (if one of polygons is a square, then we get prisms on 3-dimensional prisms); (5) the snub 24-cell; (6) a new polytope, called Grand Antiprism, having 100 vertices (all from 600-cell), 300 cells α3 and 20 cells AP rism5 (those antiprisms form two interlocking tubes). The 4-polytopes in above case (1) have form P (V ), where P is (one of six) regular 4-polytopes and the parameter V is a subset of {0, 1, 2, 3}. In Table 2 of [DDS04] are given all such embeddable 4-polytopes: α4 ({0, 1, 2, 3}) → H10 , β4 ({0, 1, 2, 3}) → H16 , 24 − cell({0, 1, 2, 3}) → H20 , 600 − cell({0, 1, 2, 3}) → H60 and 021 = α4 ({1}) = J(5, 2), β4 ({0, 3}) → 12 H12 , β4 ({0, 1, 2}) → H12 . Using the fact, that the direct product of two graphs is ℓ1 -embeddable if and only if each of them is, and the characterization of embeddable Archimedean polyhedra in [DeSt96] (see Tables 4.1, 4.2 above), we can decide about embeddability in cases 2)–4). Now, the snub 24-cell is embedded into 12 H12 (see below) and the Grand Antiprism (as well as 600-cell itself) violates 7-gonal inequality, which is necessary for embedding. It implies the following Corollary. Corollary 7.1 Besides not 7-gonal Grand Antiprism, six not 5-gonal prisms (on truncated α3 , truncated γ3 , cuboctahedron, truncated icosahedron, truncated dodecahedron and icosidodecahedron) and the case (1)(descibed above), all Archimedean 4-polytopes are embeddable.

7.4

The embedding of the snub 24-cell

The snub 24-cell was introduced by Gosset in 1897. It has 96 vertices, 288+144 edges in two orbits, 96+96+288 triangular faces in three orbits and three orbits of cells: 24+96 tetrahedra and 24 icosahedra. One orbit of edges is surrounded by one tetrahedron and two icosahedra, the other one by three tetrahedra and one icosahedron. The vertex figure (and the local graph of the skeleton) of the snub 24-cell is the tridiminished icosahedron (the regular-faced solid M7 = Nr.83 of the list [Berm71]); it is embeddable into 12 H6 . We give an embedding of the snub 24-cell into 12 H12 as 48 column-

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vectors representing some 48 vertices amongst the 96 vertices. The remaining 48 columns are obtained by adding the all-ones vector modulo 2 to each of these 48 columns. 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 1 0

0 0 0 0 0 0 1 0 1 0 0 0

0 0 0 0 0 0 1 1 1 0 1 0

0 0 0 1 0 0 0 0 0 0 1 0

0 0 0 1 0 0 0 0 1 0 1 1

0 0 0 1 0 0 0 1 0 0 1 1

0 0 0 1 0 0 0 1 1 0 1 0

0 0 0 1 1 0 0 1 0 1 1 1

0 0 0 1 1 0 0 1 1 0 1 1

0 0 0 1 1 0 1 1 1 0 1 0

0 0 0 1 1 0 1 1 1 1 1 1

0 0 1 0 0 0 0 0 1 0 0 0

0 0 1 0 0 0 0 0 1 0 1 1

0 0 1 0 0 0 1 0 1 0 1 0

0 0 1 0 0 1 1 0 1 0 0 0

0 0 1 0 1 0 1 0 1 0 1 1

0 0 1 0 1 0 1 1 1 0 1 0

0 0 1 0 1 1 1 0 1 0 1 0

0 0 1 0 1 1 1 1 1 0 1 1

0 0 1 1 1 0 0 0 1 0 1 1

0 0 1 1 1 0 0 1 1 1 1 1

0 0 1 1 1 0 1 1 1 0 1 1

0 0 1 1 1 1 1 1 1 1 1 1

0 1 0 0 0 0 1 0 0 0 0 0

0 1 0 0 0 0 1 1 1 0 0 0

0 1 0 0 0 1 1 0 1 0 0 0

0 1 0 0 0 1 1 1 0 0 0 0

0 1 0 0 1 0 1 1 1 0 1 0

0 1 0 0 1 1 1 1 0 1 0 0

0 1 0 0 1 1 1 1 1 0 0 0

0 1 0 0 1 1 1 1 1 1 1 0

0 1 0 1 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 1 0 0 1 0

0 1 0 1 0 0 0 1 0 1 0 0

0 1 0 1 0 0 0 1 0 1 1 1

0 1 0 1 0 0 1 1 0 0 0 0

0 1 0 1 0 0 1 1 0 1 1 0

0 1 0 1 0 0 1 1 1 0 1 0

0 1 0 1 0 1 1 1 0 1 0 0

0 1 0 1 1 0 1 1 0 1 1 1

0 1 0 1 1 0 1 1 1 1 1 0

0 1 0 1 1 1 1 1 0 1 1 0

0 1 0 1 1 1 1 1 1 1 1 1

0 1 1 0 1 1 1 0 1 0 0 0

0 1 1 0 1 1 1 1 1 0 1 0

0 1 1 0 1 1 1 1 1 1 0 0

0 1 1 0 1 1 1 1 1 1 1 1

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Chapter 8

Polycycles and Other Chemically Relevant Graphs

8.1

(r,q)-polycycles

Recall that a graph G is k-(vertex)-connected if a deletion of any k − 1 vertices from G does not disconnect it. A girth of graph G is the length of a maximal isometric circuit of G. A polycycle or, more precisely, (r, q)-polycycle is a simple planar 2(vertex)-connected finite or countable locally-finite graph G of girth r and maximal vertex-degree q, which admits a realization on the plane, such that: (i) all interior vertices are of degree q, (ii) all interior faces are (combinatorial) r-gons, It is shown in [DeSt02d], that (i) and (ii) imply (iii) all vertices, edges and interior faces form a cell-complex (i.e. the intersection of any two faces is edge, vertex or ∅), while neither (i), (iii) imply (ii), nor (ii), (iii) imply (i). Clearly, a polycycle is the skeleton of a 3-polytope if and only if its plane realization is 3-connected, i.e. the degree of each boundary vertex is at least three and each interior face has at most one boundary edge. The main example of (r, q)-polycycle is the skeleton of (rq), i.e. of the q-valent partition of the sphere S 2 , Euclidean plane R2 or hyperbolic plane H 2 by regular r-gons. For (r, q) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5) it is, respectively, Platonic tetrahedron, cube, octahedron, dodecahedron, icosahedron on S 2 , but with excluded exterior face; for (r, q) = (6, 3), (3, 6), (4, 4) it is the regular tiling (63), (36), (44) of R2 ; all others (rq) are regular partitions of H 2 . Call a polycycle proper if it is a partial subgraph of (the skeleton of) 75

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the regular partition (rq). Call a proper (r, q)-polycycle induced (moreover, isometric) if it is induced (moreover, isometric) subgraph of (rq). (rq) is embeddable into 21 H3 , H3 , 21 H4 , 12 H10 , 21 H6 ; Z3 , 12 Z3 , Z2 and 1 2 Z∞ (or, moreover, Z∞ ) for (r, q)=(3,3), (4,3), (3,4), (5,3), (3,5); (6,3), (3,6), (4,4) and max(r, q) ≥ 7 (or, moreover, r is even), respectively. So, any isometric polycycle is embeddable. Call a vertex-split (34), a (3, 4)-polycycle, coming from (34) as follows. Let ∇C{a,b,c,d} be (induced in (34)) 4-wheel with an apex x; replace the edges (x, a), (x, b) of (34) by the edges (x′ , a), (x′ , b), where x′ is a new vertex. See it on the right hand side of Figure 13.3. The vertex-split (35) is defined similarly. Examples of non-embeddable polycycles are: (43) − e, (34) − e, (53) − e, (35) − e, vertex-split (34), vertex-split (35) and four polycycles, given on Figures 8.1 and 8.2. Amongst above ten polycycles only vertex-split (34) and vertex-split (35) are not proper ones and amongst eight proper ones, only those on the right-hand side of Figures 8.1 and 8.2 are induced ones. Theorem 8.1 ([DeSt01], [DeSt02b], [DeSt02c]) (i) For (r, q) 6= (5, 3), (3, 5), there are exactly three non-embeddable polycycles: (43) − e, (34) − e and the vertex-split (34), (ii) except of (53) itself, any (5, 3)-polycycle is embeddable if and only if it does not contain, as an induced subgraph, neither of two proper polycycles with p5 = 6, given on Figure 8.1.

Fig. 8.1

The forbidden induced subgraphs of embeddable (5, 3)-polycycles

(iii) except of (35) and (35)−v, any (3, 5)-polycycle is embeddable if and only if it does not contain, as an induced subgraph, neither of two proper 10-vertex polycycles, given on Figure 8.2. Amongst all 39 proper (5, 3)-polycycles, the embeddable ones are: (53), three with p5 = 7 (see Figure 8.3), nine with p5 = 6 (all but two from Figure 8.1) and all 14 with p5 ≤ 5. Any outer-planar polycycle is embeddable, as well as any connected outer-planar graph.

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Polycycles and other chemically relevant graphs

Fig. 8.2

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The forbidden induced subgraphs of embeddable (3, 5)-polycycles

Fig. 8.3

The embeddable (5, 3)-polycycles with p5 = 7

Embeddable (r, q)-polycycles have the scale λ = 1 if r is even and 2, otherwise. Consider any finite embeddable polycycle with perimeter p and z closed zones of opposite (alternating left and right opposite, if r is odd) non-boundary edges. Then (see Introduction) it is embedded into H p2 +z , if r is even (it implies that p is also even), and it is embedded, with scale two, into Hp+z , if r is odd. There is a continuum of (6, 3)-polycycles, which are embedded only into Z∞ . Let us label the six edges of a hexagon consecutively by the numbers 1, 2,. . . , 6. Then opposite edges have distinct parity. Call edges with odd labels odd. Now, consider a polycycle, which is an odd tree T of hexagons. Hexagons of the odd tree are adjacent only by odd edges and there is no cycle of hexagons. For example, an infinite chain is an odd tree T0 , where every hexagon of hi , −∞ < i < ∞, is adjacent to exactly two other hexagons hi−1 and hi+1 . Let (pi , qi ) be the labels of edges, such that hi is adjacent to hi−1 and hi+1 . We obtain an infinite sequence {(pi , qi ) : −∞ < i < ∞}, pi , qi ∈ {1, 3, 5}, corresponding to T0 . Obviously, this family of sequences contains the continuum of aperiodic sequences. Note that every hexagon of T0 contains a free odd edge. Hence we can join T0 by the free odd edge to a finite or infinite branch. So, we have a continuum of infinite odd trees with finite and infinite branches.

8.2

Quasi-(r, 3)-polycycles

In [DeSt00b] the embedding of quasi-(r, 3)-polycycles, i.e. (r, 3)-polycycles without condition (ii), was considered: a few interior r′ -gonal faces with r′ 6= r are permitted. Let P be such plane realization of a graph and

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let pint denote the sequence (p3 , p4 , . . . ), where pi is the number of igons amongst interior faces of P . Such graphs, especially those with pint = (p6 ), (p4 = 1, p6 ), (p5 = 1, p6 ), (p4 = 2, p6 ), (p5 = 2, p6 ), are of interest in Organic Chemistry (namely, as polycyclic aromatic hydrocarbons); such compounds as benzene, biphenyl, fluorentene, terphenyl, indacene correspond to p6 = 2, 2, 3, 1, 1, respectively, in the five above cases. Figure 8.4 represents (in terms of the atoms of carbon C and hydrogen H, corresponding, respectively, to all and to all 2-valent vertices): benzene C6 H6 , naphtalene C10 H8 , azulene C10 H8 , indacene C12 H8 , terpheniloide C9 H5 , fluorentene C16 H10 , pentalene C8 H6 , biphenilene C12 H8 , respectively.

benzene C6 H 6 ,

naphtalene C10 H8 ,

azulene C10 H8 ,

indacene C12 H8 ,

terpheniloide C9 H 5 ,

fluorentene C16 H10 ,

pentalene C8 H 6 ,

biphenilene C12 H8

Fig. 8.4

Some small quasi-(r, 3)-polycycles

Embeddability of such graphs is useful for the calculation of some their distance-related indices and for the purpose of nomenclature. Theorem 8.2 below describes the embeddability of many quasi-(r, 3)-polycycles in terms of forbidden isometric subgraphs; in cases (iv) and (v) of this Theorem, the set of such forbidden subgraphs is infinite. An example of graphs from Theorem 8.2 (i) is coranulene C20 H10 , consisting of a pentagon, surrounded by five hexagons; it embeds into 12 H15 . Theorem 8.2 ([DeSt00b]) A quasi-(r, 3)-polycycle P is embeddable (i) for p3 = p4 = 0, p5 ≤ 1: always, (ii) for pint = (p3 ≤ 2, p6 ): if and only if each triangle has a vertex of degree 2 (i.e. P has no Figure 8.5 a) above as an isometric subgraph), (iii) for pint = (p4 = 1, p6 ): if and only if P has no Figure 8.5 b) above as an isometric subgraph,

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a

b

Fig. 8.5 The forbidden isometric subgraphs for pint = (p3 ≤ 2, p6 ) and for pint = (p4 = 1, p6 )

(iv) for pint = (p4 = 2, p6 ): only if P has no graphs Gs,t (illustrated by Figure 8.6 a), b) for (s, t) = (3, 2), (1, 1)), or Gu (illustrated by Figure 8.6 c) for u = 4, 1), or Figure 8.6 d) 8.2 d) as an isometric subgraph, (v) for pint = (p5 = 2, p6 ): only if P has no graphs Fs,t (illustrated by Figure 8.7 a), b) for (s, t) = (3, 2), (1, 1)) or Fu (illustrated by Figure 8.7 c) for u = 3) as an isometric subgraph.

a Fig. 8.6

b

Fig. 8.7

d

Some forbidden isometric subgraphs for pint = (p4 = 2, p6 )

b a

c

c

Some forbidden isometric subgraphs for pint = (p5 = 2, p6 )

Remark that embeddable graph on Figure 2.2 (see chapter 2) contains the forbidden graph F1,1 , given on Figure 8.2 b), but it is not an isometric subgraph.

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8.3

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Coordination polyhedra and metallopolyhedra

According to Wells [Well84], the chemical crystal structure is usually described by the coordination polyhedra (i.e. the convex hulls of anions forming the group of nearest neighbors of each metal ion) or by the domains of atom (i.e. Voronoi polyhedra). Coordination polyhedra are arrangements of nearest neighbors in crystals, molecules and ions. More precisely, as defined by Frank and Kasper in Chemistry, a coordination polyhedron is the convex hull of the mirror images of the center of a Voronoi-Dirichlet domain in each of its faces. Remark that the coordination polyhedron is, in general, not dual to the Voronoi domain, and different from the contact polytope of sphere packing. For example, while the Voronoi polyhedron of a point of the bcc (i.e. the body-centered cubic lattice A∗3 ) is the truncated octahedron, its coordination polyhedron is the rhombic dodecahedron. In addition to some frequent coordination polyhedra, the ℓ1 -status of some metallopolyhedra (convex hull of atoms of metallic cluster) or metal and oxygen polyhedra (see, for example, [King91; Well84]) are given in Table 8.1. The names and the numbers (as regular-faced polyhedra) are taken from the list of such 112 polyhedra given in [Berm71; Zalg69]. For example, see Figure 4.1, where the embedding of the rhombicuboctahedron (Nr.13) and the not 5-gonality of its twist (Nr.57) are given. The cuboctahedron and its twist (triangular orthobicupola Nr.47) are the coordination polyhedra of, respectively, the f cc and the hcp; their duals are space-fillers with the dual of the second one being not 5-gonal. The regular-faced polyhedra from Table 8.1 with Nr. 1, 2, 4, 32, 33, 37, 71 and 104 are the eight convex deltahedra with regular (triangular) faces and with n vertices, where 4 ≤ n ≤ 12, n 6= 11. Those deltahedra are dual of the first eight medial polyhedra Fn given in Figure 13.1 with n = 4, 8, 20, 6, 10, 16, 14 and 12. The last three ℓ1 -embeddable polyhedra of Table 8.1 are capped or 2capped Platonic solids. Remind that, i-capped tetrahedron → 12 H3+i for 1 ≤ i ≤ 4, i-capped octahedron → 12 H4+i for 1 ≤ i ≤ 8, i-capped cube → 12 H6 for 1 ≤ i ≤ 2, or i = 3 without opposite capping, and i-capped cube is not 5-gonal for 4 ≤ i ≤ 6 or i = 3 with opposite capping. The cuboctahedron and its twist (triangular orthobicupola Nr.47) are the coordination polyhedra of, respectively, the f cc and the hcp; their duals are space-fillers with the dual of the second one being not 5-gonal.

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Fig. 8.8

The edge-coalesced icosahedron

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Table 8.1 dra

Embeddability of some of the most frequent chemical polyhe-

Polyhedron (its Nr. as regular-faced)

ℓ1 -embeddability

Chemical example

T etrahedron (1) Octahedron (2) Cube (3) Icosahedron(4) Cuboctahedron (6) Truncated tetrahedron (8) Rhombicuboctahedron (13) Triangular prism (19) Square antiprism (20) Square pyramid (21) Triangular cupola (23) Gyro-elongated square pyramid (30) Triangular bipyramid (32) Pentagonal bipyramid (33) Elongated square bipyramid (35) Gyro-elongated square bipyramid (37) Triangular orthobicupola (47) Square gyrobicupola (49) Elongated triangular bicupola (55) Elongated triangular gyrobicupola (56) Twisted rhombicuboctahedron (57) Elongated pentagonal bicupola (58) Augmented triangular prism (69) Biaugmented triangular prism (70) Triaugmented triangular prism (71) Snub disphenoid (104) Octahedron + P yr3 Octahedron + 2 P yr3 T etrahedron + 2 P yr3 Edge-coalesced icosahedron (Fig. 8.8)

→ 21 H3 → 12 H4 → H3 → 12 H6 not 5-gonal not 5-gonal → 12 H10 → 12 H5 → 12 H5 → 21 H4 not 5-gonal ext. hypermetric

Rh4 (CO)12 [Os6 (CO18 )]2− CsCl [M o12 O30 (CeO12 )]8− [M o12 O36 (SiO4 )]4− [M o12 (HOAsO3 )4 O34 ]4− [V18 O42 (SO4 )]8− [Rh6 C(CO)15 ]2− [Co8 C(CO)18 ]2− F e5 C(CO)15 [W9 O30 (P O4 )]9− [Co9 Si(CO)21 ]2−

→ 21 H4 not 5-gonal → 12 H6

Os5 (CO)16 K3 ZrF7 [W10 O32 ]4−

ext. hypermetric

[V10 O26 ]4−

not 5-gonal not 5-gonal not 5-gonal

[W12 O36 (SiO4 )]4− [V10 O26 ]4− α-[W18 P2 O62 ]6−

not 5-gonal

β-[W18 P2 O62 ]6−

not 5-gonal

[H4 V18 O42 (Br)]9−

not 5-gonal

[W30 P5 O110 (N a)]14−

→

1 H 2 5

N a5 Zr2 F13

→

1 H 2 5

N2 H6 (ZrF6 )

ext. hypermetric

Eu(OH)3

not 5-gonal → 12 H5 → 12 H6 → 12 H5 ext. hypermetric

K4 ZrF8 [Ru7 (CO16 ]3− [Os8 (CO22 ]2− [Os6 (CO18 ]2− [B11 H11 ]2−

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Chapter 9

Plane Tilings

9.1

58 embeddable mosaics

We review here mosaics T (i.e. edge-to-edge tilings of Euclidean plane by regular polygons) with respect of possible embedding, isometric up to a scale λ, of their skeletons or skeletons of their duals T ∗ , into the cubic lattice Zn for some n. Main result of this chapter is the classification, given in Table 9.1, of all 58 such mosaics amongst all 165 mosaics of the catalog, given in [Chav89] and including all main classifications of mosaics. It turns out that all non-embeddable mosaics T and their duals T ∗ considered here are, moreover, not 5-gonal. In Table 9.1, non-embeddability of T or T ∗ indicated by the sign − in the columns 4 or 5, respectively. All embeddings were found by the same method of complete system of alternating zones, as in [DeSt96], [DeSt97], [CDG97]. A mosaic is denoted here by its standard face-vector ([GrSh87]), giving all stars (i.e. the list of all tiles, surrounding a vertex) for vertices of each type. Denote by v, t, e the number of types (i.e. orbits under action of the symmetry group Aut T of a mosaic T ) of vertices, tiles and edges, respectively. Call a mosaic vertex-homogeneous, if any two vertices with congruent stars belong to the same orbit of vertices. Call a mosaic tilehomogeneous, if any two congruent tiles belong to the same orbit of tiles. Call a mosaic edge-homogeneous (or strongly edge-homogeneous), if their edge figures (weak edge figures) belong to the same orbit of edges. An edge figure is an edge together with all edges, which meet it, and a weak edge figure is an edge with its two incident tiles. The 165 mosaics from [Chav89] include following classifications of mosaics: • 135 vertex-homogeneous (and 20 with v = 2), found in [Kr¨ ot69], 83

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• • • • •

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[Kr¨ ot70a], [Kr¨ ot70b]; 22 tile-homogeneous (and 13 with t = 2), found in [DbLa81]; 22 strongly edge-homogeneous, found in [Chav84]; all 11, 20, 61 mosaics with v = 1, 2, 3; all 3, 13, 26 mosaics with t = 1, 2, 3; and all 4, 4, 10 mosaics with e = 1, 2, 3.

