Clarksville, Tennessee. âI am currently an undergraduate stu- dent studying art and mathematics. Ten years ago, I was studying comput- er programming which ...
2016 Joint Mathematics Meetings
Exhibition of Mathematical Art Edited by Robert Fathauer Nathan Selikoff
Tessellations Publishing
Copyright © 2016 by Tessellations All rights reserved Copyrights to the images of and text related to the artworks belong to the respective artist Printed in the United States of America ISBN 978-1-938664-17-5 Design by Robert Fathauer
Preface 2016 is the 13th consecutive year that an exhibition of mathematical art has been held in conjunction with the Joint Mathematics Meetings. All of the exhibitions except the first are chronicled online at the Bridges Conference website (http://www.bridgesmathart.org). Robert Fathauer was the curator of the exhibition, and the exhibition website was prepared by Nathan Selikoff. The exhibition was juried by Anne Burns, Robert Fathauer, Reza Sarhangi, and Elizabeth Whiteley. Meighan Dillon edited the artists’ text. The Mathematical Art Exhibition Awards for 2015, given to recognize works that demonstrate the beauty and elegance of mathematics expressed in a visual art form, were announced at last year’s meeting. “Penrose Pursuit 2” by Kerry Mitchell was awarded Best Photograph, Painting, or Print. “Map Coloring Jewelry Set,” by Susan Goldstine was awarded Best Textile, Sculpture, or Other Medium. “15 Irregular Hexahedra,” by Aaron Pfitzenmaier, received Honorable Mention. This year’s winners will be selected at the meeting by a separate jury from the one that selected the artworks for the exhibition. Since this catalog was prepared in advance of the meeting, the winners are not identified here. The exhibition website referenced above will contain this information, though. This year’s exhibition includes the work of over 70 artists from around the world. In this catalog, the artists and their works are presented in alphabetical order, except for some small shifts to ensure that two artworks by the same artist are always presented on facing pages. We would like to thank the American Mathematical Society and the Mathematical Association of America for providing exhibition space, furnishings, and logistical support. We would especially like to thank Christine Davis, Penny Peña and Annette Emerson at the American Mathematical Society for their efforts in connection with the exhibition and awards. We would also like to thank SIGMAA-ARTS for financial support toward the printing of this catalog. Finally, we would like to thank the artists, whose creative vision and effort continue to make these exhibitions a valuable addition to the Joint Mathematics Meetings. Robert Fathauer Nathan Selikoff
2016 Joint Mathematics Meetings
Exhibition of Mathematical Art
Ellie Baker
Artist and Computer Scientist Lexington, MA “This artwork is an outgrowth of an extended research project with Susan Goldstine on applications of mathematics to bead crochet. Our book, “Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist,” outlines a new methodology for designing bead crochet patterns and describes a series of mathematically inspired design puzzles. Recently I’ve begun exploring design in fabric, a new medium with new constraints and new puzzles. Here, fabric printed with the infinitely repeating planar representation of a bead crochet pattern is sewn into an infinity scarf (topologically a torus). The construction, with hidden seams, involved stitching together opposite edges of a parallelogram and an interesting topology lesson on torus inversions.”
From Serenity to Monkey-Mind and Back (Two Twisted Tessellated Transforming Tori)
71 x 51 cm Printed polyester crepe de chine, bead crochet (glass beads and thread), 2015 This infinity scarf and bead crochet necklace are twin tori. The fabric design is (an elongated version of) the infinitely repeating planar pattern that a tiny explorer could map by charting the surface of the necklace in all directions (the universal cover of the beaded rope). The two colors, identical tessellated wave motifs, gradually transform from “calm” to “busy.” The pattern at each step has an increasing “busyness quotient” (a measure of how much the individual beads in a fundamental tile differ in color from neighboring beads). The scarf, sewn from a parallelogram to create a mobius-like twisted torus, has a small hole in one seam so that it can be turned inside out to explore the puzzling behavior of torus inversions.
6
Erika Balogh
Graphic Designer - Art Instructor University of Texas Rio Grande Valley Edinburg, Texas “My bi-cultural background has greatly shaped my identity and influenced my artwork as well. When I moved to the United States, I left behind a traditional part of my life associated with my Hungarian heritage, my childhood memories, and my family. I entered into an utterly different culture; a more modern and eclectic society that completely changed my life. However, aspects of my Hungarian traditions continue to form my artwork and my self-presentation. “Through my artworks, I explore the power of colors and symbolism. The five-pointed star, the pentagram, is a well-recognized symbol all around the world. It has been used by many companies as a trademark or as a symbol on numerous national flags.”
Never-never Land 50 x 40 cm Digital Art, 2015
My research interest include social and economic disparity, exploitation, alienation, and exploring alternatives to capitalism. In this artwork the five-pointed star represents a utopian society. The red star is a well-recognized symbol of communism. This form of socioeconomic system is a brilliant ideology, however, in reality it seems impossible to organize society. Each pentagram inscribed in a pentagon represents an individual with distinctive characteristics as they come together to generate a greater whole – a unity. But there is no perfect unity, as it is not possible to form a tiling using pentagons.
7
sarah-marie belcastro Director MathILy (serious Mathematics Infused with Levity) Holyoke, Massachusetts
“I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, whether that results in knitting a model of a mathematical object or in using mathematics to design a garment. Indeed, over my mathematical life both of these types of combinations have occurred. Most of the mathematical models I have created are only of aesthetic value and have no real function; it is rare that I am able to adapt a mathematical object for use as a garment. (It is perhaps too much to hope that I could regularly combine artistry and function in addition to knitting and mathematics.)”
Rainbow Brunnian Link Cowl
30 x 30 x 7 cm Knitted wool (various sources) and printed photographs, 2015 The central property of the Borromean rings---that removing any component unlinks the remaining components, which collectively form the unlink---generalizes to the class of Brunnian links. The Rainbow Brunnian Link Cowl has seven components rather than the three components of the Borromean rings. All linking is intrinsic, rather than introduced post-construction via grafting.
8
Brunnian Link Cowls
45 x 45 x 5 cm Knitted wool (various sources), 2015 The central property of the Borromean rings---that removing any component unlinks the remaining components---generalizes to the class of Brunnian links. Here are two different generalizations: more crossings within components, and more crossings between components. Small Borromean rings are knitted in neutral colors because of their relative plainness. The other links are knitted in richer colors so as to emphasize their richer structure, and are functional garments (cowls). Each component of the pastel link is a two-loop unknot. The brighter cowl has three times as many intercomponent crossings as the Borromean rings. All linking is intrinsic, rather than introduced post-construction via grafting.
9
Regina Bittencourt IT Consultant and Art Student Universidad Andres Bello Santiago, Chile
As a computer programmer, Regina is always working with algorithms, logic and databases. This background provides her a never-ending supply of ideas for her artwork. Sometimes, doing her job she visualizes an image out of patterns that she sees on data bases she works with. Regina is studying art to become a professional artist where she has surprised her community who didn’t expect an artwork based on the combination of math and traditional art, since she only does Mathematical Art. This time her artwork is based on Modulus 11.
Hendeka
50 x 50 cm Acrylics on canvas, 2015 A check digit is an authentication mechanism used to verify and validate the authenticity of a series of characters such as a single mistyped digit or a permutation of two successive digits, thus avoiding typing errors. The last number of every Chilean Id Card is the check digit that is calculated using Modulus 11. Each digit in the base number is assigned a multiplication factor. The sum of the products is divided by 11. If the remainder is zero, the check digit is zero. For all others, the remainder is subtracted from 11. The result is the check digit.
10
Robert Bosch
Artist/Professor of Mathematics Oberlin College Oberlin, Ohio “I am fascinated by constraints. As a mathematical optimizer, I know that in some cases, if I impose additional constraints on an optimization problem, it will become much more difficult to solve, but in other cases, it will become considerably easier. Some constraints seem to be structured in such a way that in their presence algorithms have trouble working their way to the best part of the feasible region, whereas other constraints provide the equivalent of handholds and toeholds that form an easily traversed path to an optimal solution. As an artist, I am similarly mindful of constraints. I am inspired by the words of Igor Stravinsky: ‘The more constraints one imposes, the more one frees one’s self of the chains that shackle the spirit.’”
The Jordan Curve Theorem 15 x 45 cm lasercut woods, 2015
The Jordan Curve Theorem states that when a simple closed curve is drawn in the plane, it will cut the plane into two regions: the part lies inside the curve (here, the slightly darker-colored inset piece of wood), and the part that lies outside it (here, the slightly brighter and thicker frame).
11
Tatiana Bonch-Osmolovskaya Artist, writer, philologist Sydney, Australia
“Chess is a game based on strict rules which allows billions of situations. For a long time, it inspired mathematicians, players and artists. I use 2D and 3D computer graphics in the search for intellectual wonder of perceiving a mathematical concept together with the aesthetic pleasure of viewing a beautiful image. The beauty of chess encouraged me to investigate possibilities of concave-convex surfaces, upside-down orientation, inner and outer space, color variations, fractals and so on. Here are some of these images.”
Red Queen
24 x 24 cm computer graphics, 2014 The figures of this chess cube are simultaneously inside and outside it. They are puzzled enough preparing their moves. But who is the Red Queen playing for?
12
Games playing games
26 x 30 cm computer graphics, 2014 games play games play games play games
13
Rachelle Bouchat
Assistant Professor of Mathematics Indiana University of Pennsylvania Indiana, Pennsylvania “As a mathematician, I have always been fascinated with the beauty that mathematics creates and describes. In the past several years, I taught myself to knit and crochet and have been developing mathematically inspired patterns for the fiber arts.”
Dragon Curve Double Knit Scarf 137 x 18 cm Merino Wool Yarn, 2015
This double knit scarf brings together the recursive construction of a fractal, the dragon fractal, as well as the recursive construction of an integer sequence, the Fibonacci sequence. The main panels of the scarf are based on a pattern developed from the eleventh iteration of the dragon fractal. Moreover, the striping pattern in between the main panels is illustrative of the Fibonacci sequence with color changes after 1 row, after another 1 row, after 2 rows, after 3 rows, after 5 rows, and with another color change after 8 rows. As this is a double knit pattern, the back side of the scarf is shown in the reverse color pattern.
14
Julie Brooke
Research Fellow School of Art, Australian National University Canberra, ACT, Australia Julie Brooke is an Australian painter and former biomedical scientist who investigates parallels between research in science and in the visual arts. Drawing on the traditions of optical and concrete art, she explores the potential of geometric systems and the hand-made mark to visualise memory and thought, with a particular focus on the visualisation of abstract scientific and mathematical concepts. Brooke is a Research Fellow at the Australian National University School of Art and completed a practice-led visual arts PhD in 2013 for which she was awarded the J. G. Crawford Award.
Entangled Labyrinth
30 x 30 cm Gouache, pencil and acrylic on board, 2015 During a 2014 residency in the Australia National University Applied Mathematics Department I worked with topologists to investigate how they visualise and communicate abstract concepts, focusing on the ‘entangled labyrinth.’ This complex form can be imagined as an organised tangle of labyrinthine tunnels that pass through each other without interconnecting. It is not only an abstract concept but exists in nature, providing a template for the formation of butterfly wing scales during metamorphosis. In this painting I’ve explored the metaphorical as well as the mathematical implications of this form by flattening the labyrinthine lattice into a two-dimensional grid and using complementary colours to evoke the shimmer of the butterfly’s wing.
15
Heidi Burgiel
Professor of Mathematics Bridgewater State University Bridgewater, Massachusetts “I practice mathematics because it is beautiful. Highly symmetric tilings, polyhedra and polytopes illustrate discrete groups acting on their native spaces. Colorings of these objects suggest subgroups and quotient spaces. In works like “Hyperbolic Afghan {3, 7}” we see glimpses of the kaleidoscopic landscapes described by algebraists.”
