# Mathematical ImageryMathematicians and artists create strong, stunning, works in all media and explore the visualization of mathematics

# 2018 Mathematical Art Exhibition

The 2018 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in San Diego, CA. Here on Mathematical Imagery is a selection of the works in various media, including recipients of the 2018 Mathematical Art Exhibition Awards: "A Gooseberry/Fibonacci Spiral," by Frank A Farris, awarded Best photograph, painting, or print; "Dodecahedral 11-Hole Torus," by David Honda, awarded Best textile, sculpture, or other medium; and "Excentrica" by Ekaterina Lukasheva, Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form.

40 x 40 x 10 cm, plastic, 2017. I was inspired by the Leonardo da Vinci codex, and I realized the polyhedra illustrated by Leonardo da Vinci. The polyhedra represent both Renaissance elements of surprise and modern sculpture, highlighting the relevance of Leonardo and his artistic-intellectual context in the present time. Filippo Brunelleschi invented perspective during the Renaissance period, when the use of polyhedrons and other geometric forms became more frequent, and Mazzocchio, the popular Florentine headwear of the time, also became an icon of perspective. --- Nedeljko and Milka Adžić

40 x 45 cm, inkjet print on canvas, 2017. Twenty colored spheres are surrounded by ten greenish, slinky-like toroidal helices. The helices also follow paths traced out by linear combinations of the normal and binormal vectors to the curve joining the centers of the spheres. This makes a kind of a symbiotic-geometric relationship between the solid objects and the surrounding safety net. Do you see a pattern for the colors of the spheres? --- Dan Bach

50 x 70 cm, digital print, custom software, 2017. This piece derives from work I did to generalize the so-called chaos game, which yields a Sierpinski gasket, to any regular polygon. It is a montage of fractal and near-fractal images that are the natural product of such a generalization, especially when that is cast into code. The stained-glass-like background, an "off-resonance" hexagonal analog to the stochastically produced Sierpinski gasket, is behind an "inverse" pentagonal analog, and a flare from the paths taken between chosen points of an inverse and off-resonance heptagonal analog. --- Tom Bates

24 x 12 x 12 cm, copper mesh, paper mâché, 2016. This Seifert Surface is an orientable surface whose boundary is the trefoil knot. The sculpture is based on a copper mesh frame with paper mâché and acrylic paint. Contrasting black and blue colors are used to highlight the distinct sides of this orientable surface. In creating a physical model, I found a deeper appreciation for the particular character of the trefoil knot. --- Elliott A. Best

61 x 51 cm, ink on paper, 2017. I enjoy seeking out interesting intersections between mathematics, computer science, and art. This image is an embedding of natural imagery video of flowers and a butterfly, using Locally Linear Embedding (LLE) a nonlinear dimensionality reduction technique, introduced by Saul & Roweis in 2000. LLE, an unsupervised machine learning algorithm, was applied to five frames of creative commons video of a butterfly flapping it wings surrounded by flowers utilizing open source software. -- Linda Beverly

50 x 50 cm, acrylics on canvas, 2017. This artwork mixes three math problems: It is a Simple Imperfect Squared Square of order 21, colored using the Four-Color Map Theorem and design and painted with Symmetry; the 21 squares are not of different sizes which makes it imperfect, and simple because no subset of the squares forms a rectangle or a square; and the Four-Color Map Theorem states that any map in a plane can be painted using four colors, so that regions sharing a common boundary do not share the same color. The main square has rotational Symmetry of order 4, since the tiling is invariant when rotated by 90 degrees. --- Regina Bittencourt

44 x 44 x 44 cm, wood and brass, 2017. Counting suggests that the complete bipartite graph K(5,5) might be rigid even though it has no triangles. Ben Roth and I proved that in "When is a bipartite graph a rigid framework?" (Pacific Journal of Mathematics, 90, 1981). The graph has one more edge than the number of degrees of freedom, so can be built as a tensegrity structure. In this realization four of the five vertices from each set form a tetrahedron with the fifth vertex near its center. The tetrahedra overlap so that the center vertex of each is inside the other. Eight struts join each center vertex to the four vertices of the other tetrahedron. The remaining 17 edges are cables. --- Ethan Bolker

17 x 17 x 17 cm, styrofoam, yarn, thread, 2017. I was surprised and delighted to discover a truncated octahedron when I first drew the graph whose vertices are permutations on four elements and whose edges connect permutations that swap adjacent elements (the Cayley graph of $S_4$, generated by (12), (23), and (34)). I had it in the back of my mind for years that I'd like to find a visual representation of this graph, and when I started learning temari--a Chinese/Japanese craft of embroidery on yarn/thread balls -- I realized this medium would work well. This ball is my second effort, and I am currently working on a third--experimenting with different color combinations. --- Debra K. Borkovitz