(Apropos, in Tables 1, 2 of [Chav89] should be 26, 27, (instead of 25, 26, respectively) for the number of mosaics with t = 3.) All 58 mosaics amongst above 165, such that T or T ∗ is embeddable, are given in Table 9.1 in the lexicographic order with respect to their vector (v, t, e). The pictures of those 58 mosaics are given, in the same order, in this chapter. Representatives of the orbits of vertices are indicated there by larger black circles. On the right hand side of Table 9.1 the vertex-, tile-, edge-, strongly edge-homogeneity (if any) of T is given by signs +. For each mosaic, its symmetry group Aut T is also given, using the standard crystallographic notation. Remind that the mirror image of the tiling T cannot be superimposed on the original (i.e. we have two enantiomorphic forms of T ) if and only if neither “m”, nor “g” is contained in above notation of Aut T . Remark that for any mosaic T , the embeddability of T or T ∗ into Zn or 1 2 Zn implies that n is divisible by 3 (or by 2) if Aut T contains a 3-axis (or a 4-axis) of rotation, i.e. if Aut T is one of the groups p6m, p6, p31m, p3m1, p3 (or p4m, p4g, p4, respectively). It is interesting that for the mosaics of Table 9.1 we have: (1) Nr.4, 17, 31, 37, 41 (all mosaics T with a vertex (3.6.3.6)) have T ∗ → Z3 and vice versa; (2) Nr.7, 15, 21, 23, 30, 32, 44, 47, 49, 54, 55, 57, 58 (all T with a vertex (32 .4.3.4)) have T → 12 Z4 or T → 12 Z6 (except Nr.58, containing also a vertex (34 .6)) and all 6 mosaics T , embeddable into 21 Z4 , are amongst them; (3) Nr.11, 13, 22, 25, 28, 48, 52, 53, 56, 58 (all T with a vertex (34 .6)) have T → 12 Z5 or T → 21 Z6 , and all 7 mosaics, embeddable into 12 Z5 , are amongst them. The following groups of mosaics in Table 9.1 have the same face-vector: 14,18; 21,23; 27,33,34,40; 29,38; 31,41; 35,43; 36,42; 50,51; in each of eight cases, except of the 2-nd and the 8-th, the embeddability is the same. Table 9.1 contains 41 (from 135) vertex-homogeneous, 18 (from 22) tile-

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85

homogeneous and 15 (from 22) strongly edge-homogeneous mosaics. Also it contains: • all 11 mosaics with v = 1, 12 (from 20) mosaics with v = 2 and 26 (from 61) mosaics with v = 3; • all 3 mosaics with t = 1, 11 (from 13) mosaics with t = 2 and 15 (from 26) mosaics with t = 3; • all 4 mosaics with e = 1, all 4 mosaics with e = 2 and 6 (from 10) mosaics with e = 3. Table 9.1 does not contain only the following 3 mosaics with homogeneity +, +, +, +: (32 .62 ; 3.6.3.6), (3.4.3.12; 3.122), (3.4.6.4; 4.6.12) with (v, t, e) = (2, 2, 3), (2, 3, 3), (2, 4, 4), respectively. It does not contain also the 2 mosaics with t = 2: the one (32 .62 ; 3.6.3.6) above and the second (32 .62 ; 3.6.3.6; 63)1 with (v, t, e) = (3, 2, 4) and homogeneity +, +, +, −. The embeddings given in Table 9.1 are taken from [DeSt96], [DeSt97] and [DeSt99]. Apropos, amongst Nrs. 1–11 (all Archimedean tilings in Table 9.1), all but Nrs. 7, 8 and 11, come by Wythoff kaleidoscope construction from regular tilings Nrs. 1–3 (see Table 3 in [DDS04]); amongst 13 Archimedean polyhedra all, but two snub ones, come by this construction from Platonic solids.

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Scale Isometric Polytopal Graphs Table 9.1 N r.

58 mosaics T with embeddable skeleton of T or T ∗

mosaic T ; face-vector

Aut T

T →

T∗ →

v, t, e

homogeneity

(44 )=Vo(Z

Z2 Z3 1 Z 2 3 – – Z4 1 Z 2 4 1 Z 2 3 1 Z 2 3 Z6 1 Z 2 6

Z2 1 Z 2 3 Z3 Z3 1 Z 2 ∞ – – – – – –

1, 1, 1 1, 1, 1 1, 1, 1 1, 2, 1 1, 2, 2 1, 2, 2 1, 2, 2 1, 2, 3 1, 3, 2 1, 3, 3 1, 3, 3

+,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ +,–,+,–

–

Z∞ – – – – Z3 – – – – – –

2, 2, 3 2, 2, 4 2, 2, 4 2, 3, 3 2, 3, 4 2, 3, 4 2, 3, 5 2, 4, 4 2, 4, 5 2, 4, 5 2, 5, 7 2, 6, 6

+,+,+,+ +,+,+,– +,+,+,– +,–,+,– +,–,+,– +,+,+,+ +,–,–,– +,–,+,+ +,–,–,– +,–,+,– +,–,–,– +,–,+,–

– – Z4 – – – – Z3 – – – – –

3, 2, 3 3, 2, 5 3, 3, 4 3, 3, 5 3, 3, 6 3, 3, 6 3, 4, 5 3, 4, 5 3, 4, 6 3, 4, 6 3, 4, 6 3, 4, 6 3, 4, 6

+,+,+,+ +,+,+,– +,+,+,+ +,–,+,– +,–,+,– –,–,–,– +,–,+,– +,–,+,– +,–,+,– +,–,–,– +,–,–,– –,–,–,– –,–,–,–

1 2 3 4 5 6 7 8 9 10 11

2 )=De(Z2 )

(63 )=Vo(A2 ) (36 )=De(A2 ) Kagome net; (3.6.3.6) truncated (63 ); (3.122 ) truncated (44 ); (4.82 ) dual Cairo net; (32 .4.3.4) (33 .42 ) (3.4.6.4) (4.6.12) (34 .6)

p4m p6m p6m p6m p6m p4m p4g cmm p6m p6m p6

12 13 14 15 16 17 18 19 20 21 22 23

(36 ; 32 .62 ) (34 .6; 32 .62 ) (33 .42 ; 44 )1 (36 ; 32 .4.3.4) (36 ; 33 .42 )1 (3.42 .6; 3.6.3.6)2 (33 .42 ; 44 )2 (33 .42 ; 3.4.6.4) (36 ; 33 .42 )2 (33 .42 ; 32 .4.3.4)1 (36 ; 34 .6)2 (33 .42 ; 32 .4.3.4)2

p6m cmm cmm p6m pmm cmm cmm p6m cmm p4g p6 pgg

24 25 26 27 28 29 30 31 32 33 34 35 36

(36 ; 32 .62 ; 63 ) (34 .6; 32 .62 ; 63 ) (3.42 .6; 3.4.6.4; 44 ) (36 ; 33 .42 ; 44 )1 (36 ; 34 .6; 32 .62 )2 (33 .42 ; 44 ; 44 )1 6 (3 ; 33 .42 ; 32 .4.3.4) (3.42 .6; 3.6.3.6; 44 )2 (33 .42 ; 32 .4.3.4; 44 ) (36 ; 33 .42 ; 44 )2 (36 ; 33 .42 ; 44 )3 (36 ; 33 .42 ; 33 .42 )1 (33 .42 ; 33 .42 ; 44 )1

p6m pmg p4g pmm cmm cmm p6m cmm p4 cmm pmm cmm pmm

1 Z 2 5 1 Z 2 3 1 Z 2 6 1 Z 2 3

–

1 Z 2 3 1 Z 2 3 1 Z 2 3 1 Z 2 6 1 Z 2 6 1 Z 2 4 1 Z 2 6 1 Z 2 5

–

1 Z 2 3 1 Z 2 5 1 Z 2 3 1 Z 2 6

–

1 Z 2 4 1 Z 2 3 1 Z 2 3 1 Z 2 3 1 Z 2 3

The pictures of 58 mosaics of Table 9.1 are given below in the same order.

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Plane tilings

37 38 39 40 41 42 43 44 45 46 47 48 49

(3.42 .6; 3.6.3.6; 3.6.3.6)1 (33 .42 ; 44 ; 44 )2 (36 ; 36 ; 33 .42 )1 (36 ; 33 .42 ; 44 )4 (3.42 .6; 3.6.3.6; 44 )4 (33 .42 ; 33 .42 ; 44 )2 (36 ; 33 .42 ; 33 .42 )2 (33 .42 ; 33 .42 ; 32 .4.3.4) (36 ; 33 .42 ; 3.4.6.4) (36 ; 36 ; 33 .42 )2 (33 .42 ; 32 .4.3.4; 32 .4.3.4) (36 ; 36 ; 34 .6)3 6 (3 ; 32 .4.3.4; 32 .4.3.4)

pmg cmm pmm cmm cmm pmm pmg pgg p6m cmm p2 p6 p6m

50 51 52 53 54 55

(36 ; 32 .62 ; 63 ; 63 )1 (36 ; 32 .62 ; 63 ; 63 )2 (34 .6; 32 .62 ; 63 ; 63 ) (36 ; 34 .6; 32 .62 ; 63 )4 (36 ; 33 .42 ; 32 .4.3.4; 44 ) (36 ; 32 .4.3.4; 3.42 .6; 3.4.6.4)

p6 p6m cmm pmg p4g p6m

56 57

(34 .6; 32 .62 ; 63 ; 63 ; 63 ) (32 .4.3.4; 32 .62 ; 3.42 .6; 3.4.6.4; 63 ) (34 .6; 33 .42 ; 32 .4.3.4; 32 .62 ; 3.42 .6; 3.4.6.4)

pmg p3m1

1 Z 2 6 1 Z 2 5 1 Z 2 5 1 Z 2 4 1 Z 2 6 1 Z 2 5 1 Z 2 6

cmm

1 Z 2 5

58

9.2

–

1 Z 2 3 1 Z 2 3 1 Z 2 4 1 Z 2 3 1 Z 2 3 1 Z 2 4 1 Z 2 6 1 Z 2 6

Z3 – – – Z3 – – – – – – – –

3, 4, 6 3, 4, 7 3, 5, 6 3, 5, 7 3, 5, 7 3, 5, 7 3, 5, 7 3, 5, 8 3, 6, 6 3, 6, 7 3, 6, 9 3, 7, 9 3, 8, 5

–,–,–,– –,–,–,– –,–,–,– +,–,–,– +,–,–,– –,–,–,– –,–,–,– –,–,–,– +,–,+,– –,–,–,– –,–,–,– –,–,–,– –,–,+,–

–

1 Z 2 ∞

– – – – –

4, 2, 5 4, 3, 5 4, 3, 6 4, 3, 7 4, 4, 6 4, 6, 7

–,+,–,– –,–,–,– –,–,–,– +,–,+,– +,–,+,– +,–,+,–

– –

5, 3, 7 5, 6, 8

–,–,–,– +,–,+,–

–

6, 9, 13

+,–,+,–

1 Z 2 3 1 Z 2 3 1 Z 2 3

–

Other special plane tilings

Embedding of other interesting plane partitions (by any polygons) was considered in [DeSt97]. For instance, a zonotopal Penrose tiling of the plane by two golden rhombi (or its analogue in the 3-space, by two golden rhombohedra) is embedded into Z5 (respectively, into Z6 ). An embeddability of the Penrose tiling into Z5 (a part of it is drawn on Figure 9.7 b)) is seen on Figure 9.7 a), where its five zone are shown. The Robinson subdivision of above Penrose zototopal tiling and a 10gonal T¨ ubingen triangulation (see both on Figure 9.8 a) and b) ) are not 5-gonal: the five vertices violating a 5-gonal inequality are marked on a separated graph on seven vertices. This graph is an isometric subgraph of both these triangulations (see Figure 9.8 c). The three vertices marked by small circles are taken with sign +, and the two vertices marked by small squares are taken with sign −. It is interesting that the Voronoi partition for the system of vertices of above Penrose tiling is not 5-gonal, but all found by us 5-vertex sets,

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Fig. 9.1

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Examples of mosaics, embeddable into

1 Z : 2 3

(36 ), (3.4.6.4) and (33 .42 )

violating a 5-gonal inequality, have diameter greater than 17. This Voronoi tiling is dual to the Delaunay triangulation, coming from Penrose tiling by addition of short diagonals on all rhombi, such that all obtained triangles are acute. (Recall that there are two mutually dual partitions of an n-space corresponding to any discrete system of points in this space; they are called Delaunay and Voronoi partitions, respectively. We consider these partitions for the system of vertices of above Penrose tiling.) The Voronoi partition consists of 5-, 6-, 7-gons of 3, 2, 2 types, respectively. An existence of 5 points violating a 5-gonal inequality comes from the presence of pentagons, since the skeleton of any normal partition of the plane without 3-, 4- and 5-gons, is embeddable (see [DeSt00b], Theorem 3). The following tilings are not 5-gonal (see [Well91], pages 104, 89, and also Figure 9.9 a) and b) below): a D¨ urer tiling (by regular pentagons and rhombi) and a tiling by Greek crosses (combinatorially equivalent to

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(44 ) with two new vertices on each edge). In fact, the fragment of D¨ urer tiling, depicted on Figure 9.9 a), admits unique extension to a tiling of the plane by congruent pentagons and rhombi. In this fragment we added 32nd pentagon, with respect of the fragment given in [Well91], page 104, since that original fragment is embeddable into 21 H45 (45 being the perimeter of this fragment). Also not 5-gonal √ the pinwheel tiling by the right triangles with edge lengths 1, 2 and 5 (see [Radi94]; this tiling is not edge-to-edge, but we consider it as edge-to-edge tiling by 3- and 4-gons). Let Cham(T ) denote the chamfered T, i.e. the plane tiling obtained from mosaic T by putting prisms on all its faces, followed by the removal of all original edges. Then Cham(63 ) = (63 ), Cham(36 ) is not 5-gonal and the zonotopal tiling Cham(44 ) is embedded into Z4 ; see three stylized ancient tilings, which are combinatorially equivalent to Cham(44 ) on Figure 9.10 below. The tiling Cham(44 ) is dual to (44 ), considered as an infinite chess board, with pyramids on all black squares. (44 ) provides unique (combinatorial type of) normal partition of Euclidean plane by even-gons, having only even degrees of vertices. There are none such partitions of 3-sphere and an infinity of them of the hyperbolic plane, all embeddable into Z∞ . Finally, the graph of a part of some embeddable T or T ∗ , bounded by a simple circuit of length n and containing k closed zones of interior edges, is an isometric subgraph of Hk+ n2 , if all faces are even-sided, and of 21 Hk+n , otherwise.

9.3

Face-regular bifaced plane tilings

All face-regular bifaced plane tiling of Euclidean plane were found in [Deza02]. Vertices of these tilings have the same degree k, and there are only two combinatorial types of faces (say, a- and b-gons); moreover, any a-gon (respectively, any b-gon) have the same number ta (respectively, tb ) of adjacent a-gons (respectively, b-gons). This implies that each a-gon is adjacent to a − ta b-gons and each b-gon is adjacent to b − tb a-gons. All embeddable ones amongst them are the three Archimedean tilings (4.82 ), (32 .4.3.4), (33 .42 ) (i.e. Nr.6–8 in Table 9.1, embeddable into Z4 , 1 1 2 Z4 , 2 Z3 , respectively) and the following tilings A, B and continuums D, C of tilings (Nr.3’, 13C and continuums Nr.33 and Nr.29 in notation of [Deza02]), which are embedded into 21 Z3 , 21 Z8 , 21 Z4 , 12 Z5 , respectively.

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The tiling A is the regular tiling (63 ) truncated on a “half” of its vertices, such that exactly one end of any edge is truncated. The tiling B is the Archimedean tiling (4.82 ), in which a “half” of 8-gons is cut in two hexagons by a new vertical edge, while any other 8-gon is cut in two hexagons by a new horizontal edge; any two adjacent 8-gons are cut differently. The tiling C is the continuum of all tilings, obtained from the Archimedean tiling (33 , 42 ) by taking, instead of fixed horizontal row of squares, any other (infinite in both directions) sequence of adjacent squares, going “up” and/or “right” arbitrary; all tiling then shifted accordingly. The continuum D comes by all the following decorations of the Archimedean tiling (4.82 ), considered combinatorially as the tiling of the plane by parallel horizontal zones (infinite in both left and right direction). Each zone consists of (alternated) 4- and 8-gons, such that their edges of adjacency are parallel and opposite in each 4- or 8-gon. Now decorate each zone by diagonals, cutting each 4-gon in two 3-gons or each 8-gon in two 5-gons, such that all diagonals have the same direction: either South-West to North-East, or North-West to South-East. So, we get a continuum of such decorations of the (4.82 ). The parameters (k; a, b; ta , tb ) of the seven above mentioned embeddable types of face-regular tilings are as follows: (3;4,8;0,4), (5;3,4;1,0), (5;3,4;2,2), (3;3,9;0,6), (3;4,7;0,5), (5;3,4;2,2), (4;3,5;1,3), respectively. The first three of them (the Archimedean ones) occur also in real crystal structures: they are depicted on Figures 7.20, 6.51 and 6.50 of [OkHy96] as nets, representing the paracelsian BaAl2 Si2 O8 , U3 Si2 and F e2 AlB2 , respectively.

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Fig. 9.2

Mosaics 1-11 of Table 9.1

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Fig. 9.3


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Fig. 9.4


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Fig. 9.5


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Fig. 9.6


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Fig. 9.7

Penrose rhombic tiling: a) 5 zones, b) a fragment

Fig. 9.8 a) Robinson subdivision of Penrose rhombic tiling, b) T¨ ubingen triangulation, c) their not 5-gonal isometric subgraph

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Fig. 9.9

Fragments of: a) a D¨ urer tiling, b) a tiling by Greek crosses

Fig. 9.10

Three ancient tilings, depicting Cham(44 )

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Chapter 10

Uniform Partitions of 3-space and Relatives

Recall that a uniform polyhedron is the same as a semi-regular polyhedron. A normal partition of 3-space is called uniform partition if all its facets (cells) are uniform polyhedra and group of symmetry is vertex-transitive. There are exactly 28 uniform partitions of 3-space. A short history of this result follows. In 1905 Andreini in [Andr05] proposed, as a complete list, 25 such partitions. But one of them (13′ , in his notation) turns out to be not uniform; it seems, that Coxeter [Coxe54], page 334, was the first who realized it. Also Andreini missed partitions 25–28 (in our numeration given below). Till recent years, mathematical literature was abundant with incomplete lists of those partitions. See, for example, [Crit70], [Will72] and [Pear78] (all of them do not contain partitions 24–28). The first, who published the complete list, was Gr¨ unbaum in [Gr¨ un94]. But he wrote there that, after obtaining the list, he realized that the manuscript [John91] already contained all 28 partitions. We also obtained all the 28 partitions independently, but only in 1996. The results presented in this chapter are from [DeSt00a]. All embeddable partitions in this chapter turn out to be ℓ1 -rigid and so, having scale one or two. Those embeddings were obtained by constructing a complete system of alternated zones; see chapter 1 and [CDG97], [DeSt96], [DeSt97], [DeSt98]. It turns out also that all non-embeddable partitions, considered in this chapter, are, moreover, not 5-gonal. Remind that De(T ) and V o(T ) are the Delaunay and the Voronoi partitions of 3-space associated with given set of points T . By an abuse of language, we will use the same notation for the graph, i.e. the skeleton of a partition. Remind also that P ∗ is the partition dual to the partition P ; it should not be confounded with the same notation for dual lattice.