Hyperbolic Afghan {3, 7}
7 x 44 x 44 cm KnitPicks Shine Sport yarn: 60% cotton 40% modal, 2015 “Hyperbolic Afghan {3, 7}” illustrates a tiling of the hyperbolic plane by triangles, 7 at a vertex, in crocheted cotton. Adapting techniques developed by Joshua and Lana Holden, the piece is not assembled from flat triangles but instead approximates constant curvature over its entire surface. Its coloration, inspired by William Thurston’s rendition of the heptagon tiling underlying the Klein quartic, suggests the identifications required to construct that surface as a quotient of the hyperbolic plane.
16
Daniel Raymond Chadwick Artist Scottsville, New York
“I enjoy creating images that employ ideas from mathematical and philosophical sources. My drawings consist of continuous forms, impossible objects, and stretched spaces within which discovery of the content in a new way may lead to the wonder of curiosity.”
The Schrodingers’ Cats
48 x 63 cm Mixed-media on paper, 2014 The Multiple Worlds of the Schrodingers’ Cats Erwin Schrodinger’s wave equation prompted him to write a parable about the cat in a box with a radioactive decay triggered device. He commented that the cat was in a quantum state until one opened the box; i.e., neither alive nor dead. According to the Many-Worlds theory, however, when Schrodinger opens the box the world splits in two; in one world the cat is alive but in another world the cat is dead. No cats were harmed in the production of this drawing.
17
Douglas G. Burkholder free-range mathematician Hadley, MA, USA
“Long-long ago in a place far-far away, my love of art was placed on the back burner as science and mathematics consumed my life. My love for geometry and my desire for visualization for the sake of mathematical understanding have always been central to my teaching and to comprehension of mathematical concepts. However, only decades later, have I begun to explore the artist within and attempted to merge art with my mathematics. Recent work includes searching for beauty and patterns within Conway and Radin’s non-periodic Pinwheel Tiling of the plane with 1 by 2 right triangles.”
A Radin-Conway Pinwheel Lace Sampler 50 x 50 cm Digital Print, 2015
This artwork evolved from a search for beauty and patterns within Conway and Radin’s non-periodic Pinwheel Tiling of the plane by 1x2 right triangles. The Pinwheel tiling can be created by repeatedly subdividing every triangle into five smaller triangles. This lace resulted from alternately subdividing triangles and removing triangles. Triangles are removed based upon their location in the next larger triangle. First, on the macro level, the five distinctive removal rules are applied one to each row. This removal rule is especially easy to see in the bottom row. These same five rules are then applied, on the micro level, to the columns. The remaining triangles form a sampling of twenty-five styles of lace generated by the Pinwheel tiling.
18
A Radin-Conway Quilt 50 x 50 cm Digital Print, 2015
A Radin-Conway Quilt starts by cutting a square with fringe containing almost 1,000,000 triangles from Conway and Radin’s non-periodic Pinwheel Tiling of the plane. Next, we paint about 30% of the triangles by dividing the square into five rows and five columns. A triangle is painted depending only upon its relative location in the next larger triangle of the tiling procedure and upon its row and column. Thus all triangles located in position 1 of their larger triangle are painted in the first row and are painted in the first column, etc. Thus, exactly two out of every five triangles is painted in each of the rectangles of intersections of distinct rows and columns. This provides all 30 combinations of painting the five types of tiles.
19
Anne Burns
Professor Emerita Mathematics Department, Long Island University Huntington, New York “I am interested in the connections between mathematics, art and nature, especially the concept of evolution. Thus my mathematics interests are dynamical systems, differential equations and any area that deals with states that evolve with time.”
Tangent Discs I
30 x 30 cm Digital print, 2015 An iterated Function System consisting of a group of (six) Mobius Transformations acts on six discs, five of which are tangent to the unit circle, to its two neighboring discs and to a sixth disc centered at the origin.
20
Tangent Discs II
36 x 45 cm Digital print, 2015 An iterated Function System consisting of a group of (five) Mobius Transformations acts on five discs, four of which are tangent to the unit circle, to its two neighboring discs and to a fifth disc centered at the origin.
21
Conan Chadbourne San Antonio, Texas
“My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the frequently mystical or cosmological significance that can be attributed to such imagery. Mathematical themes both subtle and overt appear in a broad range of traditional art, from Medieval illuminated manuscripts to Buddhist mandalas, intricate tilings in Islamic architecture to restrained temple geometry paintings in Japan, complex patterns in African textiles to geometric ornament in archaic Greek ceramics. Often this imagery is deeply connected with how these cultures interpret and relate to the cosmos, in much the same way that modern scientific diagrams express a scientific worldview.”
Irreducible Alignment
60 x 60 cm archival inkjet print, 2015 The Fano plane is the smallest finite projective plane, having seven points and seven lines, such that any two points uniquely determine a line and any two lines intersect at exactly one point. The 168 automorphisms of the Fano plane form a group isomorphic to PSL(2,7), the second smallest non-abelian simple group. In this image, the Fano plane is represented at the center of the image, in a projection which emphasizes the equivalence of its seven points (shown as small yellow disks) and its seven lines (shown as blue ellipses). It is surrounded by a frame containing symbols which represent the cycle structures of the 168 permutations in the automorphism group of the Fano plane.
22
Intrinsic Regularity
60 x 60 cm archival inkjet print, 2015 This image presents a visualization of the Steiner triple system S(2,3,7). This system, which is combinatorially equivalent to the Fano plane, consists of seven three-element subsets (or blocks) drawn from a seven element set such that any pair of elements occur in exactly one block, and any two blocks have exactly one element in common.
23
Tiberiu Chelcea Artist Ames, Iowa
“My work investigates the physical and tangible beauty of mass-produced technological/scientific objects and texts, as well as their relationship with our recent past and our outdated ways of doing things. As technological products become obsolete at an increasingly rapid pace, my work uses these recently discarded objects and mines them for new meanings. In particular, the works in the “Trigonopoetry” series creates paintings that incorporate new phrases from old trigonometry textbooks; these phrases are sometimes silly and sometimes profound, but never faithful to the original authoritative voice of the text.”
Terrestrial and Celestial
20 x 25 cm Acrylic on pages from old trigonometry textbook, mounted on wood, 2012 Each work from the “Trigonopoetry” series starts with pages from two trigonometry textbooks from 1920’s. Certain words forming new sentences are masked out on each leaf, and the pages are painted over. The actual composition and style for the paintings are inspired by the illustrations in the books: the style is generally hard-edge, and, for this painting, a diagram from the original pages is actually used in the composition.
24
Meaningful Result (Sometime Later)
20 x 25 cm Acrylic and paint markers on pages from old trigonometry textbook, mounted on wood, 2013 Each work from the “Trigonopoetry” series starts with pages from two trigonometry textbooks from 1920’s. Certain words forming new sentences are masked out on each leaf, and the pages are painted over. The actual composition and style for the paintings are inspired by the illustrations in the books: the style is generally hard-edge, and, for this painting, a diagram from the original pages is actually used in the composition.
25
Sandra DeLozier Coleman
Writer, poet, artist, retired mathematics professor. Niceville, Florida “Poetry, art and mathematics present common aspects to me. In the realm of poetry I find rhyme and rhythm to be as satisfying as free verse and in art I find the pattern and repetition of symmetrical drawing to be as calming as meditation. In both, making the choice to define a pattern and then adhere to the plan once begun, presents just enough of a challenge to force my mind to tune out any unpleasant clamoring of racing thoughts and for a time to let go, absorbed in the flow of the moment. The feeling is not so very different from the feeling of satisfaction that comes from completing a proof or from changing the forms of certain equations so that we can quickly analyze the geometric characteristics of the functions.”
Susurrus
30 x 36 cm Ink on paper, 2015 A delicate circle of symmetry intending to capture the feather-like murmuring sound, the whispering soft repetition of waves washing over a shore, of wind moving blossoming branches, of seagulls at once taking flight.
Teresa Downard
Mathematics Instructor and Artist Mathematics Department, Western Washington University Bellingham, Washington “So far my mathematical artwork has been about capturing some of my favorite mathematical ideas and trying to highlight the relationships present. Now I’m most interested in getting math out of the art, seeing relationships I didn’t notice before. I have a background in both mathematics and art. I practice them separately and together.”
Prime Plaid
27 x 27 cm Technical Pen on Bristol Board, 2015 This piece is about patterns in primes and composites. To setup, we have a 48x48 grid that is numbered starting in the upper left corner. The primes are white, and the composites are shaded. It is similar to a Sieve of Erastosthenes, which finds prime numbers by first removing all the multiples of 2, then the multiples of 3, then the multiples of 5, and so forth. With only the numbers 2,3, and 5, we can find all the primes below 49. 7^2=49 is the first composite not eliminated by this sieve. Notice the pattern in the composites, 11313135153131…, it changes every time we hit a p^2 for prime p. The plaid is created with a twill weave of vertical and horizontal lines with lighter shading when there are five composites in a row.
Jean Constant Consultant Hermay.org Santa Fe, New Mexico
“Mathematics and mathematical visualizations are meaningful at many scientific and technological levels. They are also an endless source of inspiration for artists. The following artworks are part of the 12-30 project – one mathematical image a day for one year, 12 mathematical visualization software, January 1st, 2015 – December 31, 2015. The 365 images portfolio is available at https:// jcdigitaljournal.wordpress.com/ and a printed compilation of the work will be available in the coming months at Hermay.org”
Unfolding of the polytope
46 x 46 x 2 cm Digital print on aluminum, 2015 02 - Unfolding polytope. The bitruncated 5-cell is a 4-dimensional polytope. The vertices of the truncated 5-cell can also be constructed on a hyperplane in 5-space. From an original script by Jeff Weeks for his geometry and topology software.
28
Poincaré’s “Pas de deux”
46 x 46 x 2 cm Digital print on aluminum, 2015 03 - Poincaré’s “Pas de deux”. The Poincare disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk’s boundary. From an original script by Eric W. Weisstein based on a program by Matthew Cook for Wolfram-Mathematica.
29
Francesco De Comité Associate Professor of Computer Science Computer Science Department, Univeristy of Lille, France Lille, France
“Manipulation of digital images, and use of ray-tracing software can help you to concretize mathematical concepts. Either for giving you an idea of how a real object will look or to represent imaginary landscapes only computers can handle. “Things become yet more interesting, when you can transform your two-dimensional dream objects in real three dimensional sculptures. You can then handle your creations, and look at them from an infinity of view angles.”
A Steiner Chain Trapped Inside Two Sets of Villarceau Circles 60 x 80 cm Digital print on cardboard, 2015
Ring cyclides are images of tori under sphere inversion. If certain conditions are fullfilled, a torus can contain a set of tangents spheres. Since the tangency property is preserved by inversion, this set of tangent spheres find its place inside the cyclide.
30
Hypocycloidal Virtual String Art 60 x 80 cm Digital Print on Cardboard, 2015
Hypocycloids are basically two-dimensionnal curves. We can add the third dimension by moving the pen up and down while drawing it. One can then imagine two points moving on this curve, and draw a line between these two points at regular intervals. Playing with the curve and the speed of the moving points makes one explore an infinite variety of shapes.
31
Doug Dunham
Professor of Computer Science Computer Science Department, University of Minnesota - Duluth Duluth, Minnesota “The goal of my art is to create aesthetically pleasing repeating hyperbolic patterns. One way to do this is to place patterns on (connected) triply periodic polyhedra in Euclidean 3-space. These polyhedra are considered to be hyperbolic since the sum of the angles around each vertex is greater than 360 degrees. The first polyhedron below is decorated with a fractal circle pattern. The second polyhedron is decorated with a butterfly motif inspired by M.C. Escher. There is a two-step connection between this polyhedron and a corresponding hyperbolic plane patterns (1) it approximates a triply periodic minimal surfaces, and (2) such a surface has the hyperbolic plane as its universal covering surface.”