9 x 28 cm, digital print on paper, 2017. A triptych of Hamiltonian-cycle portraits of William Rowan Hamilton, Lin-Manuel Miranda, and Linda Hamilton. Each has 4096 vertices. Each was produced by solving a 4096-city TSP. The optimal tours were found using the Concorde TSP Solver. --- Robert Bosch

41 x 55 cm, digital print, 2017, This is a polynomial function sin strange attractor. I created this piece in Chaoscope. I render at max iterations of 4294967295 when using this program and this design was rendered in plasma. I have used this program for 10 years now and I am always inspired to create new variations with beautiful colors and effects. My Chaoscope images are always unique as I study the images already out there and strive to always make mine different. --- Carol Branch

20 x 20 x 20 cm, nylon, glass, 2017. M.C. Escher Circle Limit woodcuts are based on a symmetry groups generated by inversions in 3 circles. The limit set of such a group is simple circle. Limit set of a group generated by inversions in spheres may be much more complicated. 4 generators groups may have rather intricate limit set which however always lies on a sphere. Only groups with 5 or more generators may have truly three dimensional structure of the limit set. To find the limit set we map each point into corresponding point in the fundamental domain. The inverse of the stretching factor of that transformation measures the distance of the point to the limit set. The isosurface of the distance data is converted into triangle mesh and 3D printed in nylon. --- Vladimir Bulatov

50 x 50 cm, digital art, 2017. Stellated Penrose is a pseudo-Penrose tiling of the small stellated dodecahedron containing 95,844 half-kites and 59,220 half-darts colored either red or black. The tiling starts from 60 half-kites each covering one isosceles triangle on the surface. The tiles are then subdivided into half-kites and half-darts for eight iterations. The final step is to label every tile A, B, C, D, or E based upon its location in the subdivision of the larger tile and then use paint-by-number to paint the tiles on each of the twelve pyramid points. Ten of the pyramid points have two of the five tile types painted red and two pyramid points have only one type painted red. For example, every tile labeled either B or E in the top pyramid point is painted red. --- Douglas G. Burkholder

60 x 60 cm, archival digital print, 2017. A Steiner quadruple system of order N, abbreviated as SQS(N), is an arrangement of N symbols into blocks of four such that every triple of symbols occurs in exactly one block. A Steiner triple system of order N-1, STS(N-1), can be derived from a SQS(N) by considering all the blocks which have a particular symbol in common, and then eliminating that symbol from those blocks. This image presents a visualization of the unique SQS(8), with an arrangement that emphasizes the construction of the derived STS(7). The inner ring of quadruples all have a single symbol (shown in grey) in common. The remaining three symbols in these seven blocks form the unique STS(7). --- Conan Chadbourne

50 x 60 cm, ink on paper, 2017. Calculated Chaos is a hand-drawn conglomeration of lines with at-a-glance symmetry. The obvious algorithmic repetitions suggest order and direction, maybe even the idea that equations could be found to generate all of the curves, but unlike computer-generated forms where equations precede the images, this somewhat chaotic image is full of deviations from precision that make movement from image to equations a mind-boggling challenge. --- Sandra DeLozier Coleman

30 x 30 x 30 cm, plastic (vinyl), 2017. This artwork is based on the observation that any polyhedron edge is connected to exactly 4 other edges (by definition, two edges are connected if they share both a face and a vertex). The flexible pieces with 4 connection points, viewed as polyhedron edges, can thus model any polyhedron. Multilayer sculptures can be created by the addition of a fifth connection point at the center of each piece which allows connections between layers. Each new layer models the rectification of the polyhedron modeled by the previous layer and has twice as many pieces. This work has four layers corresponding to a tetrahedron, octahedron, cuboctahedron, and rhombicuboctahedron, respectively; it has 6+12+24+48=90 cross-shaped pieces and 96 (red) disks. --- Mircea Draghicescu

Best photograph, painting, or print - 2018 Mathematical Art Exhibition

51 x 51 cm, digital print on aluminum, 2017. A twist on John Edmark's spirals, this pattern winds a walllpaper pattern of type p31m around the plane with the complex exponential map to create a Fibonacci spiral. The mathematical underpinnings involve a Fibonacci-like sequence of Eisenstein integers, which then determine a lattice of frequency vectors for wallpaper waves that will land correctly in the winding. The pattern is selected by "tuning" the waves: adjusting frequencies and amplitudes to find a beautiful pattern. The Western (or Sierra) Gooseberry tastes about like the eastern one, which is translucent and green, but ripens to a deep red and is covered in thorns, which make it quite inconvenient to pick. The delicious jelly is a longtime family tradition. --- Frank A. Farris

5" x 11" x 9", ceramics, 2017. My work explores the mathematics of symmetry, fractals, tessellations and more, blending it with plant and animal forms as well as inorganic forms found in nature. This negative-curvature surface is an inspired by the shapes polyclad flatworms take on in open water. It can also be thought of as a piece of a hyperbolic plane embedded in Euclidean 3-space. --- Robert Fathauer