99

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10.1

28 uniform partitions

In Tables 10.1, 10.2 of the 28 partitions, the meaning of the columns is as follows: 1: the number, which we give to the partition; 2 and 3: its numbers in [Andr05] (if any) and in [Gr¨ un94]; 4: a characterization (if any) of the partition; 5: tiles of the partition and the corresponding number of them in the Delaunay star; 5∗ : tiles of its dual; 6 and 6∗ : embeddability (if any) of the partition and of its dual. The special notations for the tiles given in the Tables 10.1, 10.2 are as follows: Cbt and Rcbt for Archimedean cuboctahedron and rhombicuboctahedron; RhDo, twisted RhDo and RhDo-v3 for Catalan rhombic dodecahedron, for its twist and for RhDo with deleted vertex of degree three; BDS ∗ for dual snub disphenoid. Remarks to Tables 10.1, 10.2: (1) The partition 15∗ is only one embeddable into Z∞ (in fact, with scale two). (2) All partitions, embeddable with scale one, are, except of 25∗ , zonohedral. The Voronoi tile of 25∗ is not centrally-symmetric. It will be interesting to find a normal tiling of 3-space, embeddable with scale 1, such that each tile is centrally-symmetric; such a non-normal tiling is given in [Shto80]: see item 35 in Table 10.3 below. (3) An embedding of tiles is necessary, but not sufficient, for embedding of whole tiling; for example, 26∗ and 27∗ are non-embeddable, while their tiles are embeddable into H4 and 21 H8 , respectively. In fact, all dual uniform partitions P ∗ in Tables 10.1, 10.2, except of 24∗ , have no non-embeddable tiles. Amongst all eleven non-embeddable uniform partitions only items 11, 15, 24 and 26 have only embeddable tiles. The same is true for tilings 30, 33 and 32∗ , 33∗ , 34∗ , 46∗ of Table 10.3. (4) Amongst all 28 partitions only items 1, 2, 5, 6, 8 have the same surrounding of edges: polygons (4.4.4.4), (4.6.6), (3.3.3.3), (3.3.6.6), (3.3.4). (5) Each of partitions 8 and 24 is the Delaunay partition of a lattice complex (i.e. bi-lattice, in which the environments of all points are identical,

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Table 10.1

Embedding of uniform partitions and their duals: partitions 1-14

3

4

1 3

22 28

3

4

11

4

5

26

5 6 7 8

2 13 14 15

1 6 8 7

9 10 11 12 13

22 6 8 12 11

27 24 18 17 13

14

11’

14

De(Z3 ), Po V o(A∗3 = bcc), sodalite De(A2 × Z) high-pressure Si V o(A2 × Z) B in AlB2 De(A3 = fcc), Cu Laves ph. M gCu2 boride CaB6 De( J-complex) perovskite (ABX3 ) zeolite rho De(4.82 × Z) De(3.6.3.6 × Z) De(34 .6 × Z) De(33 .42 × Z), boride F e2 AlB2 De(32 .4.3.4 × Z), silicide U3 Si2

8 4

5∗ of dual γ3 ∼ α3

6 emb. Z3 Z6

6∗ dual Z3 -

P rism3

12

P rism6

1 Z 2 4

Z4

P rism6

6

P rism3

Z4

1 Z 2 4

5 tiles γ3 tr(β3 )

α3 ,β3 α3 , tr(α3 ) β3 , tr(γ3 ) β3 , Cbt

8,6 2,6 1,4 2,4

RhDo ∼ γ3 P yr4 ∼ β3

-

1 Z 2 4

Z4 Z4 -

P rism8 ,tr(Cbt) P rism8 , γ3 P rism3 , P rism6 P rism3 , P rism6 P rism3 , γ3

2,2 4,2 4,4 8,2 6,4

∼ α3 ∼ P rism3 ∼ γ3 ∼ P rism5 ∼ P rism5

Z9 Z5 1 Z 2 7 1 Z 2 4

Z4 -

P rism3 ,γ3

6,4

∼ P rism5

1 Z 2 5

-


2

Uniform partitions of 3-space and relatives

1 Nr. 1 2

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Embedding of uniform partitions and their duals: partitions 15-28

3

4

7

19

De(3.122 × Z)

16 17

18 20

25 21

zeolit Linde A

18 19 20 21

17 16 21 9

9 5 10 16

oxifloride Ag7 O8 F boride U B12 De(3.4.6.4 × Z)

22

10

23

De(4.6.12 × Z)

23

19

20

selenide P d17 Se15

24 25 26 27 28

2’ -

2 3 4 12 15

De( hcp), Zn De(elong.A3 ), M oS2 De(elong.hcp), T iP silicide T hSi2 De(elongated 27)

5 tiles P rism3 , P rism12 γ3 , tr(β3 ), tr(Cbt) tr(α3 ), tr(γ3 ), tr(Cbt) γ3 , Cbt, Rcbt α3 , γ3 , Rcbt tr(α3 ), tr(β3 ), Cbt γ3 , P rism3 , P rism6 γ3 , P rism6 , P rism12 γ3 ,P rism8 , tr(γ3 ), Rcbt α3 ,β3 P rism3 ,β3 ,α3 P rism3 , β3 , α3 P rism3 P rism3 , γ3

2, 4 1,1,2 1,1, 2 2,1,2 1,3,1 2,1,2 4,2, 2 2,2, 2 1,2, 1,1 8,6 6,4,3 6,4,3 12 6,4

5∗ of dual ∼ P rism3

6 emb. -

6∗ dual 1 Z 2 ∞

∼ α3 ∼ α3

Z9 -

-

∼ BP yr3 ∼ BP yr3 ∼ P yr4 ∼ γ3

-

1 Z 2 4

-

∼ P rism3

Z7

-

∼ P yr4

-

-

twRhDo RhDo-v3 RhDo-v3 BDS ∗ ∼ P rism5

-

Z4 -

1 Z 2 7

-

1 Z 2 4

-

1 Z 2 5 1 Z 2 5


2


1 Nr. 15

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Table 10.2

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except, possibly, for an orientation): J-complex and the hcp; the tile of V o(J-complex) has form of jackstone (it explains the term ”J-complex”) and it is combinatorially equivalent to β3 . (6) Partitions 1, 3, 5 are the Delaunay partitions of lattices Z3 , A2 × Z, A3 = f cc. The partitions 2 and 4 are the Voronoi partitions of lattices A∗3 = bcc and A2 × Z. Items 10, 11, 12, 13, 14, 15, 21, 22 are the Delaunay prismatic partitions over eight Archimedean partitions of the plane; the embeddability of them and of their duals is the same as in Tables 4.1, 4.2, but the dimension increases by one. (7) In Chemistry, partitions 1, 5, 24 occur, for example, as metals polonium P o, copper Cu, zinc Zn; partitions 4, 7, 13, 20 as borides AlB2 , CaB6 , F e2 AlB2 , U B12 ; partitions 9 and 16 as zeolites rho and Linde A; partitions 3 and 14, 27 as high-pressure Si and silicides U3 Si2 , T hSi2. Finally, partitions 2, 6 (known also as Föppl partition), 8, 19, 23, 25, 26 occur as sodalite N a4 Al3 Si3 O12 Cl, Laves phase M gCu2 , perovskit (ABX3 ), Ag7 O8 F , P d17 Se15 , M oS2 , T iP . Nice pictures of all those structures and chemical details can be found in chapters 6 and 7 of [OkHy96]. (8) The ratios of tiles in the partition are as follows: 1:1 for 6, 7, 8, 10 8:1 for 12 3:2:1 for 21, 22, 25, 26

2:1 for 5, 11, 13, 14, 24, 28 2:1:1 for 17, 19, 20 3:3:1:1 for 23.

3:1 for 9 3:1:1 for 16, 18

(9) All uniform partitiomns, which come from the regular one Nr. 1 by Wythoff kaleidoscope construction, are items 1, 2, 7, 8, 9, 16, 18, 23 (see Table 4 in [DDS04]). 10.2

Other special partitions

In Table 10.3 we group some other relevant partitions. Here L5 denotes a 3-dimensional lattice of the 5-th Fedorov’s type (i.e. by the Voronoi polyhedron) and ElDo denotes its Voronoi polyhedron, called elongated dodecahedron. Remaining four lattices appear in Tables 10.1, 10.2 as Nr.1=De(Z3 ) = V o(Z3 ), Nr.5=De(A3 ), Nr.2=V o(A∗3 ), Nr.3=De(A2 × Z1 ), Nr.4=V o(A2 × Z1 ). Remark that De(L5 ) and De(A2 × Z1 ) coincide as graphs, but differ as partitions. By the notation De(Kelvin) below we denote any Kelvin packing by α3 and β3 (in proportion 2:1), which is proper, i.e. different from the lattice

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Scale Isometric Polytopal Graphs Table 10.3

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

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De(L5 ) De(D-complex) De(Kelvin) De(Gr¨ unbaum) De(elong. Kelvin) De(elong. Gr¨ unbaum) P (S1 ) P (S2 ) P (S3 ) A-19 A-20, n even A-20, n odd A-22 A-23 k −type ⊥ −type chess-type A-13′

Embedding of some other partitions α3 , P yr4 α3 , ∼ β3 α3 , β3 P rism3 α3 ,β3 , P rism3 P rism3 , γ3 S1 S2 S3 P rism∞ P rism∞ Cn × PZ Cn × PZ Aprism∞ P rism∞ ,Aprism∞ γ3 , C4 × PZ γ3 , C4 × PZ γ3 , C4 × PZ α3 , tr(α3 )

ElDo triakis tr α3 RhDo, tw RhDo P rism6 , BDS ∗ RhDo-v3 ∼ P rism5

R, twisted R

1 Z 2 4 1 Z 2 5

-

1 Z 2 5

-

1 Z 2 5

Z3 Z4 Z5 Z2 Z2 Z∞ 1 Z 2 ∞ 1 Z 2 3 1 Z 2 3 Z3 Z3 Z3 -

Z5 -

-

A3 = f cc and the bi-lattice hcp. Each proper Kelvin partition, as well as the partition 13′ in [Andr05] (given as 46 in Table 10.3, which Andreini wrongly gave as a uniform one), have exactly two vertex figures. The Voronoi tiles of the tiling 46 are two rhombohedra: a rhombohedron (i.e. the cube contracted along a diagonal), say, R, and twisted R; both are equivalent to γ3 . The same is true for each Gr¨ unbaum partition; see items 32–34 of Table 10.3. In fact, call a normal partition of the 3-space into Platonic and Archimedean polyhedra, almost-uniform if the group of symmetry is not vertex-transitive but all vertex figures are congruent. Gr¨ unbaum [Gr¨ un94] gave two infinite classes of such partitions and indicated that he do not know other examples. In our terms, they called elongated proper Kelvin and elongated proper Gr¨ unbaum partitions. Kelvin and Gr¨ unbaum partitions are defined uniquely by an infinite binary sequence characterizing the way how layers follow each other. In Kelvin partition, the layers of α3 and β3 follow each other in two different ways (say, a and b) while in Gr¨ unbaum partition the layers of P rism3 follow each other in parallel or perpendicular mutual disposition of heights. Unproper Kelvin partitions give uniform partitions 5 and 24 for sequences ...aaa... (or ...bbb...) and

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...ababab..., respectively. Proper Kelvin and Gr¨ unbaum partitions are not almost-uniform; there are even ∞-uniform ones (take a non-periodic sequence). Consider now elongations of those partitions, i.e. we add alternatively the layers of P rism3 for Kelvin and of cubes for Gr¨ unbaum partitions. Remark that RoDo − v (the Voronoi tile of partitions 25, 26, 33) can be seen as a half of RoDo cut in two, and that twRoDo is obtained from RoDo by a twist (a turn by 90o ) of two halves.The Voronoi tiles for proper Kelvin partition 31 are both RoDo and twRoDo, while only one of them remains for two unproper cases 5, 24. Similarly, the Voronoi tile of 34 (a special 5-prism) can be seen as a half of P rism6 cut in two, and BDS ∗ can be seen as twisted P rism6 in similar way. The Voronoi tiles for proper Gr¨ unbaum ∗ partition 32 are both P rism6 and BDS , while only one of them remains for two unproper cases 3, 27. Besides of two unproper cases 25, 26 (elongation of uniform 5, 24) which are uniform, we have a continuum of proper elongated Kelvin partitions (denoted 33 in Table 4), which are almost-uniform. Amongst them there is a countable number of periodic partitions corresponding to periodic (a, b)sequences. Remaining continuum consists of aperiodic tilings of 3-space by α3 , β3 , P rism3 with very simple rule: each has unique Delaunay star consisting of 6 P rism3 (put together in order to form a 6-prism), 3 α3 and 3 β3 (put alternatively on 6 triangles subdividing the hexagon) and one α3 filling remaining space in the star. Each b in the (a, b)-sequence, defining such tiling, corresponds to the twist interchanging 3 α3 and 3 β3 above (i.e. to the turn of the configuration of 4 α3 , 3 β3 by 60o ). Similar situation occurs for elongated Gr¨ unbaum partitions. Besides two uniform unproper cases 13, 28 (elongation of uniform 3, 27), we have a continuum of proper elongated Gr¨ unbaum partitions (denoted 34 in Table 4) which are almost-uniform. The aperiodic (a, b)-sequences give a continuum of aperiodic tilings by P rism3 , γ3 with similar simple rule: unique Delaunay star consisting of 4 γ3 (put together in order to form a 4-prism), 4 P rism3 put on them and 2 P rism3 filling remaining space in the star. Each b in (a, b)-sequence, defining the tiling, corresponds to a turn of all configuration of 6 P rism3 by 90o . A D-complex is the diamond bi-lattice; triakis tr(α3 ) has P yr3 on each its triangular faces. Partitions 29 and 30 from Table 10.3 are both vertextransitive, but they have some non-Archimedean tiles: P yr4 for Nr.29 and a non-regular polyhedron, having the same combinatorial type with β3 , for

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Nr.30. The partitions 35, 36, 37 of Table 10.3 are just all three non-normalizable tilings of 3-space by a convex parallelohedron, which were found in [Shto80] (see Theorem 11.3 below). The polyhedra denoted by S1 , S2 and S3 are centrally symmetric 10-hedra obtained by a decoration of the paralellopiped. S1 is equivalent to β3 truncated on two opposite vertices. P ∗ (Si ) for i = 1, 2, 3 are different partitions of 3-space by non-convex bodies, but they have the same skeleton, which is not 5-gonal. We will denote here non-compact uniform partitions, introduced in subsections 19, 20, 22, 23 of [Andr05], by A-19, A-20, A-22, A-23, respectively, and put them in Table 10.3 as Nr’s 38, 40, 41, 42. Denote by P rism∞ , AP rism∞ and Cn × PZ the ∞-sided prism, ∞-sided antiprism and the cylinder on Cn , respectively. A-19 is obtained by putting P rism∞ on δ2 and so, its skeleton is Z2 . We add, as item 39, the partition, which differs from item 38 only by other disposition of some infinite prism under net δ2 , i.e. perpendicular to those above it. A-20 is obtained by putting the cylinders on δ2 , A-22 by putting AP rism∞ on (36) and A-23 by putting P rism∞ and AP rism∞ on (33 .42 ). In subsection 20′ [Andr05] mentions also the partition into two halfspaces separated by some of ten Archimedean (and one degenerated) nets, i.e. uniform plane partitions. We can also take two parallel nets (say, T ) and fill the space between them by usual prisms (so, the skeleton will be direct product of the graph of T and K2 ) or, for T = δ2 or (33 .42 ), by a combination of usual and infinite prisms. Similar uniform partitions are obtained if we will take an infinity of parallel nets T . The partitions 43, 44, 45 of Table 10.3 differ only by the disposition of cubes and cylinders. In 43 the layers of cylinders stay parallel (k-type); in 44 they are perpendicular to the cylinders of each previous layer. In 45 we see δ2 as the infinite chess-board; cylinders stay on ”white” squares while piles of cubes stay on the ”black” ones. One can get other non-compact uniform partitions by a decoration of P rism∞ in 38, 39.

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Chapter 11

Lattices, Bi-lattices and Tiles

Remind that V o(L) and De(L) denote the skeletons of Voronoi partition and Delaunay partition for a lattice L. So, edges of these graphs are edges of the Voronoi parallelotope and of the Delaunay polytopes of L; any minimal vector of L is an edge of De(L) but, in general, not vice versa. We are interested whether the infinite graph G, where G = V o(L) or De(L), either is embedded isometrically (or with doubled distances) into a Zm for some m ≥ n, or not. We report here the results obtained in this direction for irreducible root lattices, for two generalizations of the diamond bi-lattice and for a 3-dimensional case. See [Voro1908], [RyBa79] and [CoSl88] for notions in Lattice Theory. It turns out again that all cases of non-embedding, given in this chapter, came out by violation of a 5-gonal inequality.

11.1

Irreducible root lattices

Let us consider the irreducible root lattices, i.e. An , Dn , E6 , E7 , E8 . For small dimension n, one has: De(A2 ) = (36 ) → 21 Z3 , V o(A2 ) = 3 (6 ) → Z3 ([Asso81]) and, of course, D2 = Z2 , A∗2 = A2 , D3 = A3 . Theorem 11.1 We have: (i) De(En ) is not 5-gonal for n = 6, 7, 8; (ii) for n ≥ 3, De(An ) → 21 Zn+1 , V o(An ) → Zn+1 , ∗ V o(A∗n ) → Zm (where m = n+1 2 ), De(An ) is not 5-gonal; (iii) for n ≥ 4, De(Dn ), V o(Dn ), De(Dn∗ ) are not 5-gonal. For example, De(D4 ) is not embeddable (contrary to Proposition 2 (b) of 107

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[Asso81]): take 5 points such that a = (0, 0, 0, 0), b = (1, 1, 0, 0), x = (1, 0, 1, 0), y = (1, 0, −1, 0), z = (0, 1, 0, 1) forming a not 5-gonal isometric subgraph K5 − K3 . In fact, (x, y) is not an edge, since the middle point of the segment [x, y] is the center of the square (a, x, c, y), where c = (2, 0, 0, 0), with edges (a, x), (x, c), (c, y), (y, a) from the graph De(D4 ). Now, De(D4 ) is a metric subspace of De(Dn ) for n ≥ 5. Remark also, that we have isometric embedding of Zn into De(D2n ) and of Z2 into De(A3 ). 11.2

The case of dimension 3

Now we consider five types (depending on their Voronoi polyhedron) of 3-dimensional lattices, obtained by Fedorov [Fedo1885]. Besides of Z3 , A3 = f cc and A∗3 = bcc, there are two other types of lattices having a 6-prism and an elongated dodecahedron as the Voronoi polyhedron. Let us take A2 × Z1 and, say, L5 as representatives of the lattices of these two types. Remind that De(A3 ), De(L5 ) coincide as graphs, but the partitions of R3 (for which they are skeletons) are different. Theorem 11.2 We have: (i) De(A2 × Z1 ) → 12 Z4 , V o(A2 × Z1 ) → Z4 ; (ii) De(L5 ) → 21 Z4 , V o(L5 ) → Z5 . So, the Delaunay partition of the general lattice A∗3 is unique nonembeddable one amongst De(L), V o(L) for the five Voronoi types of 3dimensional lattices. In R3 , the combinatorial type of a parallelohedron P determines the combinatorial type of the corresponding tiling by P ; also the type of a dual partition is determined by the type of its star (i.e. the configuration around a vertex). For normal partitions, we have five types of parallelohedra and five dual types of partitions (whose skeletons are of four types, this was already in [Fedo1885]); their embeddings are described in the theorem above. In [Shto80], all three types of convex parallelohedra for essentially nonnormal (i.e. non-normalizable) partitions of R3 were found (see Table 10.3 above). Denote these parallelohedra by S1 , S2 , S3 ; and denote by P (Si ), P ∗ (Si ) the tiling by Si and the dual partition for i=1, 2, 3. All Si (i=1, 2, 3)

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Lattices, bi-lattices and tiles

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are centrally-symmetric 10-hedrons obtained by decoration of a rectangular parallelepiped; their p-vectors are (p4 =10), (p4 =6, p6 =4), (p4 =4, p6 =4, p8 =2), respectively. S1 = γ3 + γ3 is (combinatorially) β3 truncated in two opposite vertices; S2 and S3 have 2-valent vertices. All P ∗ (Si ) have the same combinatorial type of the skeleton; they are partitions of R3 by non-convex bodies. Theorem 11.3 For i = 1, 2, 3, we have Si → H3+i , P (Si ) → Z2+i and P ∗ (Si ) is not 5-gonal. The two most interesting lattice complexes in 3-space are bi-lattices Jcomplex and D-complex (D-complex is called also diamond or tetrahedral packing and denoted by D3+ ). Theorem 11.4 We have: (i) De(D-complex) → 12 Z5 , but Vo(D-complex), De(J-complex), Vo(Jcomplex) are not 5-gonal; (ii) any Kelvin packing K by α3 and β3 (except A3 , but including the bi-lattice hcp) has not 5-gonal De(K), V o(K). Amongst the packings of 3-space, considered above, the tilings De(Z3 ), De(A2 × Z1 ), V o(A2 × Z1 ), De(A3 ), V o(A∗3 ), De(J-complex), De(hcp) are uniform partitions 1, 3, 4, 5, 2, 8, 24 from Tables 10.1, 10.2. Consider now the following two bi-lattices generalizing a D-complex: Dn+ := Dn ∪ (d + Dn ), where the new point d is the center of the greatest Delaunay polytope, and A+ n := An ∪ (a + An ), where the new point a is the center of a regular n-simplex, which is a Delaunay polytope of An , see [CS88]. Dn+ is a lattice if and only if n is even; D2+ = Z2 , D4+ = Z4 , D8+ = E8 . + An is always a bi-lattice; it is obtained from An by the above centering of its smallest Delaunay polytope, the regular n-simplex αn . The centering of all Delaunay polytopes of An will give A∗n . Remind that De(A+ 2) = + + 1 6 Z , A = D . V o(A2 ) = (63 ) → Z3 , V o(A+ ) = (3 ) → 3 3 3 2 2 Theorem 11.5 We have: (i) De(D3+ ) → 21 Z5 , V o(D3+ ) is not 5-gonal, De(Dn+ ) is not 5-gonal for n ≥ 5;

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110


Scale Isometric Polytopal Graphs 1 (ii) De(A+ n ) → 2 Zn+2 for n ≥ 3.