A Fractal Circle Pattern on the {3,12} Polyhedron 50 x 30 x 30 cm Printed Cardboard, 2015
This polyhedron is constructed by placing regular octahedra on all the faces of another such octahedron, so there are 12 equilateral triangles about each vertex. Each of the triangular faces has been 90% filled by a fractal pattern of circles provided by John Shier. The polyhedron consists of red and blue “diamond lattice” polyhedra and purple octahedra that connect the red and blue polyhedra. Each of the red and blue polyhedra consists of octahedral “hubs” connected by octahedral “struts”, each hub having 4 struts projecting from alternate faces. The red and blue polyhedra are in dual position with respect to each other - they form interlocking cages. Each purple connector has a red and a blue octahedron on opposite faces.
32
Butterflies on a {3,8} Polyhedron 37 x 37 x 37 cm Printed Cardboard, 2015
Unlike regular tessellations, the number of polygon sides and the number meeting at each vertex do not determine the structure of a triply periodic polyhedron. This polyhedron has the same Schlafli symbol {3,8} as the “diamond lattice” polyhedron described in the figure above, but it is a different polyhedron. It consists of snub cubes of alternating chirality positioned in a cubic lattice connected by their (non-existing) square faces. Thus a “left” snub cube is attached to a neighboring “right” snub cube by a common square. Each of the triangles is filled with red, blue, and yellow butterflies. This pattern was inspired by M.C. Escher’s Regular Division Drawing # 70. It has 3-color symmetry.
33
Frank A. Farris
Associate Professor of Mathematics Santa Clara University San Jose, California “My artistic impulse is to let the beauty of the real world shine into the realm of mathematical patterns. My method combines photographs with complex-valued functions in the plane to create images with all possible types of symmetry: Euclidean, hyperbolic, and spherical symmetries (considered as actions on the plane). For some works, I then transfer plane images back to the sphere. All these methods are explained in detail in my book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, published in 2015 by Princeton University Press.”
Mossy Frogs and Granite Bugs 51 x 61 cm Aluminum Print, 2015
In my book, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, I explain how to construct complex wave functions to create color-reversing symmetry. These frogs and bugs are shaped by waves invariant under the wallpaper group cmm, with color group p4g. Rather than color the waves with a perfectly color-reversing source image, I used a photograph where the two halves are only loosely symmetric, creating approximate color symmetry. The chosen waves could produce a tiling with two congruent shapes, but the asymmetric photo produces two slightly different tiles. I wound the original wallpaper image around the sphere with a complex exponential map and stereographic projection and assembled the ingredients in a collage.
34
An Iris Spiral
51 x 61 cm aluminum print, 2015 I photographed the irises and used complex wave functions to turn the image into a pattern with four-fold rotational symmetry. Then I applied a complex exponential mapping to wind the wallpaper around the complex plane, choosing just the right scaling to make the pattern match, while also creating five-fold symmetry. I bleached an outer ring to bring focus to the center of the spiral and to allow the original photograph of the iris to stand out. Details about wallpaper waves appear in my book, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns.
35
Robert Fathauer
Small business owner, puzzle designer, author, and artist Tessellations Company Phoenix, Arizona “I ’m endlessly fascinated by certain aspects of our world, including symmetry, chaos, and infinity. Mathematics allows me to explore these topics in distinctive artworks that I feel are an intriguing blend of complexity and beauty. The laws that govern our physical universe can be succinctly expressed by mathematical equations. As a result, mathematics can be seen throughout the natural world, and much of my work plays on mathematical forms in nature.”
Dragony Curve 60 x 45 x 3 cm Ceramics, 2015
This sculpture is based on a particular stage in the development of a fractal curve known as the ternary dragon. This ceramic piece has been mounted on a board, with standoffs, partly to make it easier to handle without breaking. The resulting construct could be viewed as either a two-dimensional or three-dimensional artwork, which echoes the manner in which fractal curves can be considered as one-dimensional (a line), two-dimensional (a plane-filling object), or something in between.
36
24 Tendrils
42 x 30 x 30 cm Ceramics and Wood, 2015 This piece combines a ceramic polyhedral form with organic fingers that are formed from willow branches. The polyhedron, known as a pentagonal icositetrahedron, has an opening in each of its 24 faces from which “tendrils” emerge that are suggestive of plant growth or possibly appendages of an insect or sea creature.
37
Gwen Fisher
Artist Bead Infinitum Sunnyvale, California “I painted these pictures using onedimensional cellular automata on a staggered grid of rectangles. I chose designs with a balance between order and chaos. In this series, I hope to achieve a sense of rhythm without exact repeats. “I start each painting by drawing a grid on canvas and choosing a set of colors. I paint the first two rows of pixels to establish my initial state. I choose a set of rules and use my initial state to determine which color to paint each cell in the next row. That row and the previous row determine the colors in the following row, and so on. The left and right sides of the canvas are identified like a cylinder. I add many layers of paint to bring out the depth and interest in the particular design.”
Pixel Painting I, Water Lilies
76 x 51 x 2 cm Acrylic Painting on Canvas, 2015 The particular rule set I used for this piece reminds me of vines with budding flowers. The pattern started to repeat near the top. You can see two copies of part of the pattern, keeping in mind that the design continues over the left and right sides like a cylinder.
38
Pixel Painting 2, Kintsugi
76 x 51 x 2 cm Acrylic Painting on Canvas, 2015 I named this piece “Kintsugi” after the Japanese art of repairing broken pottery with lacquer dusted or mixed with powdered gold in the cracks.
39
Faye Goldman
Origami Artist and Teacher Ardmore, Pennsylvania “I have been doing origami since elementary school. I was drawn to modular origami by its structure and mathematical properties. This is the medium in which I generally work. More recently, I found the Snapology technique by H. Strobl, which allows great creativity with very few rules using only strips of material. I like to use beautiful ribbon which seems to add another dimension to my work. Snapology has allowed me to dig deeply into mathematical shapes. It has provided insights into mathematical concepts and ideas. I recently published ‘Geometric Origami’ which explains the technique. The bottom line is that I make these wonderful works because they look really cool.”
Brown and Green Egg-163
13 x 10 x 10 cm Strips of polypropylene ribbon, 2013 Loosely defined, a ‘Buckyball’ is a polyhedron made of pentagons and hexagons with every vertex of degree three (three edges meeting). Buckyballs must have exactly twelve pentagons. I enjoy creating Buckyballs and their duals. I discovered that if you rearranged the twelve pentagons in a semi-regular pattern you could get interesting shapes. Thus began my series of eggs.
40
Gary Greenfield
Professor Emeritas of Mathematics and Computer Science University of Richmond Richmond, Virginia “Most of my computer generated artworks arise from visualizations of biological or mathematical processes. I want to focus the viewer’s attention on the complexity and intricacy underlying such processes.”
Doubly Intertwined 25 x 25 cm Digital Print, 2015
In general, avoidance drawings arise from several virtual drawing robots (drawbots) executing random walks based on curvature via a model introduced by Chappell subject to the condition that they may cross only their own paths. In Doubly Intertwined two drawbots avoid the canvas boundary, eight invisible barriers and each other but also have a tendency to follow nearby segments of each others paths. As a consequence, the two resulting complexly, intertwined drawbot paths are independently contractible to their respective starting points.
41
Susan Goldstine
Professor of Mathematics St. Mary’s College of Maryland St. Mary’s City, Maryland “For me, the most exciting part of mathematics is communicating it to others. I am especially interested in models that make mathematical concepts tactile or visual. This passion has led me to many artistic projects in the course of my work as a math professor and to some unexpected and delightful collaborations. “I based these artworks on Anne Lorenz-Panzer’s knitted shawl pattern Not a Drop, released under her professional label Arlene’s World of Lace. Her original shawl is rectangular with a distinctive teardrop stitch pattern. Since the drop shapes are formed by bifurcating and then rejoining the vertical lines of stitches, I became curious about ways to decouple the bifurcations and merges to form branched designs.”
Fibonacci Downpour
21 x 26 x 26 cm Merino yarn, cotton thread, embroidery hoop, 2015 In Fibonacci Downpour, the vertical stitch lines branch and form drops following a physical version of the Fibonacci recursion. The number of drops and branchings in each row are consecutive Fibonacci numbers. As the Fibonacci numbers are asymptotically exponential, the fabric falls into a more or less pseudospherical form.
42
Knit Bifurcation
55 x 24 cm Merino yarn, 2015 In Knit Bifurcation, the vertical stitch lines double at regular intervals. The teardrop shapes double as leaves and rain.
43
S. Louise Gould and Franklin Gould
Retired Professor of Mathematics Education, Adjunct Lecturer Central Connecticut State University New Britain, Connecticut “My mathematical art grows out of my experiences with my students and my explorations of mathematics, textiles, paper, and technology. I enjoy working with computer controlled machines such as the computerized embroidery sewing machine and the Craft Robo (plotter cutter) as well as traditional looms and knitting machines.”
Multiplied Snub Cube II 23 x 28 x 28 cm Printed card stock, 2015
This multiplied snub cube illustrates the geometry of the infinite polyhedron 6.3.4.3.3. Conway et al. in “The Symmetry of Things” point out how this surface can be derived from the “mu cube” or from the “mu octahedron” surfaces using the same process by which the snub cube is derived from the cube or from the octahedron. When we “snub” the mu octahedron, we see that as the four hexagons rotate and separate, their planes no longer meet at one vertex but meet, instead, in the 6 edges of a tetrahedron. That gap is then bridged by a square bounded by four triangles. Each of the four triangles shares an edge with one of the hexaons and is also co-planar with it. The resulting surface looks as if it is tiled by six-pointed stars and squares.
44
Multiplied Snub Cube I
19 x 26 x 26 cm Wool Felt and Embroidery Thread, 2015 This multiplied snub cube illustrates the organic or topological features of this Archimedean-like infinite polyhedron. It tiles the hyperbolic plane with regular hexagons, squares and triangles that meet at each vertex in the order 6.3.4.3.3 going clockwise (or alternatively, counterclockwise). Its triply-periodic surface separates Euclidean three-space into two connected congruent parts where the “negative space” is a mirror image of the “positive space.” In this infinite polyhedron the triangles lie in the same planes as the hexagons making them appear to be six-pointed stars connected to each other directly and by squares.
45
Rona Gurkewitz and Bennett Arnstein
Associate Professor of Computer Science Western Connecticut State University Danbury, Connecticut, “I enjoy the “Math without numbers, mostly” quality that makes modular origami polyhedra accessible to mathematicians and nonmathematicians, young and old folders.”
Spike Ball Family, Basic, Super and Super Duper Spike Balls 18 x 18 x 18 cm paper squares, 1990
The original Spike Ball was created by accident when trying to remember another model.This artwork is familiar and new. It looks like a bunch of cootie catchers (fortune tellers) glued together. The modules are folded from squares and contain tabs and pockets that lock together without glue to create the models. The arrangement of modules is based on polyhedra which you can see in the mountain folds, namely a cuboctahedron (12 modules), a rhombicuboctahedron(24 modules) and a truncated cube(53 modules).The modules have four points and they are aramged in groups of four modules(squares) and three modules(triangles). These models have gotten almost three million views on youtube and you can find instructions there.
46
George Hart
Research Professor Stony Brook University Stony Brook, New York “As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic.”
Sword Dancing
32 x 45 x 45 cm Wood (dyed) and cable ties, 2015 This is a model for a large wood sculpture built in February, 2015 at Middlesex University, London, consisting of two congruent but mirror-image orbs of this design, each two meters in diameter. The sixty components of the design are “affine equivalent,” meaning they can be stretched linearly to become congruent to each other. They lie in groups of three in twenty planes---the planes of a regular icosahedron which had been compressed by a factor of 1/2 along a five-fold axis.
47
Edmund Harriss University of Arkansas Fayetteville, Arkansas
“Mathematician, Teacher, Artist, Maker. I like to play with the ways that the arts can reveal the often hidden beauty of mathematics and that mathematics can be used to produce interesting or beautiful art.”
Woven Dodec
20 x 20 x 20 cm Laser cut paper, 2014 32 pieces of paper cut into two shapes connect and weave together to form a ball mixing the dodecahedron and icosahedron. Inspired by Quintron by Bathsheba Grossman.
48
Bowl for Chaim
10 x 54 x 27 cm CNC milled wood, 2015 This bowl was carved with pure mathematics. Each curve for milling was created by formulae and the GCode for milling was created using my own CAMel software. This bowl was made for Chaim Goodman Strauss as he completed his term as chair of the Department of Mathematical Sciences at the University of Arkansas. It is included with his permission.This is a tribute to Euclid, the father of geometry.
49
Maggi Harriss Great Malvern, UK
“I am fascinated by mathematical patterns and enjoy using them to make something useful. I have been teaching mathematics for longer than I care to think about and realised that the best way to understand mathematics is to enjoy it first.”
Ammann Cushion
38 x 38 x 5 cm Cotton Cross-stitch, 2009 Cushion with each tile shape for the Ammann-Beenker tiling in a different colour.
50
Trilobite and Cross Cushion 36 x 37 x 5 cm Silk Cross-stitch, 2015
The edges of Chaim Goodman-Strauss’ Trilobite and cross aperiodic tiles forming an intricate pattern.
51
Andrea Heald
Lecturer Math Department, University of Washington “As a mathematician, I like incorporating mathematical concepts into my work. When doing crochet I like creating works that have some purpose beyond aesthetics. Recently I have become very fond of creating interesting stuffed animals for my young son.”
Collatz Cornsnake
38 x 25 x 5 cm Crocheted Wool Yarn, 2015 The “scale” pattern of this snake was inspired by the Collatz conjecture. Each row of the body was derived from the row before in the following manner: if odd, it was divided by 2, if even it was multiplied by 3 and then 1 was added. Each number is expressed in ternery. Two brown stiches denote a 0, two green stitches denote a 1 and two lavender stitches denote a 2. To distinguish one number from the next, each number ends in a 00 (or 4 brown stitches). The initial round is 215 - 1.
52
Mickey (Shaw) Hubbard Artist Everett, Washington
“My inspirations are drawn from nature, mathematics and science. These inspirations are combined with my own experiences and emotions, creating a union between what is seen, what is known and what is felt internally. As an artist, my goal is to create for the viewer, visually, the concept that art, mathematics and science display a fundamental connection conveying the idea that all three encompass more than what can just be seen. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences.”
Fractals in Symmetry of Fours
25 x 20 cm Pen and ink drawing with computer colorization and enhancements., 2013 Although not a mathematician, my love of things mathematical and geometrical has always found its way into my artwork, often planned, but sometimes unexpectedly. My fascination with mathematical symmetry brought me to creating my own simple symmetric design compositions in pen and ink by creating stencil units which are then revolved around a central point. In planning this design I used a simple symmetry of four, starting with four doubled, creating a base of eight points; then on to four squared for the interior design elements. The drawing was then scanned into my computer where I added color and an abstract fractal background patterning. The final artwork provided a beautifully balanced symmetric unit based on multiples of four.
53
Hartmut F. W. Höft Professor of Computer Science, emeritus Eastern Michigan University Ann Arbor, Michigan
“In my images I visualize artistic expressions such as poetry in two or three dimensions. The pattern of colors in the first image is based on the end rhymes in a sequence of sonnets which are represented by rectangular bands that I place along a trefoil curve. I hope to show the tension between strictly adhering to the structural rules of poems and poets’ breaking them to achieve an expressive or emotional outcome. The second image shows the starting point of my design process for surfaces with a “see-through” scaffold of spirals and parabolas. This helps me evaluate the visual effects as well as expose mathematical characteristics of the surface when I change parameters in its equations.”
A Trefoil of Rhymes in Rilke’s Orpheus 50 x 50 cm Digital print on archival paper, 2015
The rhyme patterns of the sonnets in “Die Sonnette an Orpheus” by Rainer Maria Rilke determine the coloring schema in this image. I made the band 28 strips wide by mirroring the rhyme pattern of the 14-line sonnets. Therefore, the color of the first rhyme sound is at both edges of the surface. I chose the trefoil for this visualization to express the entanglement of the three characters - Orpheus, Euridice & Hades - in the Greek myth to which Rilke alludes. The trefoil curve is defined by the parametric equation 2/3 * ( sin(3t), sin(t) + 2*sin(2t), cos(t) 2*cos(2t) ) causing self-intersections when the rhyme pattern of each mirrored sonnet is drawn in the tangent-normal plane to the trefoil curve at the point where the sonnet is placed.
54
Parabolic Suspension Spindels in a Closed Spiral 50 x 50 cm Digital print on archival paper, 2015
This image shows portions of two curve patterns - parabolas and closed spirals with winding number three - on a single surface. One is a collection of entire spirals at discrete values of the parabolic parameter. The other is a collection of parabolas that are attached at fixed parameter intervals along the spiral; each parabola lies in the horizontal plane of the point of attachment on the spiral. Though not immediately obvious from the equation of this surface, f(s, t) = ( cos(3t) cos(s), sin(3t) cos(2s), sin(t) ), the image shows degeneracy into points as well as self-intersection of the surface. In my imagination the surface curves in this image evoke tension spindels suspending a central band, all protected by a swirling spiral wall.
55
Veronika Irvine & Lenka Suchanek
Graduate Student in Computer Science and Fibre Artist University of Victoria Victoria, British Columbia Bobbin lace, a 500-year-old art form, features delicate patterns formed by alternating braids. Veronika is a PhD candidate in computer science and an amateur lacemaker. She developed a mathematical model to describe bobbin lace tessellations as (D,ζ) where D is a 2-regular directed graph embedded on a torus and ζ is a mapping from the vertices of D to a braid word. From this model, Veronika has generated thousands of original patterns. Lenka, a self-taught lacemaker and designer, brings the antique tradition of handmade bobbin lace to life in pictures, sculptures and wearable art. Specializing in metal lace, she uses fine wires of copper, bronze, steel, and precious metals to create unique pieces that are traditional and modern.
Waves - Offering to the Moon
40 x 36 x 9 cm stainless steel wire, shell, driftwood cedar frame, 2015 “Waves” was designed and created by Lenka using a tessellation pattern generated algorithmically by Veronika. Lenka: “I had a beautiful frame made from old growth, driftwood red cedar and I needed a pattern that would look like the waves of the Pacific Ocean... “The model is an incredible source of designs - every graph has so many variations for working the stitches and each combination results in a different pattern. I love the experimental nature of the work. This is exactly what I was imagining! Now I want to live and work forever, because I will never run out of patterns. “I perceive this piece as the beginning of truly Canadian lace, breaking free from Old World patterns and setting out on its own, independent journey.”
56
Speculations
56 x 56 cm DMC mercerized cotton thread, 2015 These 4 pieces, designed and worked by Veronika, illustrate the use of reflective symmetry to narrow the search for aesthetically interesting patterns. To date, the approximately 500 traditional patterns used in lace have been discovered through trial and error and extensive hands-on experience - a time consuming process. By applying an intelligent search algorithm to discover the D in (D,ζ), over 100,000 workable lace tessellations have been found and we continue to look for ways to expand this search. A search filter was applied to focus on patterns with a high degree of reflective symmetry. This filter made it possible to expand the search, in a reasonable running time, from graphs with 20 vertices to graphs with over 100 vertices.
57
Chia-Chin Tsoo & Bih-Yaw Jin
Research Fellow & Professor National Center for High-Performance Computing, National Applied Research Laboratories, Hsinchu, Taiwan & Department of Chemistry, National Taiwan University, Taipei, Taiwan “We are chemists who are interested in topologically nontrivial structures inspired by fullerenes and graphene. Using the angle-weave technique of mathematical beading, we realize that it is possible to construct robust models of approximate 3D curved surfaces for arbitrary sp2-hybridized graphitic structures with only beads and strings. We show here two bead sculptures, the first model corresponds the 3D embedding of Felix Klein’s all-heptagon network and the second model the hyperbolic soccerball.”
Hyperbolic soccerball
26 x 26 x 26 cm 8mm plastic beads, 2015 Buckyball or molecular soccerball is a spherical fullerene molecule with the formula C60, in which two neighbored pentagons are separated by a single carboncabon bond. Similarly, the hyperbolic soccerbal, or the D168 Schwarzite, is a hyperbolic graphitic structure, in which two neighbored heptagons are also separated by a single carbon-carbon bond. The hyperbolic soccerball is related to the D56 Schwarzite structure by a leapfrog transformation consisting of omnicapping followed by dualization.
58
Bead model of Klein’s all-heptagon network 20 x 20 x 20 cm 8mm plastic beads, 2014
The sculpture represents a periodic graphitic structure which approximates a genus-3 negative curvature D-surface decorated with the Felix Klein’s open network consisting only of full heptagons. A hypothetical carbon allotrope based on this structure can be called a D56 protoschwarzite since there are 24 heptagons and 56 carbon atoms in a unit cell.
59
Karl Kattchee
Associate Professor of Mathematics Mathematics Department, University of Wisconsin-La Crosse La Crosse, Wisconsin “By using mathematics as part of the creative process, I can infuse my art with mathematics without necessarily representing any particular mathematical thing. On the other hand, the mathematical content may be right at the surface. Mathematical art can be conceptual, too. I use pencil, pen, pastel, paper, cardboard, scanner, camera, computer, printer, and metals to achieve my desired effects.”
DA DA Dice 4 x 8 x 4 cm Bronze, 2015
The outcome of a roll of the DA DA dice is a discrete random variable X with a singleton sample space (DADA). The probability of DADA is P(DADA)=1, so the expected value of X is E(X)=DADA.
60
45 Poppies
30 x 45 cm Digital print, 2015 This image is a classification of all closed paths, on a 6x6 grid, with the following properties: First, each path must proceed around the center of the grid and be orthogonal in the sense that every turn is 90 degrees. Also, the path must use each row and column exactly once. Finally, we require that each path be asymmetrical, and we do not distinguish between paths which differ by a rotation or flip. Each center square is colored black, and the shades of red are dictated by the winding number of each region.
61
Margaret Kepner Independent Artist Washington, DC
“I enjoy exploring the possibilities for conveying ideas in new ways, primarily visually. I have a background in mathematics, which provides me with a never-ending supply of subject matter. My lifelong interest in art gives me a vocabulary and references to utilize in my work. I particularly like to combine ideas from seemingly different areas. I coined the term “visysuals” to describe what I do, meaning the “visual expression of systems” through attributes such as color, geometric forms, and patterns. My creative process involves moving back and forth between a math concept that intrigues me and the creation of visual images that interpret that concept in interesting ways.”
Crazy Squares, Fractured Stars 50 x 50 cm Archival Inkjet Print, 2015
A geometric dissection is a subdivision of a shape into pieces that can be reassembled to create a different shape. This design is based on dissections of nine squares into regular polygrams, or stars. For each square, the sub-pieces are colored, and the corresponding star-shape is shown using the colored outlines of the reassembled pieces. For example, the rightmost square is cut into seven subparts; these pieces can be regrouped to form an 8-pointed star, as the line-figure directly to the right illustrates. Visually, the dissected squares resemble the random patchwork patterns that compose traditional Crazy Quilts. In this case, however, the pieces are not so “crazy” – they have a clear purpose, which is to create stars.