All above results are obtained by techniques given in [CDG97] and [DeSt96]. For example, not 5-gonality of De(A∗n ), n ≥ 3, is given by 5 points: a = (0, 0, 0, 0, . . . , 0), b = (1, 1, 1, 0, . . . , 0), x = (1, 0, 0, 0, . . . , 0), y = (0, 1, 0, 0, . . . , 0), z = (0, 0, 1, 0, . . . , 0) in the Selling-reduced basis of the lattice; the points a, x, x+y, b are vertices of a face of a Delaunay simplex. 11.3

Dicings

Let us consider a family of equi-spaced parallel hyperplanes that cuts Rn into slabs of equal thickness. If we take n such hyperplane families that are independent, we obtain Rn diced into equal parallelepipeds with the vertices forming an n-dimensional lattice. In general, a dicing D is an arrangement of hyperplanes formed by families of equi-spaced parallel hyperplanes that satisfies the following properties: (i) amongst the families there are n independent ones, (ii) through each vertex of the dicing there is one hyperplane from each family. The condition (i) ensures that the cells of D are polytopes. The condition (ii) implies that the vertex-set L(D) of a dicing D is a lattice and the vertex-set of any sub-dicing (satisfying (i)) coincides with L(D). The lattice L(D) is called a dicing lattice. The partition of Rn , which is cut by hyperplanes of a dicing D, is the Delaunay partition of the dicing lattice L(D). The main property of a dicing lattice is the following (see [Erda98]): the Voronoi polytope of a lattice is a zonotope if and only if the lattice is a dicing lattice. Since the skeleton of a zonotope Zm of diameter m is embeddable into Hm , the skeleton of the Voronoi partition of any dicing lattice is embeddable into Zm for the same m. Our question is: do there exist, amongst non-zonotopal centrally symmetric polytopes with embeddable skeleton, ones, which are parallelohedra (i.e. tile the space by translation) or, more precisely, Voronoi polytopes of

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lattices? If answer is yes, then the problem is, whether the corresponding Voronoi partition is embeddable into some Zm (remind that embeddable RhDo–v3 is the tile of Voronoi partitions Nr.25∗ , 26∗ from Tables 10.1, 10.2, amongst which only Nr.25∗ is embeddable) and gives, perhaps, an infinite analog of a non-realizable oriented matroid (see chapter 14). If answer is no, then we obtain a new characterization of dicing lattices as such lattices, whose Voronoi polytope has the skeleton embeddable into Hm for some m. The case of embeddability of Delaunay partition of a dicing lattice is more complicated. At first glance, it seems that the Delaunay partition formed by m families of hyperplanes should be embeddable into Zm . But this is not always so. For example, the graph of Delaunay partition of the dicing lattice A∗n , n ≥ 3, is non-embeddable. 11.4

Polytopal tiles of lattice partitions

Here the ambient discrete set of points is a point lattice. Voronoi polytopes Recall that a partition of n-space is called primitive if each vertex of the partition belongs exactly to n + 1 polytopes (tiles) of the partition. Otherwise, the partition is non-primitive. A polytope (tile) determining a primitive or non-primitive partition is called primitive or non-primitive, respectively. So, there are primitive and non-primitive Voronoi polytopes. In each dimension n ≥ 2, amongst all primitive Voronoi n-polytopes, there is only one, namely n-permutahedron, which is a zonotope. This (n + 1)!-vertex zonotope is the Voronoi polytope of the lattice An ∗; it embeds into H(n+1) . 2

For n = 2, the Voronoi polytopes (the non-primitive rectangles and primitive centrally symmetric hexagons) both are zonotopes and have the skeletons C4 = H2 and C6 → H3 . For n = 3, all five Voronoi polytopes are zonotopes (primitive truncated octahedron and four non-primitive ones: cube, P rism6 , rhombic dodecahedron, elongated dodecahedron) and so, their skeletons are embeddable with scale one into Hm (for m = 6, 3, 4, 4, 5, respectively). For n = 4, there are three primitive Voronoi polytopes. In 1937 Delaunay enumerated 2(primitive)+49=51 Voronoi n-polytopes. They are given in [Delo37]; in 1968 Shtogrin found the last one (this Nr.52 has 24 facets, which are 8-hedra of two types: 18 of one type and 6 of the other one).

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Amongst above 52 4-polytopes, 17 are zonotopes (Nr.1 and 3–19); they are embedded into Hm for m = 10, 9, 8, 8, 7, 7, 7, 7, 6, 6, 6, 5, 5, 6, 5, 4, 9, respectively. We checked that all others are not 5-gonal; moreover, only Nr.33 (unique non-zonotope with bipartite skeleton), Nr.45, Nr.48, Nr.50, Nr.51 (the regular 24-cell) and Nr.52 have only 5-gonal 3-faces. The skeleton of an i-face is not, in general, an isometric subgraph of the skeleton of a convex n-polytope (for example, a hexagon is not isometric face in a 6-pyramid), but it is so, if the polytope is a parallelohedron. Two lattices are said to be of the same L-type, if their Voronoi (and so, Delaunay) partitions are affinely equivalent. Ryshkov and Baranovskii found in [RyBa76] 221 distinct general (i.e. corresponding, in the cone of positive semi-definite quadratic forms, to domains of maximal dimension) L-types of 5-dimensional lattices. Each general L-type determines a primitive Voronoi polytope. Engel [Enge98], using a computer, found 222 distinct primitive Voronoi 5-polytopes. Using those results, the L-type, which was missed in [RyBa76], was explicitely given in [EnGr02]. Engel [Enge98] found also that there are exactly 179.377 Voronoi 5-polytopes; 80 amongst them are non-primitive ones, which are zonotopes. Note that it is a routine work to enumerate all Voronoi n-polytopes, which are zonotopes: there is a bijection between such Voronoi n-polytopes and non-isomorphic unimodular matroids of rank n. This is similar to that all n-zonotopes are in one-to-one correspondence with realizable oriented matroids of rank n (see chapter 14). Delaunay polytopes Now we consider Delaunay polytopes of small dimension and some operations on Delaunay polytopes. For n = 2, the Delaunay polytopes (triangles with acute angles and rectangles) have the skeletons C3 → 21 H3 and C4 = H2 . For n = 3, skeletons of all five Delaunay polytopes are also ℓ1 -graphs: γ3 = H3 , α3 = 21 H3 , β3 → 12 H4 , Pyr4 → 21 H4 , P rism3 → 12 H5 . All 19 types of Delaunay 4-polytopes are given in [ErRy87]. We get the following proposition, listing all embeddable ones amongst them, by direct check; the Nr.i, 1 ≤ i ≤ 16 (only in this chapter) and the letters A, B, C below are notation from [ErRy87]. Kononenko in 1997 computed that there are exactly 138 Delaunay 5polytopes and Dutour in 2002 found all 6241 Delaunay 6-polytopes. Proposition 11.1 (i) Nr.16=γ4 = α1 × γ3 = H4 ;

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(ii) P → 12 H4 for the following polytopes P : Nr.2=Pyr(P yr4 ) with G(N r. 2) = K2,2,1,1 , B=Pyr(β3 ) with G(B) = K2,2,2,1 , C=BPyr(β3 ) = β4 = 12 H4 ; (iii) P → 12 H5 for the following polyhedra P : Nr.1=Pyr(α3 ) = α4 , Nr.3=Pyr(P rism3 ), Nr.5, Nr.7=α1 × α3 = P rism(α3 ), Nr.9, Nr.13 with G(N r. 13) = T (5) = J(6, 2); (iv) P → 12 H6 for the following polyhedra P : Nr.10= α2 × α2 , Nr.11=α1 × P yr4 = P rism(P yr4 ), Nr.15= α1 × β3 = P rism(β3 ), A with G(A) = K6 (the cyclic 4-polytope); (v) P → 12 H7 for P = N r.14 = α1 × P rism3 = P rism(P rism3 ); (vi) the following polyhedra P are not 5-gonal: Nr.4, Nr.6=BPyr(P rism3 ), Nr.8=Pyr(γ3 ), Nr.12=BPyr(γ3 ). It is well known that the direct product P × P ′ of Delaunay polytopes P and P ′ is a Delaunay polytope, and G(P × P ′ ) = G(P ) × G(P ′ ). Also, for every Delaunay polytope P there is a pyramid Pyr(P ) and (if P is centrally symmetric) a bipyramid BPyr(P ), which are Delaunay polytopes (see [DeGr93]). The direct product construction preserves ℓ1 -ness of Delaunay polytopes. For example, G(P × P ′ ) → 21 Hm+m′ if G(P ) → 12 Hm and G(P ′ ) → 12 Hm′ . But the pyramid and bipyramid constructions can produce not 5-gonal Delaunay polytopes from those with ℓ1 -skeletons. Consider, for example, the following Delaunay polytopes: αn , βn , γn , 12 γn (n ≥ 5), Ambo(αn ) (n ≥ 4), the Johnson n-polytope PJ with G(PJ ) = J(n + 1, k) (3 ≤ k ≤ ⌊(n + 1)/2⌋), αn−1 × αn−1 for n ≥ 3; P rism3 , P yr4 . All of them have ℓ1 -skeletons: Ambo(αn ) → 21 Hn+1 , PJ → 12 Hn+1 , P rism3 → 21 H5 , P yr4 → 12 H4 . The pyramid and bipyramid constructions produce from them ℓ1 -polytopes in the following cases: Pyrm (αn ), Pyrm (βn ), BPyrm (βn ), Pyrm (P yr4 ). Additionally, we have the following ℓ1 -embeddings: • Pyr2 (P rism3 ) → 12 H5 , • Pyr(Ambo(αn )) → 12 Hn+1 , • Pyr(αn−1 × αn−1 ) → 12 H2n . The inclusions (i), (ii) below show that the skeletons of the mentioned there polytopes are not 5-gonal, and therefore, they are not ℓ1 -graphs:

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(i) the graph K5 − K3 is an isometric subgraph of the skeletons of the following eight types of polytopes: • • • • • • • •

Pyr(γn ), BPyr(γn ), Pyr2 (Ambo(αn )), n ≥ 6, Pyr(PJ ) (since K1,k is an isometric subgraph of J(n + 1, k)), Pyr( 12 Hn ), n ≥ 6, Pyr2 (αn−1 × αn−1 ), BPyr(Ambo(αn )), n ≥ 6, even, BPyr( 21 Hn ), n ≥ 6 even;

(ii) the graph K5 − P2 − P3 is an isometric subgraph of the skeleton of BPyr(P rism3 ). Finally, the following skeletons are hypermetric non-ℓ1 (see Proposition 4.4 of [DeGr93]): • • • • •

Pyr( 12 γ5 ), Pyr2 ( 12 γ5 ), Pyr2 (Ambo(α4 )), Pyr3 (Ambo(α4 )), Pyr3 (P rism3 ), Pyr4 (P rism3 ).

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Chapter 12

Small Polyhedra

12.1

Polyhedra with at most seven faces

In this chapter Nr.i mean only Nr.1–10, 11–44 of Figures 12.1, 12.2. The ℓ1 -status of all ten polyhedra with at most six faces and their duals is such that they are either ℓ1 -embeddable, or not 5-gonal. The ℓ1 -embeddable ones are → 21 H4 : α3 ≃ α∗3 , P yr4 ≃ P yr4∗ , BP yr3 = P rism∗3 , γ3∗ = β3 , → 21 H5 : P yr5 ≃ P yr5∗ , P rism3 ≃ BP yr3∗ , (2-truncated α3 )∗ , one with the skeleton K6 − P5 of the dual (Nr.43 in Proposition 12.1 below and on Figure 12.2); → 12 H6 : γ3 , 2-truncated α3 . Remaining are: self-dual one with the skeleton K6 − P6 , one with the skeleton K3×2 − e of the dual, its dual (Nr.44 in Proposition 12.1 and on Figure 12.2) and one with the skeleton K6 − P5 of the dual. Proposition 12.1 Amongst all 34 polyhedra (Nr.11–44 on Figure 12.2) with seven faces and their duals, we have → 21 H5 : 20∗ , 21∗ , 29∗ , 35, 36∗ , 38∗ , 39 ≃ 39∗ , 43; → 21 H6 : 12∗ , 14∗ , 16∗ , 17∗ , 18, 19 ≃ 19∗ ; → 12 H7 : 12, 20, 23; hypermetric non-ℓ1 : 13∗ ; not 7-gonal: 32∗ , 37∗ , and not 5-gonal: all others (including 40∗ = 42 and self-dual ones 34, 41). Remark 12.1 (i) Amongst above ℓ1 -polyhedra only γ3 is bipartite and only three are not ℓ1 -rigid: α3 ≃ α∗3 → 21 H3 , 12 H4 , BP yr3 → 21 H4 (two ways) and 39 ≃ 39∗ → 21 H5 , 21 H6 . All simple ones amongst above 10+34 115

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116

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1

6

Fig. 12.1

2

3

7

8

4

9

5

10

All polyhedra with at most six faces

polyhedra with at most seven faces are: α3 , P rism3 =1-truncated α3 , 2truncated α3 , γ3 and Nr.11, 12 (3-truncated α3 ), 13, 20 (1-truncated γ3 ), 23=P rism5 . (ii) 20, amongst of all 44 polyhedra with at most seven faces, are combinatorially equivalent to a space-filler: all ten, except P yr5 , with at most six faces (including all three not 5-gonal) and Nr.12, 15, 18, 20, 22, 23, 28, 30, 36, 43, 44 (amongst them Nr.12, 18, 20, 23, 43 are ℓ1 -graphs). 12.2

Simple polyhedra with at most eight faces

Proposition 12.2 Amongst all 27 cubic graphs with up to ten vertices, 19 are not 5-gonal, while remaining eight are ℓ1 -graphs: non-polytopal Petersen graph → 21 H6 and 7 simple polyhedra: i-truncated α3 → 12 H4+i for 0 ≤ i ≤ 3, i-truncated γ3 → 21 H6+i for i = 0, 1 and P rism5 → 12 H7 . Proposition 12.3 Amongst the skeletons of all 14 simple polyhedra with eight faces (i.e. with 12 vertices) four are ℓ1 -graphs (all are embedded into 12 H8 ): P rism6 , dual snub disphenoid BDS ∗ , D¨ urer’s octahedron and γ3 , truncated on two adjacent vertices. Duals of first two are not 5-gonal and of last two → 12 H6 . 4-truncated α3 and γ3 , truncated on two vertices at distance 2, are not 5gonal; their duals → 12 H7 . The five polyhedra from Figure 12.3 are respectively, not 5-, 5-, 7-, 7-, 9-gonal; the skeletons of their duals are respectively, embeddable into 21 H8 , 12 H7 , 21 H7 , 12 H6 , not 5-gonal. Remaining three polyhedra and their duals are not 5-gonal. One of the three above polyhedra (a 3-truncated P rism3 ) is the smallest

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117

Small polyhedra

11

12

16

14

13

17

15

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

Fig. 12.2

42

43

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44

All polyhedra with seven faces

simple polyhedron with p = (3, 1, 1, p6, 0, . . . ); such polyhedra exist (see page 271 in [Gr¨ un67]) if and only if p6 is odd, other than one. The D¨ urer’s octahedron above is γ3 , truncated on two opposite vertices; it is called so, because it appears in D¨ urer’s painting ”Melancolia”, 1514, staying on a triangular face. Remark 12.2 (i) The skeletons of all simple ℓ1 -polyhedra with f ≤ 8 faces (there are eleven of them) are embedded into 12 Hf ; all of them, except of α3 , are ℓ1 -rigid and γ3 → H3 , P rism6 → H4 . (ii) the polyhedron in the center of Figure 12.3 and one of three not 5-gonal simple octahedra, having not 5-gonal duals, are the two smallest 3-regular graphs with trivial group of automorphism. (iii) amongst all eleven simple polyhedra with only k-gons, k ≤ 5, as faces (duals of all eight convex deltahedra, 1-truncated α3 , 1-truncated γ3

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11 10 00 00 11 00 11 00 11 00 11 00 10 11 0 1 00 11 0 1 0 1 00 11 11 00 1010 11 00 0111 00 00 11 1010 11 00 00 11 11 00 Fig. 12.3

Some simple octahedra

and D¨ urer’s octahedron) the skeletons of only two (duals of 3-augmented P rism3 and of 2-capped AP rism4 ) are not ℓ1 -graphs.

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Chapter 13

Bifaced Polyhedra

Denote by (k; a, b; pa , pb ) and call bifaced any k-valent polyhedron, whose faces are only pa a-gons and pb b-gons, where 3 ≤ a < b and pa > 0, pb ≥ 0. Any bifaced polyhedron (k; a, b; pa , pb ) with n vertices has kn/2 = (apa + bpb )/2 edges and satisfies Euler relation n − k n2 + (pa + pb ) = 2, i.e. n = 2(pa + pb − 2)/(k − 2). Equating above two expressions for n, we obtain pa (2k − a(k − 2)) + pb (2k − b(k − 2)) = 4k.

(13.1)

This equality implies a < 2k/(k − 2). In fact, the inequality 2k ≤ a(k − 2) implies the inequality 2k < b(k − 2). Both these inequalities and 13.1 contradict to k ≥ 3. Hence only possible (k, a) are (3,5), (3,4), (3,3), (4,3), (5,3). Now, since b > a, we have b = 2k/(k − 2) only for (k, b) = (3, 6), (4, 4), and b < 2k/(k − 2) only for six simple (i.e. with k = 3) polyhedra with (a, b) = (3, 4), (3, 5), (4, 5). If pb = 0, then the above five possible values of (k, a) give (combinatorially) Platonic polyhedra Do, γ3 , α3 , β3 , Ico, respectively. If pb = 1, then bifaced polyhedra do not exist. If pb = 2, then there are no bifaced polyhedra with (k, a) = (3, 3), and for (k, a) = (3, 4), (3,5), (4,3), (5,3) polyhedra are as follows: P rismb , Barrelb (the dual of 2-capped AP rismb ), AP rismb and snub AP rismb . The last polyhedron consists of the following three circuits: outer (x1 , . . . , xb ), middle (y1 , y1′ , . . . , yb , yb′ ), and inner (z1 , . . . , zb ), where ′ xi ∼ yi , yi′ , yi+1 and zi ∼ yi−1 , yi , yi′ for all i = 1, . . . , b ordered cyclically. This polyhedron is called snub AP rismb since for b = 4 it is (combinatorially) the well known regular-faced polyhedron snub AP rism4 , which is 5-gonal, but not 7-gonal. Note that snub AP rism3 is sometimes called snub tetrahedron; it has the same skeleton as Ico, but its symmetry is T . 119

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Barrelb := (2 − AP rismb )∗ (see chapter 4) is an analog of P rismb , where the layer of 4-gons is replaced by two layers of 5-gons; it has 4b vertices. Proposition 13.1 We have: (i) P rismb → 12 Hb+2 (moreover, → H(b+2)/2 for even b), P rism∗b → 21 H4 , 21 H4 , not 5-gonal for b = 3, 4, and b ≥ 5, respectively; (ii) AP rismb → 21 Hb+1 , AP rism∗b → H3 , not 5-gonal for b ≥ 4; (iii) Barrel3 =D¨ urer’s octahedron → 21 H8 , Barrel3∗ → 12 H6 , Barrel5 = Do → 12 H10 , Barrel5∗ → 21 H6 , Barrel4 is not 5-gonal (see Figure 6.4 d)), Barrel4∗ is hypermetric nonl1 , for b > 5 both, Barrelb and its dual, are not 5-gonal; (iv) snub AP rism3 = Ico → 12 H6 , its dual → 21 H10 , snub AP rism4 is not 7-gonal, its dual is not 5-gonal, for b ≥ 5 both, snub AP rismb and its dual, are not 5-gonal. From now on we consider mainly bifaced polyhedra with pb ≥ 3. Consider first simple bifaced polyhedra, i.e. the case of k = 3. For b < 6 there are only six such polyhedra: the D¨ urer’s octahedron (i.e. γ3 truncated on two opposite vertices) and five dual deltahedra (P rism3 and duals of BP yr5 , of snub disphenoid, of 3-augmented P rism3 , of 2capped AP rism4 ). Moreover, there are only eleven simple polyhedra with P i≥3 pi = p3 + p4 + p5 : six above polyhedra, remaining three dual deltahedra (α3 , γ3 , Dodecahedron) and 1-truncated γ3 , 2-truncated α3 , having (p3 , p4 , p5 ) = (1, 3, 3), (2,2,2), respectively. The ℓ1 -status of all these polyhedra and their duals is known (see above). Simple bifaced n-vertex polyhedra with b = 6 will be denoted as an : 3n , 4n , 5n (fullerenes), for a = 3, 4, 5, respectively. Their parameter pa does not depend on n: p3 = 4, p4 = 6, p5 = 12. The polyhedra an , a = 3, 4, 5, will be considered below as well as the interesting case of (4; 3, 4; p3 , p4 ), having also constant p3 = 8. We denote last polyhedra by ocn and call them octahedrites.

13.1

Goldberg’s medial polyhedra

In 1935, Goldberg [Gold35] introduced the fullerenes (in a slightly more general setting). He mentioned that Kirkman found, in 1882, over 80 (amongst all 89) isomers of F44 . For n 6= 18, 22, he defined a medial polyhedron with

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24 n vertices as a simple polyhedron, for which all faces are ⌊6 − n+4 ⌋-gons 24 or ⌊7 − n+4 ⌋-gons. For 20 ≤ n 6= 22, medial polyhedra are the fullerenes. Therefore, we extend the notation Fn to any medial polyhedron with n vertices. Figure 13.1 reproduces those medial polyhedra, the last nine being F20 (Ih ), F24 (D6d ), F26 (D3h ), F28 (Td ), F28 (D2 ), F36 (D3h ), F36 (D6h ), C60 (Ih ) and C80 (Ih ) (C80 (Ih ) is called in [Gold35] chamfered dodecahedron). Medial polyhedra was introduced by Goldberg as putative best (isoperimetrically) approximation of a sphere within the class of polyhedra having given number f of faces. A medial polyhedron with f faces is a bifaced polyhedron (3; a = ⌊6 − 12 f ⌋, b = a + 1; pa , pb = f − pa ); it exists for any f ≥ 4, except for f = 11, 13. For f = 4, . . . , 10, 12 they are exactly dual of the eight convex deltahedra: α3 , P rism3 , P rism4 = γ3 , P rism5 , dual snub disphenoid, dual 3-augmented P rism3 , Barrel4 and Barrel5 = Do (see the case k = 3 of Table 13.1 below). The following two polyhedra given in Figure 13.2 are almost medial; we call them quasi-medial polyhedra with 18 and 22 vertices and denote ∗ them by F˜18 (C2v ) and F˜22 (C2v ). F˜18 (C2v ) is the edge-coalesced icosahedron (see [King91]) given in Figure 8.8. F˜22 (C2v ) is the one-edge truncated dodecahedron mentioned on page 274 in [SaMo97]. This polyhedron has ten pentagonal faces, which is the maximum number of faces among all simple polyhedra with 22 vertices. Both F˜18 (C2v ) and F˜22 (C2v ) and their duals are not 5-gonal. Simple bifaced polyhedra with b > 6 were considered in [Malk70]. Besides Euler’s relation (6 − a)pa = 12 + (b − 6)pb , we have:

• if a = 3, then b ∈ {7, 8, 9, 10}; • if a = 4, b ≡ 0 (mod 8), then pb is even; • if a = 5, b ≡ 0 (mod 10), then pb is even. Reference [Malk70] asserts that above necessary conditions (for existence of such a polyhedron) are sufficient, with only a finite number of exceptions. Polyhedra (3; 3, b; p3 , pb ) with b < 6 are only P rism3 and D¨ urer’s 1 1 octahedron; both are ℓ1 -embeddable into 2 H5 , 2 H8 , respectively. The polyhedra (3; 3, b; p3 , pb ) with b = 6, 8, 10 are not 5-gonal, since they contain (triangles are isolated) isometric not 5-gonal subgraph consisting of a vertex surrounded by a triangle and two even-gons. Examples of ℓ1 -graphs amongst their duals are three omni-capped Platonic solids α3 , β3 , Ico. Consider now non-simple bifaced polyhedra. There are only two classes of non-simple bifaced polyhedra, namely, (4; 3, b; p3, pb ) and (5; 3, b; p3 , pb ).