62
Catalan Connections: Level Four 40 x 60 cm Archival Inkjet Print, 2015
The Catalan numbers are a sequence of positive integers that provide answers to certain combinatorial questions. For example, in how many ways can a polygon with n+2 sides be cut into triangles? A hexagon (setting n=4) can be triangulated in fourteen different ways, so the 4th Catalan number is 14. Other types of problems also lead to the Catalan numbers: counting binary trees, balancing parentheses, finding paths through a grid, shaking hands in a circle, etc. This piece is composed of diagrams representing seven different problems; for each of these, the answer is the 4th Catalan number. The solution sets for the problems are displayed in diagonal bands. The columns indicate correspondences between elements in different solution sets.
63
Teja Krasek Artist Ljubljana, Slovenia
Teja Krasek’s theoretical, and practical, work is especially focused on symmetry as a linking concept between art and science, and on filling a plane with geometrical shapes, especially those constituting Penrose tilings (rhombs, kites, and darts). The artist’s interest is focused on the shapes’ inner relations, on the relations between the shapes and between the shapes and a regular pentagon. The artworks among others illustrate certain properties, such as golden mean relations, self-similarity, fivefold symmetry, Fibonacci sequence, inward infinity, and perceptual ambiguity. Krasek’s work concentrates on melding art, science, mathematics and technology. She employs contemporary computer technology as well as classical painting techniques.
Penrose Heart
28 x 29 cm Digital Print, 2009/2015 In the artwork we can observe a net of lines and shapes that shows interconnectedness of Penrose tiles from P2 and P3 Penrose tilings (kites and darts, and rhombs), interlaced pentagons, pentagonal stars, and golden triangles. Self-similarity is evident, and golden ratio is used as a scale factor. I dedicated this artwork to the great mathematical physicist, Sir Roger Penrose, the creator of Penrose tilings, and to all who are in love with them.
64
Kevin Lee and Alain Nicolas
nstructor Math/CSCI Normandale Community College Minneapolis Minnesota “For several years I have been writing software to create Escher-like tessellations. The goal of my new program, TesselManiac!, is to have users (especially young ones) create tessellations and explore this connection of math and art. TesselManiac! allows you to create thirty-six types of isohedral tessellations. It includes several animations, including one where the tile morphs from a base polygon tile to the final shape. “I have been exploring techniques to laser cut and engrave tiles out of wood. Different species of wood are used to color the tiling . The tiles vary in thickness to add texture. I have been collaborating with Alain Nicolas who has produced many stunning tessellations using TesselManiac!”
Viking in Wood Relief
29 x 26 x 1 cm Wood: Maple, Cherry, Walnut, Mahogany, 2015 This is my attempt at staging Nicolas’s Vikings as a wood ornament. Different species of wood of varying thickness are used in this three color design. The motif was created by Nicolas in TesselManiac! using Heesch type TCCTCC extended to include an interior mirror through the translated sides.
65
Robert Krawczyk
Professor of Architecture and Associate Dean College of Architecture, Illinois Institute of Technology Chicago, Illinois “These pieces were inspired by the wall drawings of Sol LeWitt and the findings of Father Sebastien Truchet (1657-1729). In 1704, Truchet considered all possible patterns formed by the tiling of right triangles oriented at the four corners of a square. Each tile was identical but could be placed rotated and tiled to form interesting random non-repeating patterns. Since every edge is connected to every other edge, paths are generated. LeWitt on his wall drawings used drawn arcs producing similar patterns. LeWitt instructed his crew to draw the arc in any orientation they wanted. The viewer then finds the paths. The ones included here randomly use two different tiles. Usually these patterns are rendered black and white, these are not.”
More Paths 06_0313_2
50 x 50 x 4 cm framed canvas print, 2015
66
More Paths 06_0913_1
50 x 50 x 4 cm framed canvas print, 2015
67
Heather Ames Lewis Professor of Mathematics Nazareth College Rochester, New York
“I use knitting to explore mathematical ideas. I like the challenge of research and design, and the opportunity to create a tactile article that illustrates mathematics in a fun and interesting way.”
Norwegian Frieze 44 x 51 cm Knitted yarn, 2015
This collection of traditional Norwegian designs – one for each of the seven frieze pattern symmetries – was inspired by photographs of Selbu mittens at the Nordic Heritage Museum in Seattle. There were a wide variety of designs on the cuffs of those mittens, but to find examples of all symmetries I had to expand my search to other parts of Norway. These particular patterns were found in photographs or charts described by Annichen Sibbern Bøhn (Norwegian Knitting Designs), Sue Flanders and Janine Kosel (Norwegian Handknits: Heirloom Designs from Versterheim Museum), Sheila McGregor (Traditional Scandinavian Knitting), and Terri Shea (Selbuvotter: Biography of a Knitting Tradition).
68
Hua-lun Li
Associate Professor Department of Industrial Design, Chung Hua University, Taiwan Hsin Chu, Taiwan “I try to create math works and animations without words (as few words as possible). “My dream is to bring “Proofs Without Words” to life.”
Hyperbolic Paraboloid 20 x 30 x 30 cm paper, 2015
This work demonstrates the following concept 1. Approximation of Curves by polylines 2. level curves (hyperbola) 3. saddle point 4. Asymptotes Only cutting and folding are used in this work, all cutting line is accompanied by two folding lines, one mountain one valley, and both perpendicular to the cutting line. It’s done by using Geogebra, Adobe Illustrator and laser cutter.
69
Yongquan Lu
Mathematical Artist; Computer Science Student CSAIL, MIT Cambridge, Massachusetts “I am a paper artist and am currently completing my Bachelor’s in mathematics and computer science at the Massachusetts Institute of Technology. I love working with paper and have in particular been making paper sliceform artwork since 2010. This is a medium in which paper strips are cut, folded and slotted together to form geometric configurations. These configurations have traditionally been inspired by Islamic star patterns but in theory there is no restriction. Since 2014 I have been working with my advisor Erik Demaine to develop computational tools to aid in the design of such pieces. The two pieces below demonstrate these techniques; while they were hand-assembled, their dimensions were derived via a computer program.”
Brimstone
16 x 16 x 1 cm Paper, 2015 Brimstone is an instance of paper sliceform artwork and has an underlying 1212-3 semiregular tiling. Each dodecagon’s center contains a traditional rosette motif, but I have decorated each motif with a border of diamonds, causing each strip of paper to zig-zag erratically back and forth. The three coloring of the pattern exemplifies the inherent rotational symmetry in this tiling. An interesting fact about this tiling is that there is only one type of strip in the underlying tiling (modulo rotations and translations). If the tiling was extended infinitely, any two paper strips would be indistinguishable. The assembly of this piece from the constituent paper strips took 6 hours.
70
Girih II
16 x 16 x 1 cm Paper, 2015 Girih II is an instance of paper sliceform artwork. The central structure is derived from a traditional Islamic motif, which is noted for its quasiperiodicity (like Penrose tilings!) and is discussed in Lu and Steinhardt’s 2007 paper “Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture”. I have opted to surround the motif with a border of pentagonal stars for aesthetic reasons. Observe that this pattern has 10-fold radial symmetry. There are 6 strips which are closed loops and circle around the piece before closing up; the others all extend radially from the center. The assembly of this piece from the constituent paper strips took 8 hours.
71
James Mai
Professor of Art Illinois State University Normal, Illinois “My recent work has been focused upon the development of sets of geometric forms generated from permutational and combinatorial methods. The forms in each set are at once similar, in that all forms share the same geometric features, and different, in that each form is a unique arrangement of those features. As important, each form-set is both complete, in that all permutations/combinations are present, and non-redundant, in that no two forms are the same, even after rotation or reflection. I employ color, position, scale, and grouping to show the similar and different features among the forms so that the mathematical order may be understood visually, apart from any verbal or mathematical description.”
Primordial (hexagons)
20 x 20 cm archival digital print, 2015 The 7 forms in “Primordial (hexagons)” are the complete set of partitions of the 6 vertices of a regular hexagon. The 6 points in each form are arrayed in a circle/ hexagon, and the 2 colored outlines in each form partition the 6 points in 7 distinct pairs of shapes. These 7 varieties derive from 3 simple partition types: 1 + 5 points, 2 + 4 points, and 3 + 3 points. Discounting rotations and reflections, there is one possible shape arrangement for the 1 + 5 partition (the red + green form), there are 3 possible shape arrangements for the 2 + 4 partition (the yellow + violet forms), and there are 3 possible shape arrangements for the 3 + 3 partition (the blue + orange forms).
72
Familial (octagon circuits) 20 x 20 cm archival digital print, 2015
The 12 forms in “Familial (octagon circuits)” are the complete set of circuits joining the 8 vertices of an octagon, where 4 outer edges of the octagon are preserved and the form possesses either reflective or rotational symmetry. The large white circles indicate symmetry characteristics; from top to bottom: 2 forms are rotational, 6 forms possess 1 axis of reflection, 3 forms possess 2 axes, and 1 form possesses 4 axes. Additionally, this form-set shows 5 distinct arrangements of the 4 outer edges (black lines). These 5 groups are distinguished by colors (blue, orange, yellow, red, green). This group of 12 symmetrical forms is a subset of a larger set of 202 octagon circuit-forms.
73
Vincent J. Matsko
Assistant Professor of Mathematics Mathematics Department, University of San Francisco San Francisco, California “Computer-generated art involves ideas not conceivable before the advent of modern technology. My recent work involves randomness on a large scale, experimentation with color, and integrating thousands, sometime millions, of individual elements in a single composition. “Work in this exhibition explores varying the angles in the well-known recursive algorithm for generating the Koch snowflake. An amazing variety of mathematical and artistic effects can be produced merely by changing the usual angles of 60 and 120 degrees. Varying color and line width creates interesting textures – the challenge is to create a diverse sequence of images from a simple recursive idea.”
Fractal Curve +11 +191 20 x 20 cm Digital print, 2015
The Koch snowflake is a well-known fractal represented by the string “F +60 F -120 F +60 F,” where “F” means move forward by a specified length, “+60” means turn counterclockwise 60 degrees, and “-120” means turn clockwise 120 degrees. Each occurrence of “F” is then recursively replaced with a copy of the string, which is repeated as many levels as desired. When these instructions are carried out graphically, the Koch snowflake is produced. In this piece, the same algorithm is used, but with different angles. The title is taken from the particular choice of angles used to create it. The fractal nature of such curves varies, but as the original algorithm derives from creating a fractal, a liberty is taken in calling this a “fractal image.”
74
Fractal Curve +0 +12 20 x 28 cm Digital print, 2015
75
Dorothy McGuinness Artist Everett, Washington
“After 28 years of exploring the woven form, I‘ve mastered the art of diagonal twill, with which I create forms and structures not normally found in the basketry world. My medium for this unique work is watercolor paper, which I’ve painted and cut into very narrow uniform strips to achieve the precision I seek. I am very much interested in the math and geometric constraints of the work. Using hundreds of strips of paper at a time, I explore new structural forms. The evolution of my body of work is built on taking risks, and avoiding the “known”. The risks offer challenges, which often lead to new directions. This is the excitement that keeps me working in a repetitive medium.”
Dodecahedron
25 x 25 x 25 cm Watercolor paper, acrylic paint, waxed linen thread, 2013 My 1st interpretation of a dodecahedron using diagonal twill to form a woven sculptural basket.
76
Hexahedron 2
38 x 38 x 38 cm Watercolor paper, acrylic paint, polyester thread, 2014 My 2nd interpretation of a hexahedron using diagonal twill to form a woven sculptural basket.
77
Gabriele Meyer
Senior Lecturer Department of Mathematics, University of Wisconsin, Madison Madison “I like to crochet hyperbolic surfaces. They are the intersection of my professional life as a mathematics lecturer and my art hobby. Non-flat surfaces have always presented a technical challenge, yet the most beautiful architecture and sculpture often involve such surfaces. “Hyperbolic surfaces can be naturally achieved by hyperbolic crocheting around shaped plastic line. This was my contribution to the area. The plastic line introduces the tension which makes the surfaces curve in three dimensional space. Its thickness also determines the size of the hyperbolic curves generated. My surfaces are large and best hung from the ceiling to preserve their integrity.”
Ovoid Bead with three Hyperbolic Axes as a Lamp 70 x 48 cm photograph, 2015
This object started as a hollow ovoid, top and bottom missing. I then crocheted three vertical axes down the sides. These three axes are the basis for the hyperbolic crochet. The object is hung from the ceiling. I then inserted a light tube. The photograph was taken in the dark without flash.
78
flaring red algae, lamp 70 x 48 cm photograph, 2015
The object started as a long, hollow cone with two hyperbolically curved axes spiraling down the length of the cone on opposite sides. The algae hangs from the ceiling and is attached in several places. There is a bendable light tube running through the hollow cone at the center of the algae. I took the photograph lying on the floor, in the dark without flash. The holes in the hyperbolic crochet create interesting light patterns on the ceiling.
79
Kerry Mitchell Artist Phoenix, Arizona
“My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals and numerical analysis, and combine them with image processing technology. The resulting images powerfully reflect the beauty of mathematics that is often obscured by dry formulae and analyses. “An overriding theme that encompasses all of my work is the wondrous beauty and complexity that flows from a few, relatively simple, rules. Inherent in this process are feedback and connectivity; these are the elements that generate the patterns. They also demonstrate to me that mathematics is, in many cases, a metaphor for the beauty and complexity in life. This is what I try to capture.”
Affinity 2
Digital print on aluminum panel, 2014 This image was created using a dynamic version of the Chaos Game algorithm. The Chaos Game is a simple example illustrating chaotic motion and strange attractors. It is typically implemented using three anchor points, which become the vertices of a Sierpinski triangle. The Dynamic Chaos Game allows the anchor points to move with each iteration. Here, the image comprises 25 panels. In each, the two anchor points slid along line segments, with different speeds. The parameters of the game varied slightly from panel to panel, to provide aesthetic interest while maintaining coherence. Each pixel was then colored according how frequently that point was visited during the iteration.
80
Shield 1
40 x 40 cm Digital print on aluminum panel, 2014 This image was created using a dynamic version of the Chaos Game algorithm. The Chaos Game is a simple example illustrating chaotic motion and strange attractors. It is typically implemented using three anchor points, which become the vertices of a Sierpinski triangle. The Dynamic Chaos Game allows the anchor points to move with each iteration. Here, the image comprises seven panels. In each, the three anchor points slid along line segments, with different speeds. Each pixel was then colored according how frequently that point was visited during the iteration.
81
Charlene Morrow
Emerita Faculty Member, Psychology and Mathematics Mount Holyoke College South Hadley, Massachusetts “I work mainly in the medium of modular origami. I am motivated to understand mathematical ideas by attempting to express these ideas in visual ways. I am deeply impressed by the ways that visually pleasing art often emerges through the process of trying to gain a deeper understanding of the mathematical ideas I set out to explore. I also find that new questions emerge as I look for ways to express my original questions. The dialectical process of mathematical question – visual expression – mathematical question engages my imagination and allows me to experience both the beauty and the power of mathematics.”
Pentagon Tiling 15 and 3 Evolutions 40 x 40 cm Print on paper, 2015
Colored tiling based on the newly discovered 15th tiling pentagon. The background is constructed to show the primitive 12-tile unit with an overlay to suggest one way to color the tiling so as to disguise the primitive unit. This first work serves as a departure point for exploring three variations. The first evolution uses the primitive unit, but laid out in a non-tiling format so as to create new areas between the units that, when considered themselves, constitute a complete tiling. A subsequent evolution is made with a new 12-tile “twist” unit that does not tile, but does produce a beautiful floral-like design. A third evolution is made by taking an interior space created in evolution two and creating a completely new design.
82
Wing L. Mui
Math Department Chair The Overlake School Redmond, Washington “During the day, I teach geometry to children. When I’m done with that, I make needlework samplers. I am fascinated by the form of traditional needlework samplers, where the artist stitches or embroiders (often exhaustive) collections of symbols and geometric patterns on fabric to demonstrate their skill. But I am also frustrated that the samplers often don’t intend to communicate anything else. So I design patterns based on these traditional forms and use them to intentionally enumerate, demonstrate, or celebrate mathematics.”
Simple Symmetric Square Sampler
25 x 25 cm blackwork embroidery on 18 count Aida cloth, 2015 This blackwork sampler showcases all the possible symmetries that can be stitched on Aida fabric. The outline is the smallest simple perfect squared square, which is 112 stitches on each side and divided into 21 subsquares. The largest square is filled by patterns representing the 7 frieze groups. The 2 smallest squares contain the empty set and a point (a French knot). The last 18 squares contain the 12 wallpaper groups and 6 rosette groups without 3-fold symmetry. Some of the patterns are based on historic designs, though most are original. This piece was inspired by Susan Goldstine’s challenge for a patient reader to crochet a simple perfect squared sqaure in her chapter in Crafting by Concepts, edited by belcastro and Yackel.
83
Andrew Morse Artist Dana Point, California
“In early 2012, Andrew began coding Ruby plugins for SketchUp that produced three-dimensional models based on mathematical functions. His idea was to build visually intriguing models based on equations of three variables, each representing a dimension in space. The same equation was then used to feedback on the initial model and augment its basic shape. This process created stunning, and often unexpected, results that can be seen in each individual piece. Each work of art you see here is based on a mathematical function. Each piece is shown as a 2D representation of a model in 3D space.”
Black Hole
46 x 61 cm Printed digital image on paper, 2012 Programmatically created using the base function z = x3+cos(y).
84
The Bull
46 x 61 cm Printed digital image on paper, 2012 Programmatically created using the base function z = cos(x)*y.
85
Robert Orndorff “My paper folding art is math plus paper. Folded paper is simply tangible math. These particular figures can be folded from most types of paper. No tools are needed. Each is folded from a simple uncut rectangle. Do you listen to music or perform it yourself? Likewise, I invite you to contemplate these figures or fold them yourself. For centuries, artists have used math, technology or special tools. My work is part of that tradition. I will give a talk at the meeting, and will share the crease patterns for these figures with anybody who is interested.”
Five Examples from “Three & Twenty Bivalve Mollusks:” Cosmopolitan Scallop; Giant Clam Nos. 4 and 5; Hypar Seabed; and Geoduck 51 x 28 cm Paper, Straight Pins and Cork, 2012-15
Five small three dimensional figures from my book , mounted like butterflies with straight pins to a cork base in a two dimensional frame. These figures make use of both planar and curved faces. Some of the math behind such curved face figures was only presented for the first time last summer. The behavior at each of the crease pattern vertices is governed by a set of equations. Some vertices fold flat; others do not. Designing figures like these requires not only math but also trial and error, in a process that goes back and forth between the computer and the hands any number of times over days or weeks.
86
OSU Triptych No. 2
20 x 46 cm Paper and Acrylic, 2015 This is a permanent manifestation of an ephemeral artwork, namely, one solution for a specific one-straight-cut problem. Such problems are usually stated as follows: How must one fold a paper rectangle into a flat figure such that one straight cut through all of the layers will produce a given planar straight-line graph? Here the problem has been solved with paper and then represented in acrylic. To a significant degree the work relies on transmitted and reflected light, and so it never looks the same twice. The figure (the letters “OSU”) has been divided into three frames. The crease patterns for the left and right letters are pedestrian but that for the central letter is sublime.
87
David Reimann
Professor of Mathematics and Computer Science Albion College Albion, Michigan “I am interested in creating patterns that convey messages at multiple levels and scales using a wide variety of mathematical elements and media. I enjoy creating art that uses text to form geometric patterns and geometric patterns that form text. I am also very interested in using tessellations and other modular forms as the basis of interesting 2D patterns and 3D shapes.”
Walnut Star
38 x 38 x 38 cm Walnut veneer and brass fasteners, 2015 This form is based on the small rhombicosidodecahedron, an Archimedean solid with 120 edges. The underlying polyhedral edges have been replaced by 4.75 cm squares made from laser-cut paper-backed walnut veneer and connected at their corners with brass split-pin fasteners. The 62 faces (squares, hexagons, and pentagons) and 60 vertices of the underlying polyhedron are transformed into open negative space. The expansion of linear edges into squares results in a sphere-like shape with 20 knobs.
88
Radiance
50 x 50 cm Digital Print, 2015 The artwork depicts 10000 yellow paths emanating from five equidistant starting points. Each path flows away from a starting point and terminates with its endpoint tangent with a ray from the center.
89
Larry Riddle
Professor of Mathematics Agnes Scott College Decatur, Georgia “I have been working with needle crafts since graduate school. I have also been interested in fractals and fractal geometry for more than 20 years. I have I combined these mathematical and artistic interests to create cross stitch and backstitch pieces to illustrate the beauty and mathematics of fractals associated with iterated function systems. As a mathematician I like to seek fractal images that have symmetry or illustrate some interesting mathematical idea. I must be sure that the fractal can be represented accurately on a canvas that permits only vertical, horizontal, and diagonal stitches of a fixed size. Fractals that are built from squares or from lines rotated by multiples of 45 degrees work particularly well.”
Levy Dragon Inside Tapestry
31 x 31 cm Back stitch embroidery on 10 point canvas, 2013 The fractal now known as the Levy Dragon was first studied by Paul Levy in 1938, well before computers were able to draw the image or the name “fractal” had even been invented! The dragon is constructed from a line segment with a basic motif that replaces that segment with two sides of an isosceles right triangle. That motif is then repeated on each of the new sides of the triangles. The Levy Inside Tapestry consists of four copies of the Levy Dragon built from the edges of a square. The iteration steps are repeated on each of the four sides of the square with the initial triangle motif placed inside the square. This back stitch design shows the twelfth iteration for this inside construction.
90
Levy Dragon Outside Tapestry
31 x 31 cm Back stitch embroidery on 18 point canvas, 2013 The Levy Dragon Outside Tapestry consists of four copies of the Levy Dragon built from the edges of a square. The iteration steps are repeated on each of the four sides of the square with the initial triangle motif placed outside the square. This back stitch design shows the twelfth iteration for this outside construction.The outside tapestry was done in two shades of blue to better show each of the four copies of the Levy dragon.
91
Reza Sarhangi
Mathematics Professor Department of Mathematics, Towson University Towson, Maryland “I am interested in Persian geometric art and its historical methods of construction, which I explore using the computer software Geometer’s Sketchpad. I then create digital artworks from these geometric constructions primarily using the computer software PaintShopPro.”
Kokabi Stars 50 x 50 cm Tile, 2015
Kokabi Star (the great pentagram) can be constructed using the lines of the 10/3 star polygon. Patterning this star can be achieved using different approaches. Some of the presented stars in this artwork have been made based on the actual tiling on existing buildings. Some others have been constructed based on old treatises and scrolls. Some of the patterns have been created using the traditional compass-straightedge process. Modularity is another approach in this regard. Moreover, the two decorated quasicrystal patterns of Star and Sun (the only two quasicrystal patterns with global five-fold rotational symmetry) and their striking relationships with Kokabi Star have been presented. Is this relationship a theorem?