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For b ≤ 6, all possible (k; a, b) are (4;3,6), (4;3,5), (4;3,4), (5;3,6), (5;3,5), (5;3,4). In each of those cases there is an infinity of such polyhedra. Namely, we have: (5; 3, 4; p3, p4 ) exists for any p4 > 1, [Fisc75]; (4; 3, 4; p3, p4 ) exists for any p4 > 1, [Gr¨ un67], p.282; an infinity of examples is constructed for each of remaining four cases in [Gr¨ un96] (all of them, except tetrakis truncated octahedron (4;3,6;24,8), are not ℓ1 -embeddable). Remind, that if P is a polyhedron (k; a, b; pa , pb ) with n vertices, then Ambo(P ) is a 4-valent polyhedron with pa a-faces, pb b-faces and, in addition, n k-faces. So, Ambo(P ) is bifaced if and only if k ∈ {a, b}. All possible cases for Ambo(P ) and its parameters are as follows: (i) k = a = 3, 4, 5, (ii) Ambo(P ) = (3, 4)2n if P = (3, 4)n , (iii) Ambo(P ) = (4; 3, 5; 20 + 5p5 , 12 + 5p5 ) if P = (5; 3, 5; p3 , p5 ). (In (iii), we use 13.1 giving p3 = 20 + 5p5 .) If we consider only P with b ≤ 6, then we have in addition to (ii) and (iii): (i.1) Ambo(P ) = (3, 4)9 if P = P rism3 = (3; 3, 4; 2, 3), (i.2) Ambo(P ) = (4; 3, 5; 14, 6) if P is the D¨ urer’s octahedron, n (i.3) Ambo(P ) = (4; 3, 6; n + 4, 2 − 2) if P = 3n . Remark that cases (ii), (iii), (i3 ) of the Ambo(P )-construction produce also an infinity of bifaced polyhedra with k = 4, a = 3 and b = 4, 5, 6. The dual of a bifaced polyhedron (k, a, b) has vertices of two valences a and b. We denote the numbers of vertices with these valences by va and vb . Self-dual polyhedron with p = (pa , pb ), 3 ≤ a < b, exists, according to [Juco70], if and only if pi = vi , i = a, b, a = 3 and pa = p3 = n − pb , pb = n−4 b−3 . So, it is α3 , P yrb for pb = 0, 1. For b = 4, the sub-case n ≡ 1 (mod 4) is realized by k-elongated P yr4 having p = (p3 = 4, p4 = 4k + 1); so, all above polyhedra are embedded into a half-cube. But the polyhedra with b = 4 and n = 6, 7, 8 are not 5-gonal (the skeleton is K6 − P6 for n = 6). Also the gyrobifastigum (a regular-faced polyhedron) has p = (p3 , p4 ) = v = (v3 , v4 ) = (4, 4), but it is not self-dual; it and its dual are not 5-gonal. Another nice example of a bifaced polyhedron with two vertex-valences is the following chimera of cube and dodecahedron. It has p = (p4 = 6, p5 = 12) and v = (v3 = 20, v4 = 6); it is the fundamental polyhedron of the smallest known (by the volume) closed hyperbolic space (see [Week85]). Neither it, nor its dual with the skeleton K6 − K3 − K2 , are 5-gonal.

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Bifaced polyhedra

Table 13.1 All k-valent polyhedra with only a-gonal and b-gonal faces, 3 ≤ a < b ≤ 6, pa > 0, pb > 0 (a, b) \ k (5, 6) (4, 6) (3, 6)

3 5n (fullerene) exists iff p6 ≥ 2 4n exists iff p6 ≥ 2 3n exists iff p6 ≥ 4, even

(4, 5) (3, 5)

only four dual deltahedra only D¨ urer’s octahedron

(3, 4)

only P rism3

Table 13.2 ton k 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5

13.2

a,b 3,5 4,5 4,b 4,6 4,6 4,6 5,6 5,6 5,6 5,6 3,b 3,4 3,4 3,4 3,4 3,4 3,4 3,6 3,4 3,5

4 – – p3 = 8 + 2p6 , n = 6 + 3p6 – p3 = 8 + p5 , n = 6 + 2p5 p3 = 8, n = 6 + p4 exists iff p4 ≥ 2

5 – – p3 = 20 + 8p6 , n = 12 + 6p6 – p3 = 20 + 5p5 , n = 12 + 4p5 p3 = 20 + 2p4 , n = 12 + 2p4 exists iff p4 ≥ 2

Known bifaced polyhedra with embeddable skelepa , pb 2,6 4,4 b,2 6,8 6,12 6,12 12,3 12,10 12,12 12,30 2b,2 8,4m 8,3 8,4 8,10 8,10 8,18 24,8 32,6 80,12

polyhedron D¨ urer’s octahedron dual snub disphenoid P rismb (incl. b = 3) tr(β3 ) Cham(γ3 ) twisted Cham(γ3 ) fullerene 526 (D3h ) fullerene 540 (Td ) fullerene 544 (T ) Cham(Do) AP rismb 2-capped (C4 × Pm+1 ) Ambo(P rism3 ) with symmetry D2 6-elong. of above 6-elong. of 2-(C4 × P2 ) rhombicuboctahedron tetrakis tr(β3 ) snub γ3 snub Do

embedd. into 1 H 2 8 1 H 2 8 1 H 2 b+2 H6 H7 H7 1 H 2 12 1 H 2 15 1 H 2 16 1 H 2 22 1 H 2 b+1 1 H 2 2m+4 1 H 2 6 1 H 2 6 1 H 2 8 1 H 2 8 1 H 2 10 1 H 2 12 1 H 2 9 1 H 2 15

ta , tb 0,4 1,2 2,0 0,3 0,4 0,4 -,0 2,2,3 0,4 2,0 2, -,1 -,0 1,1,0,2,3 -,0 -,0

Face-regular bifaced polyhedra

Examples of ℓ1 -polyhedra amongst non-simple bifaced polyhedra are: rhombicuboctahedron=(4;3,4;8,18)→ 21 H10 , tetrakis truncated β3 = (4; 3, 6; 24, 8) → 21 H12 , snub γ3 = (5; 3, 4; 32, 6) → 12 H9 , snub Do = (5; 3, 5; 80, 12) → 12 H15 , and, amongst their duals, both Catalan zonohedra.

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Scale Isometric Polytopal Graphs Table 13.3 Known bifaced polyhedra with embeddable skeleton of the dual k 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4

a, b 3,4 3,5 3,6 3,6 3,6 3,6 3,6 3,7 3,7 3,7 3,8 3,8 3,8 3,9 3,10 5,6 5,6 5,6 3,4 3,4

pa , pb 2,4 2,6 4,4 4,6 4,6 4,12 4,16 6,6 8,12 8,12 8,6 12,12 12,12 16,12 20,12 12,4 12,8 12,20 8,6 8,4m

4 4

3,5 3,5

20,12 16,8

polyhedron P rism3 D¨ urer’s octahedron tr(α3 ) Cham(α3 ) twisted Cham(α3 ) 4- tr(Do) tr( omni-capped α3 ) 6- tr(γ3 ) 8- tr(Do) 8- tr(Do) tr(γ3 ) 12- tr(Do) 12- tr(Do) 16- tr(Do) tr(Do) fullerene 528 (Td ) fullerene 536 (D6h ) 560 = tr(Ico) cuboctahedron 2-(C4 × Pm+1 )= (C4 × Pm+2 )∗ icosidodecahedron 4-cap of tr(β3 )

embedd. into 1 H 2 4 1 H 2 6 1 H 2 7 1 H 2 8 1 H 2 8 1 H 2 10 1 H 2 11 1 H 2 10 1 H 2 14 1 H 2 14 1 H 2 12 1 H 2 18 1 H 2 18 1 H 2 22 1 H 2 26 1 H 2 7 1 H 2 8 1 H 2 10 H4 Hm+3

ta , tb 0,2 0,4 0,3 0,4 0,4 0,5 0,0,4 0,5 0,5 0,4 0,5 0,5 0,5 0,5 3,0 2,0,3 0,0 2, -

H6 H7

0,0 2,3

Duals of above six ℓ1 -polyhedra are not 5-gonal. Only three Archimedean polyhedra are not bifaced: rhombicosidodecahedron → 21 H9 and two large zonohedra. The lists of known ℓ1 -polyhedra amongst bifaced polyhedra and their duals are given in Tables 13.2 and 13.3; they compiled from [DeGr99] and [Deza02]. It turns out that all known bifaced ℓ1 -polyhedra (or having ℓ1 -dual) are a- or b-face-regular. A polyhedron, having only a- or b-gonal faces, is called a-face-regular if each a-gonal face is adjacent to the same number ta of a-gonal faces; it called b-face-regular if each b-gonal face is adjacent to the same number tb of b-gonal faces. The classification of such bifaced polyhedra is addressed in [DeGr01] and [BrDe99]. Remark that the polyhedron 2-capped(C4 × Pm+1 ) is a-face-regular with ta=3 = 2; it is b-face-regular only if m = 1, 2 (with tb=4 = 2, 3 for m = 1, 2, respectively). Remark 13.1 (i) Last column in Tables 13.2, 13.3 give, when relevant, above numbers ta , tb All face-regular (i.e. admitting both, ta and tb ) bifaced

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polyhedra of any degree k (see [Deza02]) appear amongst polyhedra, given on those Tables. (ii) For all ℓ1 -polyhedra of Table 13.2 (except of P rism3 , D¨ urer’s octahedron and 2-capped C4 × Pm+1 ) the dual polyhedra are not 5-gonal. (iii) Many of ℓ1 -polyhedra P from both Tables 13.2 and 13.3, are embedded into 21 H2d , where d is the diameter of P . But, for example, the diameter of dual truncated α3 is two and of dual truncated dodecahedron is four.

13.3

Constructions of bifaced polyhedra

Consider some operations transforming a bifaced polyhedron P into another bifaced polyhedron. Let P have the parameters (k; a, b; pa , pb ). If b = 2k, then the dual of omnicapped P is called in [FoMa95] leapfrog of P . It has parameters (3; a, b; pa, pb + |V (P )|) and k|V (P )| vertices. If b = 2k = 6, then remind, that the chamfering of P (replace all edges by hexagons) has parameters (3; a, b; pa , pb + |E(P )|) and 4|V (P )| vertices. All 5n , 4n , 3n with icosahedral, octahedral, tetrahedral symmetry, respectively, are characterized in [Gold37]. They have n = 20(a2 + ab + b2 ), 8(a2 + ab + b2 ), 4(a2 + ab + b2 ), respectively, for a ≥ b ≥ 0 The cases b = 0 and a = b correspond to the full symmetry Ih , Oh , Td . The leapfrog (and chamfering) of Do, γ3 , α3 correspond to the case a = b = 1 (a = 2, b = 0, respectively); above leapfrogs are truncated Ico, β3 , α3 , respectively. Clearly, each of classes of all 5n , 4n , 3n is closed under operations of leapfrog and chamfering. Let P , P ′ be two Platonic polyhedra having (k, l, n, f ) and (k ′ = k, l′ , n′ , f ′ ), respectively, as size of faces, valence, the number of vertices and the number of faces. Consider the convex polyhedron P + f P ′ obtained by adjoining a copy of P ′ on each face of P . Then (P + f P ′ )∗ is a bifaced polyhedron with parameters (k; l′ , l(l′ − 1); f (n′ − k), n). The case P ′ = α3 corresponds to omnicapping of simplicial P . While, clearly, α3 + iα3 , β3 + iα3 are ℓ1 -embeddable for 0 ≤ i ≤ f , α3 + iβ3 for 2 ≤ i ≤ 4 and β3 + iβ3 for 1 ≤ i ≤ 8 are not ℓ1 . Given a polyhedron P , let P + P yrq (P + P rismq , P + AP rismq ) be the polyhedron defined by the join of new vertex to all vertices of a q-gonal face (the join of P rismq , AP rismq to a q-gonal face, respectively). See Corollary 4.1 in chapter 4 about the cases when P yrm + P rismm is embeddable. Remark that AP rismn → 21 Hn+1 but AP rismn + AP rismn

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is not 5-gonal. The golden dodecahedron of [HiPe89], i.e. Do with AP rism5 on each of its 12 faces, is not 5-gonal. The embedding of the elongation P + P rismm of P along an isometric m-gonal face, implies the following embedding. Let P be a polyhedron with a vertex v such that P − v → 12 Hm . Then P , truncated on the vertex v, is embedded into 12 Hm+2 . For example, this truncation of P on two opposite caps is embedded into: 21 H10 if P is Ico (i.e. 2-capped AP rism5 ); 12 H9 if P is 2-capped AP rism4 (which is an extreme hypermetric); H4 if P is β3 = AP rism3 ; 21 H11 if P is 5∗24 (i.e. if P is 2-capped AP rism6 , which is not 5-gonal). Amongst 92 regular-faced polyhedra, 19 are elongated P (i.e. obtained from P by adjoining or inserting a prism), 27 are i-augmented P i.e. icappings of P . For example, pentakis of AP rism5 and of Do are embeddable, pentakis of P yr5 and of P rism5 are not 5-gonal, tetrakis of P rism3 and of AP rism4 are hypermetrics non-ℓ1 .

13.4

Polyhedra 3n and 4n

Recall, that we denote an n-vertex simple bifaced polyhedron with b = 6 12 , i.e. p3 = 4, p4 = 6, by an , a = 3, 4, 5. The equation 13.1 gives pa = 6−a p5 = 12. The embedding for 3n , 4n and their dual is presented below. Proposition 13.2 (i) Besides of the cube, all ℓ1 -4n are: the P rism6 , the truncated octahedron, the chamfered cube and the twisted chamfered cube. (ii) The octahedron is the unique ℓ1 -embeddable dual 4n . (iii) The tetrahedron is the unique ℓ1 -embeddable 3n . (iv) Besides of the tetrahedron, all known ℓ1 -3∗n are: the triakis tetrahedron 3∗12 → 12 H7 , the duals of the chamfered tetrahedron and the twisted one, that is, both 3∗16 → 12 H8 , the 4-capped on disjoint facets Ico 3∗28 → 21 H10 and the dual of the leapfrog of the triakis tetrahedron 3∗36 (Td ) → 12 H11 . Proof. To prove that above polyhedra are not ℓ1 -embeddable, we simply exhibit (see Figure 13.3) not 5-gonal configurations contained in their skeletons. The coefficients bi of the violated 5-gonal inequality are again, respectively, 0 for a black vertex, −1 for a square one, and 1 for a white circle.

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Theorem 2 of [GrMo63] gives, that 3n exists for any n ≡ 0 (mod 4), except of n = 8, and provides complete description of their skeletons. Since 34 = α3 → 12 H3 , 12 H4 , then it is not ℓ1 -rigid. The ”would-be” 38 is 2-, but not 3-connected and it is also not 5-gonal. There are exactly N3 (n) polyhedra 3n for 1 ≤ n4 ≤ 30, where N3 (n) is given below: n 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 N3 (n) 1 0 1 2 1 2 2 4 3 3 2 7 3 4 5 n 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 4 N3 (n) 8 3 7 4 9 7 6 4 14 6 7 8 12 5 13 Theorem 3.4 in [Sah94] implies, that N3 (n) = O(n). Clearly, α3 is unique 3n with abutting triangles. So, any 3n , n > 4, is not 5-gonal, since it contains an not 5-gonal isometric subgraph, given on the left-hand side of Figure 13.3. We know six ℓ1 -polyhedra 3∗n ; a computation in [Duto02] showed that no others exist for n ≤ 136. Except of α3 and the unique 312 = truncated α3 , any 3n is a 4-vertex P truncation of a simple polyhedron with i≥3 pi = p4 + p5 + p6 (so, p5 = P 12 − 2p4 from the Euler equality i≥3 (6 − i)pi = 12) and n − 8 vertices (so, n = 28 + 2p6 − 2p4 ). In particular, 3n is 4-vertex truncation of a fullerene 5n−8 if and only if this 5n−8 has four vertices of type (5,5,5) at pairwise distance at least three; so, any pair of triangles is separated by more than one hexagon. Such 5n−8 has either four isolated triples of pentagons, or two isolated clusters of six pentagons. 3∗n → 21 Hm if and only if 5∗n−8 → 12 Hm−4 . For small values of n we have (see Figure 13.4; marked vertices indicate truncation): (1) exactly seven of 3n come as 4-vertex truncation of dual deltahedra: unique 312 from α3 , both 316 (chamfered and twisted chamfered α3 ) from γ3 , the unique 320 from dual snub disphenoid, both 324 from dual 2-AP rism4 and one (of two) 328 from Do. The second one 328 comes from a simple polyhedron with 12 faces, such that p = (p4 , p5 , p6 ) = (4, 4, 4). (2) exactly six 3n come as 4-vertex truncation of fullerenes 5m , 20 ≤ m ≤ 36: one 328 from 520 , a 332 from 524 , a 336 from 528 (D2 ), another 336 from 528 (Td ), a 340 from 532 (D2 ), a 344 from 536 (D2 ); each of these six fullerenes has a unique, up to a symmetry, set of four vertices at pairwise distance ≥ 3.

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Theorem 1 of [GrMo63] gives that 4n exists for any even n ≥ 8 except n = 10. Clearly, 4n is bipartite, and there is an infinity of centrally symmetric 4n . Hence it either → Hm or is not 5-gonal. For 4 ≤ n2 ≤ 30, there are N4 (n) polyhedra 4n , where N4 (n) is given below: n 2

4 5 6 7 8 9 10 11 12 13 14 15 16 17 N4 (n) 1 0 1 1 1 1 3 1 3 3 3 2 8 3 n 18 19 20 21 22 23 24 25 26 27 28 29 30 2 N4 (n) 7 7 7 5 14 6 12 12 13 10 23 12 19 Theorem 3.4 in [Sah94] implies that N4 (n) = O(n3 ). The unique ℓ1 polyhedron among 4∗n is 4∗8 = β3 → 21 H4 , since other 4∗n contain the following not 5-gonal isometric 7-vertex subgraph: C{1,...,6} with chords (1,3), (4,6) plus a new vertex connected with vertices 1,3,4,6. Clearly 48 = γ3 , 412 = P rism6 , truncated β3 is 424 (one of three), two 432 are Cham(γ3 ) and twisted Cham(γ3 ) (see Figure 13.5). Those five polyhedra are the only ℓ1 -4n (see [DDS05]). They are embedded into H3 , H4 , H6 , H7 , H7 , respectively. The first three are Voronoi polyhedra, Cham(γ3 ) is a non space-filling zonohedron, twisted one is not centrally symmetric (it is only one amongst above five, which, in terms of next chapter, is not an equicut graph). Chamt (γ3 ) is ℓ1 only for t = 1. For n ≡ 2 (mod 6), the dual of the column of n−2 6 octahedra β3 gives a 4n ; for n > 8, it and its dual are not 5-gonal. Examples of 4n without abutting pairs of 4-gons are Chamt (γ3 ) ( t ≥ 1) and duals of: tetrakis cube, cuboctahedron, twisted cuboctahedron (i.e. the triangular ortobicupola Nr.47), gyro-elongated triangular bicupola and snub cube, having, respectively 8a2 (a ≥ 2), 24, 32, 32, 44, 56 vertices. Last two are not 5-gonal. ”Dual tetrakis” above, means a truncation on six 4-valent vertices of the dual polyhedra. On the other hand, any 4n with each pair of 4-gons separated by at least three edges (for example, Chamt (γ3 ), t ≥ 2) comes as 6-edge truncation (put 4-gons instead of edges) of a fullerene 5n−12 . Hence such a 4n has no abutting pair of 4-gons and, moreover, the corresponding 5n−12 has six pairs of adjacent pentagons. Many 4n come as 6-edge truncation of 5n under weaker conditions: Cham(γ3 ) from 520 , tr(β3 ) from α3 . Also truncated β3 is 6-(disjoint) edges truncation of 412 = P rism6 , P rism6 is 2-(disjoint) edges truncation of 48 = γ3 , γ3 is 2-(disjoint) edges truncation of α3 . Suitable 6-(disjoint) edges truncation of 524 (dual 2-AP rism6 ) gives a 436 and so, on (see Figure 13.5).

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Bifaced polyhedra

13.5

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Polyhedra 5n (fullerenes) revisited

Theorem 1 of [GrMo63] gives that 5n exists for any even n ≥ 20 except n = 22. The polyhedra 5n , i.e. the bifaced polyhedra (3; 5, 6; 12, p6), are called fullerenes in Chemistry; see chapter 2. In fact, all 5n are the cases f = 2 + n2 ≥ 12 of medial polyhedra (cf. chapter 13.1). The number N5 (n) of polyhedra 5n is O(n9 ) from Theorem 3.4 in [Sah94] (based on a result from [Thur98]). All known ℓ1 -embeddable 5n are Do = 520 → 21 H10 , 526 → 12 H12 , 540 (Td ) → 21 H15 , 544 (T ) → 21 H16 and Cham(Do) = 580 (Ih ) → 12 H22 . The last one is the dual pentakis icosidodecahedron; its twisted version is not ℓ1 -embeddable. All known ℓ1 -5∗n are Ico = 5∗20 → 12 H6 , hexakis truncated α3 = 528 (Td ) → 21 H7 , hexakis AP rism26 = 5∗36 (D6h ) → 12 H8 and pentakis Do = 5∗60 (Ih ) → 12 H10 . In fact, no other ℓ1 - embeddable 5n , ℓ1 -5∗n exist for n < 60 and they are not expected for other n. All seven above fullerenes, such that it or its dual is ℓ1 -embeddable, have the highest symmetry for its number of vertices and all, except the dual of 528 (Td ) of diameter 3, have the corresponding embedding into 21 H2m , where m is the diameter of the skeleton. The 528 (Td ) is also unique not antipodal amongst them. Some interesting classes of 5n were mentioned above in chapter 2: 510(t+1) = (2-AP rismt5 )∗ , 512(t+1) = (2-AP rismt6 )∗ , t ≥ 1, and those (with four isolated triples of pentagons) coming from collapsing of four triangles in some 3n . 13.6

Polyhedra ocn (octahedrites)

Those are 4-valent bifaced polyhedra on n vertices with parameters (4; 3, 4; p3 , p4 ). So, by 13.1, p3 = 8 and n = 6 + p4 . Besides of 3n , 4n , 5n , it is the only case of bifaced polyhedra, for which pa is fixed for given (k, b). [Gr¨ un67], page 282, gives the existence of ocn for any n ≥ 6, except n = 7. Remark that dual octahedrites oc∗n are exactly the case d = 3 of almost simple cubical d-polytopes in terms of [BlBl98]. Actually, [BlBl98] characterizes those d-polytopes for d ≥ 4. It is not difficult to enumerate all almost simple cubical 2-dimensional complexes; they are P2 × Pn , for any finite n and P2 × Z+ , P2 × Z, i.e. infinite in one or both directions

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(respectively) paths of squares and the cube with deleted vertex or edge. But the variety of such d-polytopes becomes too rich for the exceptional case d = 3. The numbers Noc (n) of polyhedra ocn for 6 ≤ n ≤ 30 are given below: n 6 7 8 9 10 11 12 13 14 15 16 17 18 Noc (n) 1 0 1 1 2 1 5 2 8 5 12 8 25 n 19 20 21 22 23 24 25 26 27 28 29 30 Noc (n) 13 30 23 51 33 76 51 109 78 144 106 218 Clearly, Ambo(ocn ) is a oc2n . Another operation, namely, inserting into some ocn , a ring of m 4-gons along a simple straight-ahead circuit (defined in chapter 1), produces a ocn+m ; let us call it m-elongation, see chapter 1. For example, 2-P rismt4 =BP yr4t+1 = oc4t+6 → 21 H2t+4 is, m times iterated, 4-elongation of 2-P rism4 . Iterated 3-elongations of β3 , 4elongations of AP rism4 and of 2-P rism4 give ocn for n = 6 + 3m, 8 + 4m, 10 + 4m with any natural m. First examples of octahedrites ocn are: β3 = oc6 → 21 H4 , AP rism4 = oc8 → 12 H5 , Ambo(P rism3 )=3-elongated β3 = oc9 → 21 H6 , 2-P rism4 = oc10 → 21 H6 ; the second oc10 is also embedded into 12 H6 . All but one of the polyhedra on Figure 13.6 are not 5-gonal: the oc11 , four polyhedra oc12 , (cuboctahedron=Ambo(β3 )=4-elongated AP rism4 , twisted cuboctahedron = Nr.47, another two times 3-elongation of β3 and another oc12 ) and amboAP rism4 = oc16 . Now, rhombicuboctahedron oc24 → 21 H10 but its twisted version (”14th Archimedean solid”) is not 5-gonal. Both these polyhedra are 8-elongations of the twisted Ambo(AP rism4 ), which is a oc16 → 21 H8 , and of Ambo(AP rism4 ), which is not 5-gonal (see both smaller polyhedra on Figure 6.3). Above oc16 is also 6-elongation of 2P rism4 ; 6-elongation of the second oc10 is another oc16 , which is embedded into 21 H8 also. Amongst the duals of above ocn , all embeddable are zonohedra: (cuboctahedron)∗ → H4 and (2 − P rismt4 )∗=P rismt+1 → Ht+3 , for any 4 integer t ≥ 0. Apropos, (Ambo(P rism3 )∗ = (oc9 )∗ is the smallest convex polyhedron with odd number of faces, all of which are quadrilaterals; it is not 5-gonal.

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Bifaced polyhedra

Fig. 13.1

Some Goldberg’s medial polyhedra

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Fig. 13.2

Quasi-medial polyhedra F˜18 (C2v ) and F˜22 (C2v )

Fig. 13.3

Not 5-gonal configurations in 3n and 4∗n

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Bifaced polyhedra

a) would-be 38

b) both 316

c) 328 (T )

d) 328 (D2d )

5

5

5 5 5 5

5

5 5

5 5

5

5 5

5

5

5

5 5

5

5

5

5

5

5

5

5

5

f) 4-truncation of 532 (D2 )

5 5

5 5

e) 4-truncations of both 528

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5 5

5

5 5

g) 4-truncations of 540 Fig. 13.4

5

5 5

5

Some graphs 3n

5

5 5

5

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a) γ3

b) P rism6

c) truncated β3

d) a 436 (as edge-truncation)

e) unique 414

f) unique 416

g) unique 418

h) a 420

2

7 3 5

3

5

1

1

5 4 2 6

7

6

6 3

7

7

5 3

2

7

6

1

3

4

5 3

3

1

7

1

6

2 4

6

4

2

7 4

2 1

4

5 5 6

2

4

i) twisted Cham(γ3 ) Fig. 13.5

j) Cham(γ3 ) Some graphs 4n

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Bifaced polyhedra

a) cuboctahedron

b) twisted cuboctahedron

c) two other oc12

d) Ambo(AP rism4 )

e) twisted Ambo(AP rism4 ) Fig. 13.6

f) the oc11 Some ocn

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Chapter 14

Special ℓ1-graphs

14.1

Equicut ℓ1 -graphs

In the first half of this chapter we follow [DePa01], where proofs of results below can be found. A graph G is an equicut graph if it admit an l1 -embedding, such that the equality holds in the left-hand side of the inequality (1.2) of chapter 1, concerning the size s(dG ) of this embedding. Below s(dG ) means the size of such equicut embedding. This means that, for such a graph, every S in the equality (1.1) of chapter 1 corresponds to an equicut δ(S), i.e. satisfy aS 6= 0 if and only if S partitions V into parts of size ⌈ n2 ⌉ and ⌊ n2 ⌋, where n = |V |. Remind that a connected graph is called 2-connected (or 2-vertexconnected) if it remains connected after deletion of any vertex. Lemma 14.1 An equicut graph with at least four vertices is 2-connected. This lemma implies the following Corollary 14.1 For any equicut graph G with n ≥ 4 vertices, we have 2−

n 1 ≤ s(dG ) ≤ ⌈ n2 ⌉ 2

with equality on the left-hand side if and only if G = Kn , and on the righthand side if and only if G = Cn . The condition n ≥ 4 is necessary in the statements above. Indeed, s(dP3 ) = 2 > s(dC3 ) = 32 . Note, that P2 and P3 are the only equicut trees. Also, W (C5 ) = 15 < W (P{123452} ) and s(dC5 ) = 25 < s(dP{123452} ), where 137

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P{123452} denotes the circuit on 2,3,4,5 with an extra edge attached to the vertex 2. Remark 14.1 G is an equicut graph if there is a realization with the binary matrix F with the column sums ⌈ n2 ⌉ or ⌊ n2 ⌋. If, instead of this condition, we asked that any row of F has exactly k 1’s, then we obtain other special ℓ1 -graph. Namely, one which is embedded isometrically, up to scale λ, into the Johnson graph J(m, k). It was observed by Shpectorov, that such graphs can be recognized in polynomial time using the algorithm in [DeSh96], but we are not aware of any similar characterization of the equicut graphs. It is easy to see, that any graph G, which is embedded isometrically, up to scale λ, into hypercube Hm , is embedded also isometrically, up to scale λ, into Johnson graph J(2m, m). In fact, let any vertex v of G be addressed by corresponding subset Av of a given m-set; then one can address v by the union of Av and the image of its complement in a bijection of the given m-set on some m-set, which is disjoint with given one. The columns “J(m, k)?” of Tables 4.1, 4.2 give all embeddings of Q into J(m, k), where Q is anyone amongst Platonic polyhedra, semi-regular polyhedra or their duals. Call an equicut ℓ1 -graph an antipodal doubling if its realization in Hm A O (i.e. the above (0,1)-matrix F ) has the form F = , where A J − A J′ is n2 × m′ (0,1)-matrix, J, J ′ are matrices consisting of 1’s only and O is n ′ 2 × (m − m ) matrix consisting of 0’s. If, moreover, the matrix A is a realization, with the same scale λ, of a graph G′ , then it is straightforward to check, that J ′ has λ(d(G′ ) + 1) − n columns, where d(G) is diameter of G. Note that Double Odd graph DO2s+1 (see, for example, DO5 on Figure 14.1) with s ≥ 3 is an example of antipodal doubling with the matrix A not corresponding to the realization of a graph G′ , for any decomposition of F into the above form. Remark 14.2 An antipodal doubling is exactly an ℓ1 -graph, that admits an antipodal isomorphism, i.e. it has a central symmetry (for any vertex, there is exactly one other on the distance equal to the diameter) and the mapping of all vertices into their antipodes is an isomorphism. Antipodal extensions of arbitrary ℓ1 -metrics was considered in [DeLa97], chapter 7.2. In order to investigate, when one can construct an ℓ1 -graph from an ℓ1 -graph via the antipodal doubling (see Theorem 14.1 below), let us introduce the following definition. For a graph G = (V, E), define its diametral

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Special ℓ1 -graphs

12 s

124 s

125s

s14 123 s 15

25 s

s24

s

s 145

245s

s 234

23 s 135

s

s

45 s 34

s 235 s 35

Fig. 14.1

134 s

13 s

s 345

Double Odd graph DO2s+1 with s = 2

doubling as the graph 2G with the vertex-set V + ∪ V − (where V + and V − are two copies of V ) and the adjacency is as follows: ua is adjacent to v b if a = b and (u, v) ∈ E, or if a 6= b and dG (u, v) = d(G), where a, b ∈ {+, −}. Lemma 14.2 The subgraphs of 2G, induced on V + and V − , are isometric to G if and only if dG (u, v) ≤ 2 + dG (w1 , w2 )

for any u, v, w1 , w2 ∈ V,

satisfying dG (u, w1 ) = dG (v, w2 ) = d(G).

Lemma 14.3

(14.1)

Let G satisfy the condition of above lemma. Then d2G (u+ , v − ) = d(G) + 1 − dG (u, v)

(14.2)

if and only if any geodesic in G lies on a geodesic of length d(G). Certain properties of G are inherited by 2G. Lemma 14.4 Let G be a graph satisfying equations (14.1) and (14.2). Then 2G satisfies d(2G) = d(G) + 1, equations (14.1) and (14.2).

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If d(G) = 2, then G satisfies (14.1). Moreover, G 6= Kn satisfies (14.2), unless it has an edge (u, v) with Gu = Gv , where Gu is the subgraph of G induced by the neighborhood of the vertex u. In particular, the strongly regular graphs, considered below, satisfy (14.1) and (14.2); note, that ℓ1 graphs form a rather small sub-family of strongly regular graphs. Not always the graph 2G defines uniquely the graph G, from which it was constructed. It can be that 2G′ = 2G for G 6= G′ . See below for many examples of this situation. But the graphs G and G′ are related by the following graph operation. The diametral switching of a graph G with respect to S ⊂ V is a graph G′ that is obtained from G by retaining the edges that lie within (S × S) ∪ ((V − S) × (V − S)) and replacing the set of edges from S ×(V −S) with the set {(u, v) ∈ S ×(V −S) : dG (u, v) = d(G)}. Note that Seidel switching is an operation, that coincides with the diametral switching for graphs of diameter two. Theorem 14.1 Let G be an ℓ1 -graph. Then 2G is an ℓ1 -graph if G satisfies (14.1), (14.2) and s(dG ) ≤ d(G) + 1.

(14.3)

Moreover, if 2G is an ℓ1 -graph, then it holds: (i) d(2G) = d(G) + 1, 2G satisfies equations (14.1), (14.2) and (14.3) with equality, A O (ii) all ℓ1 -realizations of 2G are equicut of the form with J − A J′ λ(d(G) + 1) − m columns, up to permutations of rows and columns, and taking complements of columns, where A is an ℓ1 -realization of G with scale λ. Remark 14.3 K4 − P3 , K4 − P2 , the Dynkin diagram E6 are examples of ℓ1 -graphs satisfying (14.1), (14.3), but not (14.2). Affine Dynkin diagram ˜6 is an example of a graph that does not satisfy (14.1) and (14.3), but E does satisfy (14.2). Note that any graph G with diameter D(G) = 2 satisfies (14.1) and (14.2). Certainly, not all of them are ℓ1 -graphs, for instance, K2,3 . Also not all ℓ1 -graphs of diameter two satisfy (14.3), for instance, K1,4 . In general, for any ℓ1 -graph G = (V, E) with |V | ≥ 4 one has D(G) ≤ s(dG ) ≤ D(G) + |V | − 3.

(14.4)

The equality at the right-hand side of (14.4) holds if and only if G is a star, as can be seen by applying Theorem 1.1 of chapter 1.

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Theorem 14.1 generalizes the situation for the cocktail-party graph Kn×2 , considered in [DeLa97], chapter 7.4, to arbitrary ℓ1 -graphs. It implies, that the minimal scale of an ℓ1 -embedding of 2G equals the minimal λ, such that the metric λdG is embedded isometrically into Hm with m = λ(D(G) + 1). In particular case of G = K4a , Lemma 7.4.6 of [DeLa97] gives for such a minimal λ, the inequality λ ≥ 2a, with the equality if and only if there exists a Hadamard matrix of order 4a. Now we list many examples in no particular order. An equicut ℓ1 graph is called a doubling if it is an antipodal doubling. Moreover, if it is obtained from a graph G0 as in Theorem 14.1, we give such a representation G = 2G0 . All equicut graphs with at most six vertices are: Cn (2 ≤ n ≤ 6), Kn , (n = 4, 5, 6), P3 , 4-wheel and the octahedron K3×2 . Amongst these eleven graphs only Kn is not ℓ1 -rigid and only C4 , C6 and K3×2 are doublings. The scale of the direct product G × G′ of two l1 -graphs is the least common multiple of the scales of G and G′ , and the size will be the sum of their sizes. Moreover, G × G′ is ℓ1 -rigid if and only if G and G′ are. Lemma 14.5 If l1 -graphs G, G′ have even number of vertices each, then G × G′ is an equicut graph if and only if they are. A Doob graph (the direct product of a number of copies of Shrikhande graph and a number of copies of K4 (see, for example, [BCN89], page 27) is an example of an equicut graph obtained via Lemma 14.5. It is a (non ℓ1 -rigid and non-doubling) ℓ1 -graph of scale two. Embeddable distance-regular graphs. Here we again freely use notation from [BCN89]; also, a significant use is made of [KoSp94]. A graph of diameter d is called distance-transitive graph, if it is connected and admits a group of automorphisms, which is transitive, for any 1 ≤ i ≤ d, on the set of all pairs of its vertices, being on the distance i. More general, a connected graph is called distance-regular graph, if it is regular and, given two vertices x and y, the number of vertices at distance i from x and at distance j from y depends only on the distance d(x,y). Any distance-regular graph of diameter two is called strongly regular graph. All hypermetric polytopal strongly regular graphs are: • (n × n)-grids Kn × Kn ,

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• • • •

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triangular graphs T (n) = J(n, 2), the skeleton Kn×2 of βn , G(121 ) = 21 H5 and G(221 )=the Schläfli graph;

only the last one, the Schläfli graph, is not ℓ1 -embeddable. Reference [Kool90] proved that all finite distance-regular graphs, embeddable with scale one into an Hm are the distance-transitive ones: • γm , • C2m and • for odd m, Double Odd graphs DOm . In [KoSp94] all ℓ1 -embeddable distance-regular finite graphs are found. Moreover, the Petersen graph and the Shrikhande graph are both equicut graphs of scale two and size 3; both are ℓ1 -rigid and are not doublings. The Double Odd graph DO2s+1 is an equicut graph of scale one and size 2s + 1. The halved cube 12 Hm is an equicut graph of scale two and size m. It is not ℓ1 -rigid only for m = 3, 4; it is a doubling if and only if m is even. The Johnson graph J(2s, s) is a non ℓ1 -rigid doubling of scale two and size s. Further, the following graphs are distance-regular equicut graphs. (1) Any Taylor ℓ1 -graph: 12 H6 , J(6, 3), C6 , H3 , icosahedron. They are all doublings of diameter three and size 3; they can be constructed using Theorem 14.1 above. (2) Any strongly regular ℓ1 -graph, except J(s, 2) (s ≥ 5) and any (s × s)grid H(2, s) with s odd. That is, C5 , the Petersen graph, 12 H5 , the Shrikhande graph, Hm with m even, Ks×2 . (3) Amongst distance-regular graphs G with diameter greater than two and µ > 1: 12 Hm with m > 5, H(m, d), J(s, t) with t > 2, icosahedron and Doob graphs. (4) All Coxeter ℓ1 -graphs except J(s, t) with t < 2s : J(2s, s), icosahedron, dodecahedron, Ks×2 , 12 Hs , Hs , Cs (s ≥ 5). (5) All cubic distance-regular ℓ1 -graphs: K4 , Petersen graph, H3 , DO5 , dodecahedron. Also all amply-regular ℓ1 -graphs with µ > 1 are equicut graphs. Yet another example is given by the 12-vertex co-edge regular subgraph of the Clebsh graph 12 H5 , (see [BCN89], chapter 3.11, page 104). It is an equicut

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graph of size 52 , scale two, non-doubling. Reference [Macp82] showed that any infinite distance-transitive graph G of finite degree arises in the following way: the vertices of G are p-valent vertices of T , two of them being adjacent if they lie at distance two in T , where T is an infinite tree, in which the vertices of the bipartite blocks have degrees p, q, respectively. (In other words, G is the halved subgraph of the infinite distance-biregular tree T .) The graph induced by all neighbors of fixed vertex of G, is the disjoint union of p complete graphs Kq−1 . It is easy to see that, in general, G → 21 Z∞ . In the special case q = 2, it is just the infinite p-regular tree and it is embedded into Z∞ , except the case p = q = 2, when it is the infinite path PZ → Z1 . Some equicut graphs, which are doublings of ℓ1 -graphs (see some in chapter 7.2 of [DeLa97]). (1) (2) (3) (4) (5) (6) (7) (8)

C2s = 2Ps . Ks×2 = 2Ks . Hs = 2Hs−1 . J(2s, s) = 2J(2s − 1, s). 1 1 2 H2s = 2 2 H2s−1 . P rism2s = 2C2s AP rism2s+1 = 2C2s+1 . Do is the doubling of C{1,2,...,9} with an extra vertex connected to the vertices 3, 6 and 9 of the circle. (9) Ico is the doubling of a 5-wheel; as well, it is the doubling of the graph obtained from hexagon C{1,2,...,6} by adding edges (2,4), (2,6) and (4,6). (10) J(6, 3) is the doubling of the Petersen graph, in addition to the 4th item above for s = 3. (11) 21 H6 is the doubling of the Shrikhande graph and of (4 × 4)-grid H2×4 (more precisely, of its realization in 12 H6 ), in addition to the 5th item above for s = 3. In the items 9, 10, 11, we have (diametral) switching-equivalent graphs G, such that 2G is a Taylor ℓ1 -graph; see the definition preceding Theorem 14.1. This situation, in general, is well-known. For instance, the Gosset graph (it is a Taylor graph, which is not an ℓ1 -graph) can be obtained as the diametral doubling of one of five non-isomorphic, but switching-equivalent, graphs. For definitions and discussion of this situation in more general setting see, for example, [BCN89], pages 103–105.