92
Henry Segerman
Assistant Professor Department of Mathematics, Oklahoma State Department Stillwater, Oklahoma Henry Segerman is an assistant professor in the Department of Mathematics at Oklahoma State University. His mathematical research is in 3-dimensional geometry and topology, and concepts from those areas often appear in his work. Other artistic interests involve procedural generation, self reference, ambigrams and puzzles.
Stereographic projection (grid)
10 x 9 x 9 cm 3D printed nylon plastic, lamp, 2014 The light rays from the lamp are partly blocked by the shrinking design on the sphere; the resulting shadow is a regular tiling of the plane by squares. This illustrates how stereographic projection transforms the sphere, minus the north pole, into the plane. Note how shapes are slightly distorted near the south pole, and dramatically distorted near the north pole.
93
Saul Schleimer and Henry Segerman
Reader (SS), Assistant Professor (HS) Mathematics Institute, University of Warwick (SS), Department of Mathematics, Oklahoma State University (HS) Coventry, United Kingdom (SS), Stillwater, Oklahoma (HS) Saul Schleimer is a geometric topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and additionally in visualization of ideas from these fields. Henry Segerman is an assistant professor in the Department of Mathematics at Oklahoma State University. His mathematical research is in 3-dimensional geometry and topology, and concepts from those areas often appear in his work. Other artistic interests involve procedural generation, self reference, ambigrams and puzzles.
Klein quartic
17 x 17 x 17 cm 3D printed nylon plastic, lamp, 2015 The Klein quartic K, given by x³y + y³z + z³x = 0, naturally lives in two-dimensional complex projective space. As proved by Klein, the intrinsic conformal structure on K is covered by the (2,3,7) triangle tiling of the hyperbolic plane. The 336 triangles that tile K show that it is the maximally symmetric genus three surface. This sculpture is a projection of the Klein quartic to three-dimensional space. The projection retains a tetrahedral symmetry from the full group of order 336. Our construction is based on a parametrization of K due to Ramanujan. We used a hill-climbing algorithm to search a space of bihomogeneous polynomials to find a projection that balances the surface being embedded against its quasiconformal distortion.
94
(2,3,5) triangle tiling
10 x 9 x 9 cm 3D printed nylon plastic, lamp, 2014 In the plane, the three angles of a triangle must add up to π: 180 degrees. However on the sphere there is a triangle with angles (π/2, π/3, π/5). As shown in the sculpture, the sphere is tiled by 120 copies of this triangle. The LED is positioned at the north pole of the sphere. The resulting shadows are the stereographic projection of the triangles to the plane. Note how the angles of the tiling are faithfully reproduced.
95
Carlo Séquin
Professor of Computer Science University of California Berkeley, California “Stimulated by the LEGO-Knot project, I aimed to design a set of modular parts that permits to compose not only various handle-bodies, but also single-sided surfaces of higher genus. The modular parts employed in my sculptures are tubular 3-way junctions, where one of the tubular stubs exposes the opposite side of the surface shown by the other two stubs. Depending on how the parts are connected, the resulting compositions remain orientable or become singlesided; in the latter case they correspond to connected sums of multiple Klein bottles; which I call “Super-Bottles.” For some assemblies, the resulting surface remains 2-sided (σ = 2); the genus then drops to half the value that it would be for the single-sided Super-Bottle.”
Reconfigurable Super-Bottle of Genus 10/σ
36 x 30 x 28 cm ABS plastic, printed on an FDM machine, 2015 The eight parts demonstrate four different geometries in which one leg of a tubular 3-way junction can be made to switch surface orientation. The eight parts can be put together in hundreds of different ways. In most cases the resulting surface is single-sided (σ = 1), but in a few cases it is still double-sided (σ = 2). The genus of the resulting surface is 10/σ. The configuration shown is a non-orientable surface of genus 10, corresponding to the connected sum of five Klein Bottles, with a total of twelve punctures.
96
Snap-together Super-Bottle of Genus 4/σ
16 x 20 x 14 cm ABS plastic, printed on an FDM machine, 2015 The two identical parts of which the sculpture is composed can be put together in three different ways. In two cases the resulting surface is single-sided (σ = 1) and in the third case it is double-sided (σ = 2). The genus of the resulting surface is 4/σ. The configuration shown is a non-orientable surface of genus 4, corresponding to the connected sum of two Klein Bottles, with two punctures. The insets show the two individual parts and an assembly of them resulting in a 2-hole torus of genus 2 (with two punctures).
97
Ben Sigma
Student San Francisco State University “If a computer is the bicycle for the brain, where are our hang gliders? I am interested in exploring unrealizable spaces across our highest bandwidth interfaces.”
Tessercraft
45 x 45 x 45 cm Interactive VR installation using an Oculus Rift and joystick, 2015 We manage to visualize 3D objects and spaces effectively on a 2D screen. Can we teach ourselves to visualize 4D objects and spaces on a 3D virtual reality viewer? This piece explores whether intuition can be built around higher dimensional spaces through direct interaction with four spatial dimensions and one time dimension. Tesseracts, 16-cells and other 4D polyhedra are visible in a visceral way. A stereoscopic projection maps the fourth spatial dimension to a 3D slice that can be rotated. Color and texture indicate 4D distance and orientation. Nearby objects are easy and intuitive to interact with while distant objects in the 4D direction become translucent but still visible.
98
Anđelka Simić
Teacher of Mathematics and Computer Science High School “Branislav Petronijević” Ub, Serbia “Geometry, which is studied during education, describes very well characteristics of space, but it is limited by direct experience and we meet problems when we try to show threedimensional object on paper. Sooner or later, each of us wonders if there are the fourth, fifth, sixth or higher dimensions. Artists use a different perspectives and techniques in their works in order to show more realistically three - dimensional space and objects in it. I found new dimensions and pictures of my thoughts when I started working with optical ornaments. Now, using optical ornaments on jewelry I can share my experience with others.”
Transparent hypercube - jewelry 3 x 3 x 3 cm Printed transparent foils, 2015
As the basic element we can find only one black-and-white antisymmetric square with the sets of diagonal lines. Using combination of these squares, which are made from transparent foils, we can form cubes very easily. After that, we pack cubes, like we packed squares for cube, and then we form hypercube. I made this transparent hypercube jewelry from cubes by the same principle which we used to make cubes from squares. During the first observation, our perception chooses new forms that are clearly recognizable when we observed the figure. When we add the source of light, like projects at school or disco lights, we clearly see the new dimensions that leave us breathless.
99
Clifford SingerTM Artist Las Vegas, Nevada
“In that I have selected an Inverted Tractrix for the mathematical basis of the new work, it is my intent to revisit Tractrix after numerous years. Tractrix Inverse with Circle represents asymptotic curve or asymptotic line a priori the primary mathematics to formalize an expanse in geometric construction. The geometric construction procedures are basically algorithmic. The collinear intersects enables visual truths in the geometric pictorial field.”
Tractrix Inverse with Circle
30 x 60 cm Digital Print on Vinyl, 2015 © “I work with the one dimension of color, (that of value). Consider white and black, as there is an additive identity and additive inverse property between white and black according to Wittgenstein and Goethe where both are color. White lines on a black field are expressions of a norm that borders between logic and the empirical with the addition of gray value to balance the spatial experience. The following list of color is present in this geometric work: black, white and gray. Basically, black and white anchors the visual field.”
100
Panagiotis Stefanides Chartered Engineer [UK] Hellenic Aerospace Industry [ex] Athens, Greece
“This work, presented to various conferences, is a proposed interpretation of Plato’s Timaeus Scalene Orthogonal Triangle by Panagiotis Stefanides. It is noted here that, a similar, constituent part of this triangle but not the same, is the Kepler triangle discovered by Magirus. Quadrature of the circle by compass and ruler is achieved based on the special quality of this triangle [ a quadrature triangle] and its relationship with circle, the parallelogramme and the square.”
CIRCLE’S QUADRATURE
32 x 23 x 3 cm AUTOCAD DRAWING FRAMED PHOTO, 2009 Autocad used: Geometry and Vector definition by Panagiotis Stefanides assisted for the Computerized AutoCad Drawing by Dr. Giannis Kandylas. More information: http://www.stefanides.gr/Html/GOLDEN_ROOT_SYMMETRIES.htm
101
Andrew James Smith Artist, Retired Installer Design@Riverside, School of Architecture, University of Waterloo Cambridge, Ontario, Canada
“As an artist, for five decades, I have been preoccupied with creating designs based on arrangements of regular polygons. The first of a series (that would quickly span over 300 variations in ten years) came to me one morning while camping in the hills overlooking my home and the Pacific Ocean. It soon became apparent there were two principals to organizing the shapes: 1. each polygon sharing a side, or having a side parallel with the others, or 2. each polygon sharing an apex, or having an apex aligned with others. A few of the designs exhibited both attributes; one I call the Protogon, exhibited a portion of both attributes, but because of its uniqueness failed to be included within my “Five Variables”, until I scanned the whole opus.”
The Protomid Sculpture
43 x 28 cm Photographic Pigment Print on Archival Paper, 2014 Eventually being separated from my paper studio, I once again focused on content. By the new millennium, I started scanning my polygon designs, and eventually the overlooked Protogon. Influenced by working as a gallery installer in the School of Architecture at the University of Waterloo, I began extruding ones I had converted into vector graphics. I first used Google Sketchup on a Mac to project my 2-D version into the three dimensional view I call the Protomid. I then carved a 18 x 21 x 21 inch sculptural version of the Protomid from 24 pieces of basswood, sealed with BIN (a white pigmented shellac) and took this photograph. The ribbon of shared sides clads the Protomid as it wraps around its elevation.
102
Protogon Red Line
43 x 43 cm Digital Pigment Print on Archival Paper, 2013 This view of my Protogon is comprised of a section of the theoretically infinite field of regular polygons with sides of equal length, and shapes that share a side with consecutive shapes. The unique red-lined spiral on the Protogon traces its shared sides. My first sketch of it was just a doodle on lined notebook paper in 1968. Commencing in 1969, I made more accurate drawings of the Protogon and other designs using a pencil, protractor, straight edge, and compass on paper (some of which I made by hand). Then, for three decades, my involvement with these designs was eclipsed by the very technique I was developing to express them - paper making. I had invented systems of color watermarking and held exhibits of my pulp-paintings.
103
Bogdan Soban “Computer generated images created by use of commercial graphics programs are usually recognizable by the style. In order to achieve different results it is necessary to develop proper software. Installation of commonly known software algorithms and formulas in proper programs again does not guarantee diversity. In order to achieve the high diversity of results it is necessary to look for the original methodological approaches and algorithms. My efforts in the recent period is focusing on finding innovative methods of image processing that ensure that digital origin of pictures could be hidden as much as possible. I use exclusively my own programs developed in Visual Basic programming language and some of them are free accessible on my web.”
structure
30 x 40 cm digital print on canvas, 2015 This image was created using multi-level deformation process. Deformation method is based on deformation of previous generated images, which typically consists of geometric objects. The geometric elements are one, two and threedimensional structures, colored by randomly chosen color palette and mostly tinted surfaces. Then the image is elaborated by deformation algorithm, which is the core of such a program. During the elaboration process each pixel is colored with color of one of its neighbor. Which neighbor is chosen is of course the matter of the deformation algorithm calculation. Based on the same basic image, the program can produce an infinite variety of images different to each other using the same base image.
104
eruption
30 x 40 cm digital print on canvas, 2015 This image was created using multi-level algorithmic approach which is characterized by a program built-in multi-level algorithmic structure. At each level there is a set of built-in mathematical expressions and formulas, which are formed more or less intuitive. In these expressions are entering two types of variables: variables which represent the genetic code and are constant for each new program cycle and variables as a result of the process of drawing which currently change their values continuously. The number of possible types of images is equal to the product of the number of algorithms in each level. The programs generally draw abstract paintings and any association to the image of the real world is purely coincidental.