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Equicut polytopes. The skeletons of many ”nice” polytopes are equicut graphs. Below we list several such examples. All five Platonic solids have equicut skeletons; all, except the tetrahedron α3 , are ℓ1 -rigid. All except the cube γ3 (of scale one) have scale two. The sizes for α3 , β3 , γ3 , Ico and Do are 32 , 2, 3, 3 and 5, respectively. The skeleton of any zonotope is a doubling, and it has scale one; so, it is ℓ1 -rigid. Amongst all ℓ1 -embeddable semi-regular polyhedra, we have: (1) all zonohedra (i.e. 3-dimensional zonotopes) are as follows: the truncated octahedron, the truncated cuboctahedron, the truncated icosidodecahedron and P rism2s (s > 2) with sizes 6, 9, 15 and s + 1, respectively; (2) all other doublings are as follows: the rhombicuboctahedron, the rhombicosidodecahedron and AP rism2s+1 (s > 1) with scale two and sizes 5, 8 and s + 1, respectively; (3) all remaining equicut polytopes are: the snub cube, the snub dodecahedron and AP rism2s (s > 1); all have scale two and sizes 29 , 15 2 and s + 12 , respectively; (4) the remaining P rism2s+1 (s > 1) has scale two and size s + 32 ; it is not an equicut graph. Amongst all Catalan (dual Archimedean) ℓ1 -embeddable polyhedra we have: (1) all zonohedra are as follows: dual cuboctahedron and dual icosidodecahedron of sizes 4 and 6, respectively; (2) the only other doubling is dual truncated Ico of scale two and size 5; (3) all remaining cases (duals of tr(γ3 ), tr(Do), tr(α3 ) and P rism3 ) are non-equicut ℓ1 -graphs of scale two and sizes 6, 13, 72 and 2, respectively. All Platonic, semi-regular and dual to semi-regular ℓ1 -polyhedra are ℓ1 rigid, except the tetrahedron and the dual P rism3 . All not equicut graphs amongst them (see the columns “cuts” in Tables 4.1, 4.2) are: P rismn for odd n and duals of four semi-regular polyhedra (P rism3 and truncated ones of tetrahedron, cube, dodecahedron). Examples of regular-faced polyhedra (from the list of 92 polyhedra) with equicut skeletons (all have scale two) are: Nr.75 (biaugmented P rism6 ) of

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size 4 (a doubling) and two non-doublings: Nr.74 (augmented P rism6 ) of size 4 and Nr.83 (tridiminished Ico) of size 3. The regular ℓ1 -embeddable polytopes of dimension greater than three, have equicut skeletons. They are as follows: αn , γn and βn . There are just three semi-regular ℓ1 -embeddable polytopes of dimension greater than three (see [DeSh96]). Two of them have equicut skeletons: 12 H5 and the snub 24-cell. The latter is a 4-dimensional semi-regular polytope with 96 vertices. The regular 4-polytope 600-cell can be obtained by capping its 24 icosahedral facets. Its skeleton has scale two and size six; it is a doubling. Three of the chamfered Platonic solids have ℓ1 -skeletons: Cham(γ3 ) is a zonohedron of size 7, Cham(Do) has an equicut (non-doubling) skeleton of scale two and size 11, Cham(α3 ) has non-equicut ℓ1 -skeleton of scale two and size 4.

14.2

Scale one embedding

Remind first (see chapter 1) that a finite ℓ1 -graph is embeddable with scale one if and only if it is bipartite and 5-gonal. Small bipartite polyhedral graphs Amongst six bipartite polyhedral graphs with at most nine faces, there are two embeddable ones, the cube 48 and the 412 = P rism6 ; both are zonohedra, embeddable in H3 and H4 , respectively. Other four are the oc∗8 , oc∗9 , 414 and the not 5-gonal 12-vertex graph on Figure 14.2 a). Amongst five bipartite polyhedral graphs with ten faces and at most 13 vertices, there are two embeddable ones, a oc∗10 and RhDo-v3 (see Figure 14.2 c) and d); they are also Nr.13 (for t=1) and 2 in Table 14.1. Other three are the second oc∗10 , the AP rism∗5 and the not 5-gonal 13-vertex graph on Figure 14.2 b). Both above embeddable graphs are embedded into H4 , but only the graph on Figure 14.2 d) is not centrally symmetric. Remind that we allocate to polyhedra, presented only by their combinatorial type, the maximal possible symmetry. For example, the graph on Figure 14.2 c) and, in general, any polyhedron C4 × Pt+2 , t ≥ 1 from Table 14.1 is seen as a centrally-symmetric one, but it is not zonotope, since it should have some trapezoidal faces in order to be realisable as a convex polyhedron. A non-convex realization of C4 × P3 (as two adjacent cubes) tiles the 3-space non-normally; see the partition Nr.35 in Table 10.3) and the graph

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of this tiling is embedded into Z3 . Nr.15–16 of Table 14.1 are Voronoi t-polytopes for the lattices Zt and A∗t ; the polyhedra Nr.1, 3 and 4 are Voronoi polyhedra of the lattices A2 × Z1 , A3 and L5 (see chapter 11). Recall that Nr.2 of Table 14.1 (RhDo-v3 , i.e. RhDo with a deleted simple vertex) is the tile of three Voronoi partitions from Tables 10.1, 10.2 and 10.3: Nr.25, which is embeddable into Z4 , and not 5-gonal ones Nr.26 and 33. Apropos, two 13-vertex graphs on Figure 14.2 b) and d), are exactly two smallest non-Hamiltonian bipartite polyhedral graphs, which can be realized as the Delaunay tesselation of their vertices (see Figure 10 in [Dill96]).

a

c Fig. 14.2

b

d

Some small bipartite polyhedral graphs

Zonotopes Zonotopes are affine projections (called shadows in [Coxe73]) of hypercubes γm . (Apropos, any convex polytope is affine projection of a simplex αm and any centrally symmetric convex polytope is affine projection of a cross-polytope βm .) Remind that a dicing can be seen as a lattice, the Voronoi polytope of which is a zonotope (see [Erda98]). The following result was proved in [BEZ90].

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Proposition 14.1 The skeleton of any zonotope of diameter m is isometrically embeddable into the skeleton Hm of γm , whose affine projection it is. Remark 14.4

Examples of zonotopes are:

(1) five of Archimedean and their dual (Catalan) polyhedra are zonohedra: Nr.5, 9, 11 and 3, 6 of Table 14.1; (2) All five (combinatorially different) Voronoi polyhedra are zonohedra: cube γ3 = H3 , rhombic dodecahedron RhDo → H4 , P rism6 → H4 , elongated dodecahedron ElDo → H5 , truncated β3 → H6 ; (3) All five golden isozonohedra of Coxeter (zonohedra with all faces being rhombic with the diagonals in golden proportion) are: 2 types of hexahedra (equivalent to γ3 ), rhombic dodecahedron, rhombic icosahedron P z(5) and triacontahedron (i.e. dual icosidodecahedron). (4) Curious family of eight 3-zonotopes, embeddabble into Hm , 3 ≤ m ≤ 10, and having m(m − 1) faces, is given in [ScYo00]. (5) Besides P rism2m and γm , some examples of infinite families of zonotopes (Nr.12–16 in Table 14.1) are polar zonohedra P z(m) → Hm (γ3 , RhDo, rhombic icosahedron for m = 3, 4, 5, respectively, see [Coxe73]) and m-dimensional permutahedra (for m = 2 and 3, it is C6 and truncated β3 , respectively). The tope graphs of oriented matroids An oriented matroid on E is a pair M = (E, F), where E is a finite (melement) set and F is a set sign vectors on E (called faces), which satisfy some special axioms (see, for example, [Fuku95]). A natural order on signed vectors (on E) defines the rank of a face f as the length of any maximal chain from 0 to f . The maximal faces are called topes; the common rank r of topes is called the rank of the oriented matroid M . The top graph of M is the graph, denoted by T (M ), having the topes as vertices, with two of them being adjacent if they have a common face of rank r − 1. The tope graph uniquely defines the oriented matroid. A graph is called antipodal, if for each its vertex a there is an unique vertex a′ such that a′ has larger distance from a than any of neighbors of a′ . If the skeleton of a polytope is antipodal, then the maximal symmetry of this polytope contains central symmetry. In those terms, the following results by Fukuda and Handa (see [Fuku95]) connects isometric subgraphs of hypercubes with above tope graphs of an oriented matroids M = (E, F)

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Scale Isometric Polytopal Graphs Table 14.1

Some isometric polyhedral subgraphs of hypercubes

Nr.

Nr. vertices

deg.

polyhedron

emb. in

Aut

zonotope?

1 2 3 4 5 6 7 8 9 10 11

12 13 14 18 24 32 32 32 48 56 120

3 3,4 4 3,4 3 4 3 3 3 3,4 3

P rism6 RhDo − v3 (cuboct.)∗ = RhDo ElDo tr(β3 ) (icosidode.)∗ = triac. Cham(γ3 ) tw. Cham(γ3 ) tr(cuboct.) Fukuda’s polyhed. tr(icosidode.)

H4 H4 H4 H5 H6 H6 H7 H7 H9 H9 H15

D6h C3v Oh D4h Oh Ih Oh D3h Oh Ci Ih

+ non-CS + + + + + non-CS + no +

12 13 14a

4t > 12 4(t + 2) > 8 t(t − 1) + 2

P rism2t = C2t × P2 C4 × Pt+2 P z(t), t > 3 odd

Ht+1 Ht+3 Ht

D2th D4h Dtd

+ no +

14b

t(t − 1) + 2

P z(t), t > 4 even

Ht

Dth

+

15 16

2t (t + 1)!

3 4 3, t>4 3, t>4 t t

t-hypercube permutahedron

Ht H“t+1”

+ +

2

of rank r ≥ 3 on m-element set E: (i) T (M ) is an r-connected antipodal graph and T (M ) → Hm ; (ii) A graph is the top graph of an oriented matroid of rank three on m-element set if and only if it is the skeleton of a centrally symmetric polyhedron, which is isometric subgraph of Hm . If T (M ) is the skeleton of a zonotope, then this zonotope is rdimensional and has diameter m. We are interested now by construction of non-zonotopal centrally-symmetric polyhedra. Every Voronoi polytope is centrally symmetric with centrally symmetric facets. But a zonotope additionally has centrally symmetric faces of all dimensions. The skeleton G(P ) of a zonotope P is the top graph of an oriented matroid M (P ), which is realized by the zonotope P . Hence such oriented matroid is called realizable oriented matroid, or linear one. A face of dimension k of P realizes an oriented matroid of rank k. Consider a 3-dimensional zonotope, i.e. a zonohedron P . Let F and F ′ be two opposite faces of P . Let F has 2k edges, k ≥ 3. These edges are parallel to k vectors. These k vectors are linearly independent and form a circuit C(F ) of the matroid M (P ). Let {(i, i′ ) : 1 ≤ i ≤ 2k} be 2k pairs of antipodal vertices of F and F ′ , respectively. Denote by Q(G) the graph

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obtained from G := G(P ) by adding two new vertices v, v ′ and new edges (v, i), (v ′ , i′ ) for i = 2j + 1, 0 ≤ j ≤ k − 1. Clearly, Q(G) is antipodal and G is an isometric subgraph of Q(G). If G → Hm , then Q(G) → Hm also. (In fact, let a(1) = ∅, a(2j + 1) = {1, j + 1}, 1 ≤ j ≤ k − 1; then a(v) = {1} is uniquely determined.) If G is a 3-connected planar graph, then so, is Q(G). Hence Q(G) is the skeleton of a polyhedron, which we denote by Q(P ). It is centrally symmetric. Usually, if P is a zonohedron, so is Q(P ). But now, the set of vectors, representing F and F ′ , is not a circuit of the oriented matroid M (Q(P )). Consider elongated dodecahedron ElDo and rhombic dodecahedron RhDo. One can check, that Q(P rism6 ) is RhDo, Q2 (ElDo) is the rhombic icosahedron P z(5), Q4 (tr(β3 )) is the triacontahedron, i.e. dual icosidodecahedron. (Here Qm (P ) := Q(Qm−1 (P )).) But there are cases, when the polyhedron Q(P ) is not a zonohedron and hence, it does not realizes an oriented matroid. An example was given by Fukuda in [Fuku95] (using a hexagonal faces F and F ′ , on which new vertices were put); see Figure 14.3. 1

6 4

9 6

2

4

6

2

4

8 2

9

8 3

6

8 1 5 2

1

5 9 3

5

9

9

5

8

3

1

3

7 6

7 4

3 2

5 4 3

5

1

2 6

4

8 1

6 3

4

5 2

8

5

3 8

9

6 6

7

4

1 8

7

7 7

2

7 9

7

5 6

4

2

3

3

5 4

8

9

9 9

3

8 2

5 5 3

1

Fig. 14.3 A non-zonotopal centrally symmetric polyhedron, whose skeleton is embeddable into H9

This polyhedron contains 56 vertices, 18 hexagonal and 18 square faces. It is an example of a non-zonotopal centrally symmetric polyhedron, whose

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skeleton is embedded into H9 . This polyhedron is a minimal one in the following sense: any centrally symmetric 3-polytopal isometric subgraph of Hm , m ≤ 8, is (being the tope graph of an rank-three oriented matroid with at most eight elements) the skeleton of a zonotope. Fukuda constructed it as (non-Pappus) extension of an oriented rankthree matroid on eight points. As a linear extension of the same matroid, he obtain a dual zonohedron of diameter nine, having 54 vertices, 20 hexagonal and 12 square faces: just delete two marked opposite vertices from 56-vertex polyhedron depicted in Figure 14.3 above. Remark 14.5 See [DeSt96; DeSt97] for similar isometric embeddings of skeletons of infinite polyhedra (plane tilings) into cubic lattices Zn . For example, regular square tiling regular and hexagonal tiling are embeddable into Z2 and Z3 ; they are Voronoi partitions of the lattices Z2 , A2 . The Archimedean tilings (4.8.8), (4.6.12) and dual Archimedean tiling [3.6.3.6] are embeddable into Z4 , Z6 , Z3 . Also Penrose aperiodic rhombic tiling is embeddable into Z5 and dual mosaic Nr.12 (from Table 9.1 above) is embeddable into Z∞ . The tiling on Figure 14.4 b embeds also into Z∞ . Clearly, any lattice plane tiling, such that its Voronoi partition is embeddable into a Zm , is a dicing. In other words, it is zonohedral, i.e. having only centrally symmetric faces. Any zonohedral plane tiling embeds into a Zm , m ≤ ∞. A zonohedral plane tiling needs not to be periodic (for example, Penrose aperiodic tiling by two golden rhombuses) or to be a projection of Zm , m < ∞ (see, for example, the tiling on Figure 14.4 a). The tiling [3.6.3.6] above is non-zonohedral, since its quadrangular faces are not centrally symmetric.

a Fig. 14.4

b Two embeddable tilings

Some generalizations of planar scale 1 embeddable graphs

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Let G be a bipartite plane graph. If it is infinite, we suppose that its vertices have bounded degree and that it is discrete, i.e. any ǫ-neighborhood of any point on the plane contains only finite number of its vertices. Call such a graph G admissible (or strictly admissible) if there exist a mapping f of its vertices into vertices of an hypercube Hm , m ≤ ∞, such that its edges (x, y) are mapped into edges (f (x), f (y)) of Hm and the images of the circuits, bounding its interior (or, respectively, all) faces are isometric subgraphs of the same Hm . Using a result from [DSS86], we have that G is admissible (or strictly admissible) if and only if every of its zone (i.e. non-extendible sequence of opposite edges) is simple, i.e. it has no selfintersections. (In strictly admissible case the exterior face is included; so, G is considered as a graph on the sphere and zones can go through the exterior face.) In fact, if a zone self-intersects in a face, then the circuit, bounding this face, will be not isometric in Hm . On the other hand, if G is a plane quadriliage, i.e. all its interior faces are 4-gons, then it was proved in [DSS86] that it is admissible (or strictly admissible) if and only if all its zones are simple. The general case of admissible (or strictly admissible) graph G can be reduced to above one by partition of all interior (or, respectively, all) faces into 4-gons; so that the zones and their simplicity (or not) are preserved. In terms of zones, we have that G is scale one embeddable (i.e. an isometric planar subgraph of a hypercube) if and only if it is admissible and, moreover, every its zone is convex. In fact, zones are belts, corresponding to convex opposite cuts, such that exactly one of them goes through any edge. A partial subgraph of an hypercube is always bipartite; its vertices are mapped injectively into those of the hypercube. Clearly, there are following irreversible implications for possible properties of graphs with respect to the hypercubes: isometric (i.e. scale 1 embeddable) → induced → partial → bipartite, isometric planar → admissible → bipartite planar. (We take for each graph its strongest property; for example, the 8-cycle, which is only a partial subgraph of H3 , is considered isometric since it is isometric subgraph of H4 .) On Figure 14.5 we have seven counterexamples. In the last graph g), two vertices, marked by a black circle, are the points of self-tangency of the zones of eight 4-gons around them; the images of two vertices, marked by a white circles, coincide in H6 .

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a) the dual of AP rism4 is bipartite planar, but non-admissible and non-partial one

b) K2,3 is admissible in H2 , but non-partial

c) the cube with deleted edge is admissible and partial in H3 , but non-induced

d) admissible and induced in H4 , but non-isometric

e) isometric, but not strictly admissible

f) induced in H6 , but not admissible

g) strictly admissible in H6 , but non-partial Fig. 14.5

Seven counterexamples

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Chapter 15

Some Generalization of ℓ1-embedding

15.1

Quasi-embedding

Here we consider quasi-embedding or, more exactly, t-embedding of given graph G, i.e. our usual embedding, but for t-truncated distance min(t, dG ), where t is less than the diameter of the graph G. We say that a metric d is t-embeddable if there is an ℓ1 -embeddable metric d′ such that d′i,j = min(di,j , t). This notion was introduced in [DeSh96], where it was shown that the polynomial algorithm given there for the recognition of ℓ1 -graphs can be extended to the recognition of t-embeddable graphs. In what follows, describing a t-embedding of a polyhedron P , we associate to every its vertex v a subset a(v) of a set N . Usually we take as N the set of all faces, which are k-gons for a fixed k. We say that a face F is reachable by an m-path from a vertex v if there is an m-path of length m from the vertex v to a vertex of the face F . We use here notation C60 for the truncated icosahedron 560 (Ih ). Even if C60 is not ℓ1 -embeddable, we still can quasi-embed it into 12 H20 in the following sense. According to [DeSh96], the truncated icosahedron (of diameter 9) has a unique 7-embedding into 12 H20 : associate each vertex to 2+2+3 hexagons (amongst all 20) reachable by 0-, 1-, 2-paths, respectively. It is also the unique 3-embedding, but not unique 2-embedding: for example, associate every vertex to two its hexagons. The unique 3-embedding of C60 into 21 H20 is called quasi-C60 . Moreover, quasi-C60 is an ℓ1 -rigid (but not graphic) metric. The Figure 15.1 illustrates the 7-embedding of C60 . The construction is as follows: we can associate a coordinate of R20 to each of the 20 hexagons. Then a vertex v (for example the white atom of Figure 15.1) is mapped into the vertex φ(v) of the half-cube 21 H20 (in odd representation), whose 153

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non-zero coordinates correspond to the seven hexagons of C60 containing a vertex, whose distance to v is less than three (the seven grey hexagons of Figure 15.1). This give us a 7-embedding of C60 into the half-cube 21 H20 . All distances eight and nine in C60 became seven in quasi-C60 (recall that C60 has diameter 9). The automorphism group of quasi-C60 (i.e. all permutations of the 20 coordinates indexed by the 20 hexagons of C60 , which preserve quasi-C60 ) is equal to the one of C60 , which is the Coxeter group H3 isomorphic to {1, −1} × A5 . The Figure 15.2 illustrates one automorphism of quasi-C60 : the reversing of the spiral Hamiltonian path on the 20 hexagons of C60 .

Fig. 15.1

Embedding up do distance 7 of C60 (Ih ) into

1 H 2 20

Icosidodecahedron (of diameter 5) has unique ([DeSh96]) 4-embedding in associate every vertex to 1+1+2 pentagons (from all 12) reachable by 0-, 1-, 2-paths. It is also a unique 3-embedding, but there is another 2-embedding: associate every vertex to two its triangular faces. Cuboctahedron (of diameter 3) has at least two following 2-embeddings: into 21 H6 : associate every vertex to its two square faces, into 12 H8 : associate every vertex to its two triangular faces; this one, as well as two above ones are t-embeddings into 21 H2d(P )+2 , where d(P ) is the diameter of P . The vertex figure (and the local graph of the skeleton) of the snub 24cell is the tridiminished icosahedron (the regular-faced solid M7 = N r. 83 of diameter 3). It is embedded into 21 H6 and has a 2-embedding into 12 H7 . Any simple polyhedron has a 2-embedding in the tetrahedral graph 1 2 H12 :

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Some generalization of ℓ1 -embedding

Fig. 15.2

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155

The spiral Hamiltonian automorphism of quasi-C60

J(n, 3) (so in 12 Hn ): associate every vertex to its three faces. If the diameter of a simple polyhedron is at least 3, then it is 3-embeddable if and only if sizes of its faces are from the set {3, 4, 5}. Examples of this procedure are: (1) dodecahedron (of diameter 5) has a 3-embedding into Johnson graph J(12, 3); also it has a 5-embedding into 21 H10 , (2) α3 → J(4, 3) → 21 H4 (not unique), ∗ (3) M25 → J(8, 3) → 12 H8 , (4) a 2-embedding of P rismn , which turns out to be P rismn → 21 Hn+2 (→ H n+2 for even n). 2

Another procedure: fix a 5-wheel ∇C5 in the skeleton of an icosahedron and associate every vertex v to the set of all vertices of the 5-wheel at distances 0 and 1 from v; we get the (unique) 3-embedding of the icosahedron. The dual of F60 (Cs )r , considered on Figure 22 of [DDG98], admits ∗ a 4-embedding into 12 H10 . More precisely, F60 (Cs )r has diameter 5 and all distances are preserved except those between four opposite pentagons: (38, 48), (1378, 1478), (45, 35) and (2459, 2359). Instead of five, those distances became four in 21 H10 . See Figure 15.3, where the 4-embedding of ∗ F60 (Cs )r is given by labels of facets of F60 (Cs )r . Additionally, a not 5-gonal configuration of F60 (Cs )r is given, in Figure 15.3. The following Theorem (see [DFS03]) describe completely t-embeddings for fully icosahedral (i.e. with symmetry Ih ) fullerenes and their duals. Theorem 15.1

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156

Scale Isometric Polytopal Graphs 34

3479 3478

3459 345679 234569

346789 134678

12 18

25 0

35

2359

1378

45

38

34

3478

2569

48

3459

1678

3479 1256

1478

2459

1268

25

18 12 0

∗ (C ) into Fig. 15.3 The 4-embedding of F60 s r ration of F60 (Cs )r

1 H 2 10

and a not 5-gonal configu-

(i) C20k2 (Ih ), k ≥ 1, is (2k + 7)-embeddable into 12 H12k−2 ; its diameter is 6k − 1, (i’) (C20k2 (Ih ))∗ , k ≥ 2, is 2k-embeddable into 12 H6k ; its diameter is 3k, (ii) C60k2 (Ih ), k ≥ 1, is (6k + 1)-embeddable into 21 H20k ; its diameter is 10k − 1, (ii’) (C60k2 (Ih ))∗ , k ≥ 1, is (3k+2)-embeddable into 21 H10k ; its diameter is 5k. In fact, let us indicate for each of four above cases, the complete collection of zones, giving a t-embedding. Each zone will be without selfintersections; so any two of them will be either parallel, i.e. disjoint, or intersect exactly in two faces. We call a zone of faces of a fullerene special or pure, according to if it contains or no some pentagons. The dimension of the half-cube is the number of special zones plus twice the number of pure ones. This dimension turns out to be twice the diameter of the graph in the cases (i), (i’) and (ii’). Remark that this t-embedding is isometric only for k = 1 in the cases (i), (i’), (ii’) and for k = 2 in the case (i). In the case (i’) above there are 6k alternating zones of length 10k each; they are partitioned

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into six parallel classes. In the case (ii’) above there are 10k alternating zones of length 18k each; they are partitioned into ten parallel classes. In the case (i) above there are 6(k − 1) pure zones of length 5k each; they are partitioned into six parallel classes; there are ten special (alternating) zones of length 6k each. Each special zone consists of six pentagons, separated by (k − 1)-strings of hexagons. Each zone have equal number (six for pure and three for special one) of pentagons inside and outside of it. In the case (ii) above there are 10(k − 1) pure zones of length 9k each; there are 20 special (alternating) zones of length 9k each. Each special zone is not alternating; it consists of three pentagons, separated by (3k − 1)-strings of hexagons. Each pure zone have six pentagons inside and outside of the ring formed by the zone; each special one has exactly three pentagons inside or outside of it. All 20k zones are partitioned into ten parallel classes and each class consists of k − 1 pure zones, taken in a “sandwich” by two special ones. 15.2

Lipschitz embedding

Here we consider another relaxation of our main notion of l1 -embedding. The mapping f : X → Y of metric spaces X, Y is called Lipschitz, if for any vertices a, b of X, holds

1≤

dX (a, b) ≤ C, dY (f (a), f (b))

where C is a constant, called distortion. Bourgain showed in 1985 that any metric on at most n points is Lipschitz-embeddable into ℓk1 (for a finite dimension k) with distortion C = O(log n); Linial, London and Rabinovitch showed in 1994 that, moreover, k = O((log n)2 ). For example, K1,3 has distortion > 1. Aharoni in 1978 showed that any lp with p > 2 is not Lipschitz-embeddable into l1 . It looks plausible that the path-metric of any infinite hypermetric graph (moreover, any 5-gonal metric) is Lipschitz-embeddable, up to a scale, into a Zm , m ≤ ∞. 15.3

Polytopal hypermetrics

The convex cone of all hypermetrics on m points is denoted by Hypm . Call a hypermetric d on m points of rank k, if the intersection of all faces of

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Hypm , to which d belongs, has dimension k; call a hypermetric extreme hypermetric, if it has rank 1, i.e. it belongs to an extreme ray of Hypm . A polytope P × P ′ is hypermetric if and only if both P, P ′ are hypermetric; it is non-embeddable if and only if at least one of P, P ′ is nonembeddable. (But, for example, any pyramid P yr(P ) over polytope P with at least 28 vertices, is embeddable if and only if it is hypermetric.) Any hypermetric, but not ℓ1 -embeddable graph of rank k has at most 56k vertices; the skeleton of the direct product of k copies of the 7dimensional Gosset polytope 321 realizes equality in this upper bound. We denote below by Gi , 1 ≤ i ≤ 26, the graphs from [DeGr93] related (but not as path-metrics) to extreme hypermetric on seven points. Remark that extreme hypermetric graphs G1 and G2 are skeletons of 4-dimensional pyramids with bases P yr5 and 2-capped α3 , respectively. All hypermetric but non-ℓ1 graphs with at most seven vertices are known (see [DeGr93]); they are 12 graphic metrics amongst of 26 extreme hypermetrics on seven points. Proposition 15.1 Amongst of all hypermetric non-l1 graphs with at most seven vertices, the polytopal ones are only: 3-polytopal G4 (see Figure 15.4, d)) and 4-polytopal G1 = ∇2 C5 = K7 − C5 , G2 = K7 − P4 . Any, except of K2 , hypermetric is extreme if and only if it generate 221 or 321 . So, the number of vertices of any extreme hypermetric graph is within the interval [7,56] and any polytope, such that its skeleton is extreme hypermetric, has dimension within the interval [6,7]. Call an extreme hypermetric graph of type I (of type II) if it generates the root lattice E6 (E7 , respectively). A graph G of type I has diameter two, since it is an induced subgraph of the Schläfli graph G(221 ) of diameter two, and in this case ∇G is an extreme hypermetric of type II. Clearly, G(P yr(P )) = ∇G(P ), dim(P yr(P ))=dimP +1. (Remark that P yrk (C5 ) → 12 H5 for k ∈ {0, 1}, it is extreme hypermetric for k ∈ {2, 3}, and it is 7- but not 9-gonal for k = 4.) Examples of polytopes, whose skeletons are extreme hypermetric but are not represented as ∇G′ for an extreme hypermetric graph G′ are: of type I: (1) the polyhedra Nr.30, 71, 106 = M22 and one with skeleton G4 all having 9, 9, 10 and 7 vertices, respectively (see Figure 15.4); (2) the 4-polytopes with skeletons G1 , G2 and both with seven vertices; (3) the 6-polytopes: with skeletons K9 − C6 = G(P yr3 (P rism3 )),

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a) 1 − AP rism4 ,

b) triaugmented P rism3 ,

c) sphenocorona M22 ,

d) extreme hypermetric graph G4

Fig. 15.4

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Examples of hypermetric non-embeddable polyhedra

∇2 T (5) = G(P yr2 (Ambo(α4 ))), ∇ 21 H5 and the Shläfli polytope 221 having, respectively, 9, 12, 17 and 27 vertices; of type II: (1) the polyhedra Nr.37, 107 = M21 + P yr4 (2) and the 7-polytope 321 with 10, 11 and 56 vertices, respectively. So, we have two following series of inscribed polytopes, such that their skeletons are extreme hypermetrics: G1 = ∇2 C5 < ∇2 G(021 ) < ∇G(121 ) < G(221 ) (of type I), ∇3 C5 < ∇3 G(021 ) < ∇2 G(121 ) < G(321 ). Some examples of extreme hypermetrics, which are not polytopal, but are close to polytopal in a sense: (1) 7-vertex graph G18 of type I is a planar graph of a skew polyhedron. (2) The skeleton of the stella octangula (the section of γ3 by β3 with vertices in the centers of faces of γ3 ), which is a non-convex polyhedron; it is

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14-vertex graph Gso . It contains the extreme hypermetric G4 as an induced subgraph. Gso is an isometric subgraph of the Gosset graph G(321 ) and, since it has diameter 3, it is of type II. 3 3 (3) Antiwebs AW92 , AW12 , are of type I, and AW14 is of type II; AW92 is projectively-planar and it becomes planar, if we delete one edge. Finally, one can enumerate isometric subgraphs (and polytopal ones between them) of a given half-cube 12 Hn , which are not isometric subgraphs of its facet. For example, the following graphs are such isometric polytopal subgraphs of the Clebsh graph 12 H5 = G(121 ): (1) C5 , K5 = G(α4 ) (together with four graphs from 2) below), they are only such subgraphs of 21 H5 with at most 6 vertices); (2) K6 − C6 = G(P rism3 ), K6 − P5 (polyhedral), K6 − P4 = G(2-capped α3 ), ∇C5 = G(P yr5 ) (amongst of all ten 6-vertex isometric subgraphs of 12 H5 ); (3) the skeletons of the following polyhedra with at least 7 vertices: Nr.27 (1-capped P rism3 ), Nr.60 (augmented P rism3 ), Nr.70 (biaugmented P rism3 ), 1-capped β3 , AP rism4 ; (4) the skeletons of 4-polytopes: P yr(P rism3 ), α1 × α3 , G(021 ) = T (5), T (5) − K1 , T (5) − K2 , 1-capped (on a facet β3 ) 021 ; (5) the skeletons of 5-polytopes: P yr2 (P rism3 ), P yr(ambo-α4 ). 15.4

Simplicial n-manifolds

An interesting relaxation of our embeddings of polytopes is to consider scale-isometric embedding into hypercubes and cubic lattices of 1-skeletons of simplicial and cubical complexes more general than the boundary complexes of polytopes. For example, the simplicial complex on {1, 2, 3, 4, 5} with the facets {1, 2, 3}, {1, 2, 4}, {1, 2, 5} has the skeleton K5 − K3 ; the cubical complex on {1, 2, 3, 4, 5, 6} with the facets {1, 2, 3, 4}, {2, 3, 5, 6}, {1, 4, 5, 6} has the skeleton K3,3 . So, both are not 5-gonal. While the general case of simplicial complexes is too vast, we have some partial results in terms of links of (n − 2)-faces. For any n-simplex S containing fixed (n − 2)-face S ′ , there exists unique edge e, such that S is the join of S ′ and e. The link of (n − 2)-face S ′ is the cycle formed by such edges e, for all n-simplexes S, containing S ′ . Theorem 15.2 Let M be a closed simplicial n-manifold of dimension n ≥ 3. Then we have:

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(i) M is not embeddable, if it has an (n − 2)-face belonging to at least five n-simplexes and such that its link is an isometric cycle in the skeleton. (ii) M is embeddable, if any of its (n − 2)-faces belongs to at most four (i.e. 3 or 4) n-simplexes. If a (n − 2)-face belongs to at least six n-simplexes (so, to at least six (n − 1)-simplexes), then the skeleton of M is not 5-gonal, since it contains the isometric subgraph K5 − K3 . For example, De(A∗3 ) is not 5-gonal, because it has six tetrahedra on some edges. If a (n − 2)-face belongs to exactly five n-simplexes, then the skeleton of M contains the isometric subgraph K7 − C5 (i.e., the skeleton of 4-polytope P yr(P yr5 )), which is 5-gonal but is not embeddable. For example, the regular 4-polytope 600-cell, which is a closed simplicial 3-manifold, has five tetrahedra on each edge. The condition of isometricity of the link is necessary (it was missed in a result from [DeSt98]). For example, there exists an embeddable 3-manifold having an edge, which belongs to five tetrahedra; its skeleton is K7 − P2 . In order to check (ii) for n = 3, for example, consider closed simplicial 3-manifolds, such that any edge belongs to at most four tetrahedra. There are exactly five such manifolds. Their skeletons are K5 = G(α4 ), K2×4 = G(β4 ), K6 (a 3-dimensional submanifold of α5 ), K6 − e, K7 − 2e. All those five skeletons are embeddable into 12 Hm with m = 5, 4, 6, 8, 8, respectively. One can show, that the skeleton of any n-manifold in the case (ii) is the graph Ki − tP2 (i.e. the i-clique with t disjoint edges deleted), such that either (i, t) = (2n + 2, n + 1), or ⌋. n ≥ t + 1 and n + t + 2 ≤ i ≤ n + t + 1 + ⌊ n−t+1 2 In fact, any subgraph of the skeleton of (n+1)-cross-polytope, containing Kn+2 − P2 , will appear as such skeleton; all of them are embeddable graphs of diameter two. For example, K7 − P2 appears as the skeleton of an nmanifold of type (ii), but only for n = 4.

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Index

(r, q)-polycycle, 75 (s × s)-grid, 142 AP rismkm , 50 An , 107 k Barrelm , 51 D-complex, 105 Dn , 107 Dn+ , 109 De(L), 107 E6 , 107 E7 , 107 E8 , 107 J-complex, 101 P rismm n , 45 RhDo-v3 , 100 k T owerm , 50 V o(L), 107 ℓ1 -embeddable graph, 5 ℓ1 -graph, 4 ℓ1 -metric, 6 ℓ1 -rigid graph, 6 tr(cube), 45, 46 tr(cuboctahed.), 45, 46 tr(dodecahedron), 45, 46 tr(icosahedron), 45, 46 tr(icosidodecah.), 45, 46 tr(tetrahedron), 45, 46 1 -truncation, 53 2 1 -truncation, 53 3 i-capping, 54 i-layered dodecahedron, 30 i-truncation, 54

n-bipyramid BP yr(P ), 17 n-cross-polytope βn , 17 n-cube γn , 17 n-permutahedron, 111 n-simplex αn , 17 n-sphere, 35 n-wheel, 3 s(3, 4, 3), 71 s(3, 4, 3, 3), 54 t-embeddable graphs, 153 t-truncated distance, 153 tr(octahedron), 45, 46 1-AP rismm , 50 120-cell, 40 2-AP rismm , 50 24-cell, 40 5-gonal inequality, 9 600-cell, 40 adjacent, 1 alternating cut, 11 alternating zone, 13 ambo-polytope, 18 antipodal doubling, 138 antiweb AWnk , 49 aperiodic sequences, 77 Archimedean 4-polytopes, 72 Archimedean polyhedra, 16 augmented sphenocorona, 64 azulene, 78 basic polyhedra, 63 171

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172



bcc, 101 belt, 10 benzene, 78 bi-lattice, 100 bicupolas, 68 bifaced polyhedra, 120 Bilunabirotunda, 69 bipartite graph, 1 biphenyl, 78 border line, 14 boride F e2 AlB2 , 101 boride U B12 , 102 boride CaB, 101 buckminsterfullerene, 25 Cairo net, 86 Campanus sphere, 46 capping, 18 carbon C, 78 Cartesian product G1 × G2 , 3 Catalan polyhedra, 16 cell, 35 centering, 109 central circuit, 12 chamfering, 18, 59 circuit Cn , 2 coated vesicles cells, 31 cocktail-party graph Kn×2 , 2 convex n-polytope, 15 convex cut, 6 convex subset, 4 coordination polyhedra, 80 copper Cu, 103 coranulene, 78 cube γ3 , 16 cubic lattice graph Zn , 2 cubical complex, 160 cuboctahedron Cbt, 63 cupola Cupn , 49 cut {S, S}, 5 cut cone, 5 cut semimetric, 5 D¨ urer’s octahedron, 15 Delaunay partition, 19 Delaunay polytope, 19

deltahedron, 15 density, 36 diametral doubling, 139 diametral switching, 140 diamond bi-lattice, 105 dicing lattice, 110 distance-regular graph, 141 distance-transitive graph, 141 ditrigonal dodecahedron, 39 dodecadodecahedron, 39 dodecahedron Do, 16 Doob graph, 141 Double Odd graph DO2s+1 , 142 dual polytope, 15 Dyck map, 38 edge figure, 83 Edge-coalesced icosahedron, 82 edge-homogeneous, 83 elliptic distance, 47 elongated dodecahedron ElDo, 15 elongation, 17 equicut graph, 137 Euclidean n-space, 35 Euclidean plane R2 , 75 Eulerian plane graph, 12 extreme hypermetric graph, 158 F¨ oppl partition, 103 face-regular, 89 fcc, 101 fluorentene, 78 fullerene, 25 fundamental region, 36 geodesic, 4 girth of graph, 75 golden isozonohedra, 147 golden number, 72 golden rhombi, 87 golden rhombohedra, 87 golden truncation, 54 Gosset graph G(321 ), 160 Gr¨ unbaum partition, 104 Grand Antiprism, 73 great dodecahedron, 36

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Bibliography

great icosahedron, 36 grid distance, 48 grid graph, 48 gyro-elongated P yr4 , 64 gyrobifastigium, 64 Hadamard matrix, 141 half-cube graph 12 Hn , 2 half-cubic lattice graph 21 Zn , 2 Hamming distance, 48 Hamming graph, 48 hcp, 102 hebesphenocorona, 70 hexagonal sheet, 31 hexakis, 54 high-pressure Si, 101 Hilbert cube, 21 honeycomb, 16 hydrogen H, 78 hyperbolic n-space, 35 hyperbolic plane H 2 , 75 hypercube graph Hn , 2 hypermetric graph, 8 hypermetric inequality, 8 icosahedron Ico, 16 icosidodecahedron, 63 incident, 1 indacene, 78 induced subgraph, 1 infinite chess-board, 106 infinite dimensional cube, 20 infinite distance-transitive graph, 143 infinite prism, 106 infinite regular polyhedra, 38 irreducible root lattices, 107 isometric subgraph, 4 Johnson n-polytope, 16 Johnson graph J(n, k), 3 k-hedron, 15 Kagome net, 86 Karlsruhe distance, 45 Kelvin partition, 104 Klein map, 38

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Koester’s graph, 51 labyrinths, 38 lattice, 107 lattice complex, 100 Laves phase M gCu2 , 103 leapfrog, 125 Lee distance, 47 left opposite edge, 11 left-right circuit, 12 line graph, 3 Lipschitz-embeddable, 157 medial graph, 18 medial polyhedra, 121 metallic cluster, 80 metallopolyhedra, 80 metric, 4 minimal energy, 31 minimal scale, 4 mosaic, 83 Moscow distance, 45 Moscow graph, 45 multipartite graph Kn1 ,n2 ,...,nk , 2 naphtalene, 78 non-compact uniform partitions, 106 NP-complete, 9 octahedrites, 120 octahedron β3 , 16 octicosahedric polytope, 71 omnicapping, 54 opposite cut, 11 oriented genus, 38 oriented matroid, 148 outerplanar graph, 14 oxifloride Ag7 O8 F , 102 paracelsian BaAl2 Si2 O8 , 90 partial subgraph, 75 path Pn , 2 path-metric dG , 4 Penrose tiling, 87 pentacle, 39 pentagonal rotunda, 63

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pentagram, 39 pentakis, 54 pentalene, 78 perfect matching, 2 perovskite (ABX3 ), 101 Petersen graph, 142 pinwheel tiling, 89 plane graph, 10 Platonic solids, 16 polar zonohedra P z(m), 147 polonium P o, 103 polycyclic aromatic hydrocarbons, 78 polyhedron, 15 polytopal graph, 6 preferable fullerene, 25 projective plane, 38 quadriliage, 151 quasi-(r, 3)-polycycles, 77 quasi-embedding, 153 quasi-medial polyhedra, 121 quasi-regular n-polytope, 16 radar-discrimination distance, 45 realizable oriented matroid, 148 regular n-polytope, 16 regular compounds, 39 regular maps, 37 regular partition (rq), 76 regular skew polyhedra, 38 regular star-honeycombs, 40 regular-faced n-polytope, 16 regular-faced polyhedra, 63 rhombic dodecahedron RhDo, 149 rhombicicosahedron, 69 rhombicosidodecahedron RIDo, 63 rhombicuboctahedron Rcbt, 63 root lattices, 107 root system E8 , 71 Schl¨ afli notation, 36 Schl¨ afli graph G(221 ), 158 Seidel switching, 140 selenide P d17 Se15 , 102 semi-regular n-polytope, 16 semimetric, 3

Shrikhande graph, 142 Siamese dodecahedron, 63 silicide T hSi2 , 102 silicide U3 Si2 , 101 simple n-polytope, 15 simplex α3 , 16 simplicial n-manifold, 160 simplicial complex, 160 size s(dG ), 6 skeleton G(P ), 6 snub AP rismb , 119 snub 24-cell, 71 snub cube, 45, 46 snub disphenoid, 63 snub dodecah., 45, 46 sodalite, 101 sphenocorona, 64 sphenomegacorona, 70 sphere S 2 , 75 spherical wavelets, 60 star-polytope, 35 stella octangula, 39 stellated k-gon, 48 stellation, 54 strain energy, 31 strongly regular graph, 141 suspension ∇G, 3 terphenyl, 78 tetrakis, 54 Tikhonov cube, 21 tile-homogeneous, 83 tiling, 16 top graph, 148 torus, 38 triacontahedron, 147 triakis, 54 triangle inequality, 3 Triangular hebesphenorotunda, 69 triangulation, 12 triaugmented P rism3 , 64 tridiminished icosahedron, 70 twisted Rcbt, 44 twisted RhDo, 100 twisted chamfered cube, 126 twisted cuboctahedron, 130

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Bibliography

uniform polyhedra, 39 vertex figure, 16 vertex-homogeneous, 83 vertex-split (34), 76 vertex-split (35), 76 vertex-transitive, 72 Voronoi partition, 19 Voronoi polytope P (x), 19 Wiener number, 4 zeolit Linde A, 102 zeolite rho, 101 zinc Zn, 103 zone, 5 zonotope, 148

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