105
Robert Spann Washington, DC
“I am intrigued by analogs between the compositional rules and color theory principles that artists use and the mathematical/statistical properties of images. A digital image is a map from the unit square to a set of k colors. I start with a set of desirable mathematical and statistical properties for an image and then produce maps which have these properties. Currently, I am experimenting with combinations of equations (formed using Discrete Cosine Transforms) that have different symmetries and/ or parities. I use the resulting maps to produce candidate images for further refinement. I then refine these candidate images using digital manipulation based on my own aesthetic judgments.”
Frolic
40 x 50 cm Digital Print, 2015 I start with two maps, f(x,y) and g(x,y); each a map from the unit square onto the set of integers 1,..,n and 1,..,m respectively. Each is constructed using Discrete Cosine Transforms. I combine these two maps to form a map from the unit square onto the integers 1,..,mn. Each integer is assigned a color. In Frolic, f(x,y) is symmetric with respect to a diagonal mirror and is also symmetric with respect to a 180 degree rotation. The function g(x,y) has odd parity with respect to a vertical reflection and no symmetry or parity with respect to horizontal reflections or diagonal mirrors. The result is an image which has structure, but no symmetries. The initial image is further refined based on my own aesthetic judgments.
106
Time Travel
40 x 50 cm Digital Print, 2015 I start with two maps, f(x,y) and g(x,y); each is a map from the unit square onto the set of integers 1,..,n and 1,..,m respectively. Each is constructed using Discrete Cosine Transforms. I combine these two maps to form a map from the unit square onto the integers 1,..,mn. Each integer is assigned a color. In Time Travel f(x,y) has odd parity with respect to a diagonal axis. The function g(x,y) has even parity with respect to horizontal reflections, odd parity with respect to vertical reflections and odd parity when rotated through 180 degrees. When the resulting image is rotated 180 degrees, the shapes do not change and each color is mapped to a neighboring hue. There are no symmetries with respect to horizontal or vertical reflections.
107
Sarah Stengle Visual Artist Saint Paul, Minnesota
“Apollonius of Perga did highly sophisticated mathematics with simple drawing tools while exploring ratio, proportion, and intersection—all of which I associate with visual art. I decided to use the imagery found in his work on Conics in a series of drawings called Postcards from Perga in which I am imposing Apollonius of Perga’s mathematical work onto postcards. My intention is to integrate the postcard image with the lines of the proof in such a way as to evoke emotions rarely associated with mathematical proofs, or for that matter, postcards. Mathematical imagery from the classical era is nearly timeless, and acts as a gentle foil, quietly amplifying the postcard’s ephemeral nature.”
Postcard from Perga, Book 1, Proposition 54: Asbury Park Boardwalk 20 x 30 cm Ink and pencil on vintage postcard, 2015
This is part of an ongoing project in which I am imposing Apollonius of Perga’s mathematical work with conics onto postcards. This image from Asbury Park is postmarked 1915, and the proof chosen for superimposition is an extremely symmetrical and calm image inscribed within a circle. Without defining the connotations, the image can be seen to resemble celestial charts of bygone eras, the crosshairs of a gun site, a kite within a kite, or a sort of ancient religious symbol. There is stillness to this image. The symmetry of the superimposed proof amplifies that sense of stillness and intensifies the strangely static nature of the crowds at the beach in this postcard image from exactly one hundred years ago.
108
Anna Ursyn
Professor University of Northern Colorado Greeley, Colorado “My computer graphics explorations serve as a point of departure for a series of prints or sculptures. I explore the dynamic factor of line. “Generative art results in precise images with perfect lines that follow premeditated transformations. I started working with computers by programming. “I could include color, shade, patterns, apply clipping algorithms, rotate and paste content into other images, zoom and transform. Then, photosilkscreen and photolithograph gave me a new level of color combinations.”
Shrine of Knowledge
10 x 8 cm wood and 3d print, as archival print, 2015 Platonic forms and Platonic shapes are juxtaposed with a model of the Universe created by Johannes Kepler, who used it to depict his vision of how the Universe could look like. This composition underlies the basic forms seen as divine forms by Plato.
109
Cliff & Danny Stoll Chief Bottle Washer Acme Klein Bottle Oakland, California
“We make Klein bottles and other topological manifolds for the math community. Much fun - mathematicians suggest designs, and we create glass shapes. Recently, we’ve been playing with linked glass Klein bottles and other objects. A half-twist Mobius loop can be mapped onto a Mobius loop with three twists. This works for Klein Bottles as well: By connecting any odd number of Klein Bottles, you’ll make a shape that’s homeomorphic to a Klein bottle. What happens if you try make a Mobius Loop with two half twists? Or connect an even number of Klein bottles? Now that’s a different story...”
Two Linked Klein Bottles 28 x 10 x 10 cm Glass (borosilicate), 2013
Two Klein bottles linked together do not form a Klein Bottle! This double-Klein bottle is two sided and orientable ... it’s homeomorphic to a torus! It’s made from two “standard” glass Klein bottles, cut using a diamond saw, and connected by welding in a glass lathe. This was commissioned by cosmologist Paul Steinhardt for the cover of his book, “Endless Universe: Beyond the Big Bang”
110
Triple Klein Bottle (externally linked) 34 x 34 x 15 cm Borosilicate Glass, 2014
Three connected Klein Bottles, linked externally to form a single manifold -- this is homeomorphic to a “standard” R3 immersed Klein Bottle. We built this by welding together three stock Klein bottles. The borosilicate glass has a low coefficient of expansion, so we can apply a torch to one seam without creating expansion cracks. This weighs about 500 grams, and has 3mm thick walls. Like our other Klein bottles, this is R3 immersed (not embedded), Riemannian, unbounded, affine, Hausdorffian, compact, and nonorientable.
111
Paul Watkins
Student Austin Peay State University Clarksville, Tennessee “I am currently an undergraduate student studying art and mathematics. Ten years ago, I was studying computer programming which easily led to a study of math. Years later, I was introduced to computer glitching as an art form. Today, I mostly do work in glitch art and gif animation, with a focus on internet culture and based on the relentless bombardment of input we experience in modern daily life. As I continue to study mathematics, my ideas are often heavily influenced by the math around us everywhere.”
Trefoil
10 x 35 x 30 cm Gypsum cement cast, 2015 While moving into my upper level undergraduate studies, knot theory has become an intriguing topic. This work is quite blatantly inspired by the images I’ve seen introduced early in the theory. I have sought to make ideas tangible and introduce knot theory to people typically uninterested in the math behind it by creating several knot works recently. After several attempts, I was able to construct this solid trefoil knot cast from a mix of plaster and cement.
112
Anduriel Widmark Artist Denver, Colorado
“I create forms to explore the possibilities existing within a generated reality. Shaping and manipulating space allows a context for relationships to be questioned. Abstraction expands reality and presents an opportunity to look outside of a regular pattern of seeing. Relationships between underlying forces exemplify the inseparability of structure and narration. Structures and voids are used to organize a lattice around these distinctions and connections.”
Selenited Hyperboloid
18 x 18 x 18 cm flame worked borosilicate glass, selenite sphere, 2015 A simple hyperboloid, made of straight lines, is holding a sphere made from selenite. The selenite and doubly-ruled glass surface highlight intriguing patterns and a dynamic space. This hyperboloid consists of 12 lines welded together to create a perfect space to cradle the sphere, highlighting the structure’s overall simplicity and elegance.
113
Elizabeth Whiteley Studio Artist Washington DC
“As a continuation of my explorations in the geometry of pattern design, I am studying line group symmetries. “Textbook examples tend to use friezes to demonstrate the seven line groups. Those examples do not answer a basic question for artists: How do you turn the corner? That is, if you want to create an illustrated rectangular border based on the symmetry of a particular line group, what happens at the vertices? “My research uncovered solutions in a book on tessellations for quilt makers. I applied that guidance to borders on my original drawings. The corners are transitional focal points which indicate a change of axial direction as the viewer’s eye moves horizontally or vertically around the border.”
Ginko 1
51 x 41 cm Silverpoint Drawing on Coated Paper, 2015 This original hand-drawn image includes a border which illustrates Group t (sometimes noted as p) from the seven line groups. The generator, or fundamental region, is a 2-leaf motif formed of a diagonal mirror image of a single leaf, within an implied square. The operation is translation of that generator. A property of the Group t border is that in order to have the corner turn symmetrically, there must be a mirror of the generator at the center of each of the 4 sides. The corners will then match. My drawing process began by coating the paper with a metalpoint ground based on a Renaissance bone ash formula. I drew with a stylus of sterling silver wire. The rough texture of the ground allowed tiny particles of metal to adhere.
114
Calladium 1
41 x 51 cm Silverpoint Drawing on Coated Paper, 2015 This original hand-drawn image includes a border which illustrates Group t2mm (sometimes noted as p2mm) from the seven line groups. The generator, or fundamental region, is a single leaf which becomes a 4-leaf motif. It demonstrates three reflections. Curiously, there is the optical illusion that the generators moving vertically are different from the generators moving horizontally. In fact, they are identical. My drawing process began by coating the paper with a metalpoint ground based on a Renaissance bone ash formula. For the drawing, I used a stylus of sterling silver wire. The rough texture of the ground allowed tiny particles of metal to adhere.
115
Shirley Yap
Associate Professor of Mathematics California State University East Bay San Francisco, California “I enjoy the technical and artistic challenge of bringing mathematics out of pages and computer screens and into people’s hands. Installations that are interactive and exploratory are particularly inspiring. Although I use computer aided design programs and mathematical software to create both two and three dimensional pieces, I often begin creating with equations and sketches on paper. My media include wood, metal, clay, and plastic.”
Leaves of Glass : Bernstein polynomials
5 x 20 x 20 cm Etched acrylic mounted on stained birch, 2015 Bernstein polynomials form a basis with many interesting properties, including the ability to be defined recursively in an intuitive way. This fact, as well as a theorem of Mabry which computes the envelope of a family of Bernstein curves, led me to experiment with the polynomials within certain parameters. The results were surprisingly beautiful, and led to a familiar curve stitching model as well as similar, but more complex, envelopes given by quadratics and cubics. I designed the piece and built the wood mounting to facilitate interaction and exploration of the piece by the viewer.
116
Jing Zhou
Associate Professor of Art and Design Monmouth Univeristy West Long Branch, New Jersey “As a Chinese artist living in the Western world, I am aware of art, literature, philosophy, and mythology from both cultures. Developing a personal visual language that expresses universal ideas, I create artworks for the stories and aesthetics of each project, and for making visible those concepts reflecting my personal experiences. My artwork explores our common humanity, diverse society, and my inner voyage. Creating artwork required me to realize my nature, restudy my culture, and adapt new thinking, which resulted in a new perspective on life. It has challenged me to constantly solve visual problems, learn new techniques, and explore the splendid human heritage. My artistic creation is a process of deciphering my life journey.”
Randomness {Ch’an Mind Zen Mind Digital Print Series} 45 x 30 cm Digital Print on Paper, 2012
Inspired by the space-time uncertainty principle in String Theory and Quantum Mechanics, the symbolic elements in this image represent different energies: the earth, mankind, and this ever-changing universe. As all life originates from water, the x-ray image of a spiral tun shell in this image signifies the earth and mother nature. The shell is mapped by mathematical patterns. Indicating human civilizations and irregularities of nature, hundreds of colorful random dots are actually part of the desert irrigation from the Middle East. Horizontally flowing cross the space, many crescent shapes symbolize the uncertain state of subatomic particles.
117