45 x 45 x 2 cm, birds-eye maple veneer, stained and finished with tung oil, 2017. The flowing birds-eye maple grain, in contrast to the rigid, symmetric, laser-cut depiction of the polygon and vertices, hints at the topological freedom permitted in deforming the polygon and gluing the edges to obtain the embedding of a graph in a compact surface. Gluing the 90-gon's vertices and edges according to the numbering yields the complete graph on 10 vertices. Including the polygon interior, the result is an embedding in a compact orientable surface with 10 vertices, 45 edges, and 1 face. The Euler characteristic and genus relationship V-E+F = 2-2g yields g = 18. Thus the embedding is in an 18-hole torus. Since the graph's complement is the single open-disk polygon interior, this is a maximal 2-cell embedding. --- Robert Franzosa

30 x 30 x 15 cm, polypropylene ribbon, 2016. I love using the Snapology technique to make polyhedra. This series of models, based on Platonic solids, is built using only rhombi. The grouping of rhombi, which represent each face of the base solid, share a vertex (star) in the center made of 2n rhombi, where n is the number of sides of the model's Platonic face. An additional 2n rhombi are added around the vertex. Then a final rhombus is added to each edge. The vertices of the initial polygon have 3 or 4 rhombi. I find it is fascinating that with these simple rules, vertices appear with 3, 4, and 5 rhombi. --- Faye E. Goldman

37 x 30 x 20 cm, glass and crystal beads, thread, clasp, ear wires, 2017. In our paper "Building a better bracelet: wallpaper patterns in bead crochet," Ellie Baker and I prove that 13 of the 17 wallpaper groups generate designs in bead crochet rope, but two of them only occur in fairly trivial patterns. This seamless necklace smoothly transitions between designs for all 13 groups. The earrings (inner loop), bracelet (outer loop), and clasp (back) are bead woven with planar wallpaper designs that generate the patterns on the necklace. "Serpentine Symmetries" is handmade and contains over 7700 beads. --- Susan Goldstine

15 x 20 cm, digital print, 2017. I label the edges of a 30 x 40 grid with zeros and ones. I treat each row of edges and each column of edges as inputs to a chaotic one-dimensional cellular automaton. After 500 iterations of the automata I use the edge labels to assign hexadecimal digits to the cells. I assign colors to the digits to create a composition. I generate artworks by repeating the process 25,000 times invoking a hill-climbing algorithm to maximize the number of occurrence of the hex digits 5 (brown) and 9 (yellow) in the central region. The choice of digits induces rules governing how browns and yellows can be positioned with respect to each other. Rule 30 is used for rows of edges. Rule 54 is used for columns of edges. --- Gary Greenfield

43 x 23 x 23 cm, wire, 2017. The interpretation of mathematical ideas, nature, and the relations and transitions between them, are at the core of my work as an artist. "Kleine Fröschlein" is a study of a Klein bottle, a perfect playground for a couple of frogs to have some fun with a game of tag. Will they ever catch each other? --- Zdeňka Guadarrama

12 x 12 cm, cotton thread, 2012. This trivet consists of 12 rings. Each ring has the same number of consecutive segments as the corresponding Fibonacci with the first two rings being solid and the last having 144 distinct segments. Colors for segments were chosen so that no two adjacent segments have the same color. --- Andrea Heald

210 x 150 cm, cotton fabric, 2017. This design begins with an elementary geometric concept that illustrates the relationships between areas of squares formed from the sides of a right triangle. Iterating this concept leads to a beautiful fractal which, when looking deeper and deeper, leads to an abundance of beauty and intrigue. What area of the quilt is covered by purple fabric, by blue fabric, ... by red fabric? This quilt showcases the beauty that is possible through angles: a rainbow of color and a pattern of squares. --- Natalie Hobson

Best textile, sculpture, or other medium - 2018 Mathematical Art Exhibition

34 x 37 x 37 cm, cardstock paper, 2016. The inspiration for this piece was to create a model with an internal structure. One of the challenges with my work is that as my pieces have gotten larger, the issue of weight vs. support has become an issue. I wanted to see if I could build something that was mathematically and visually pleasing, but also structurally sound. The overall shape is based upon a dodecahedron. At the center of each face of the dodecahedron, the surface sinks inward like a funnel. Each of the 12 funnels joins at the center with another, smaller dodecahedron-based structure. Topologically speaking, the piece can also be considered an 11-holed torus. --- David Honda

50 x 50 cm, archival inkjet print, 2017. A knight's tour on an 8x8 grid is a path that visits every cell once and is made up entirely of knight's moves. If a point on the tour is chosen as "1", continuing along the path assigning consecutive numbers produces an 8x8 array of numbers from 1 to 64. In certain cases, the array turns out to be "magic." Some closed knight's tours can be numbered magically in several ways; a tour discovered by Jaenisch generates five different Magic Knight Tours. This piece shows these five MKTs expressed in a color-coded base 8 system. The numbering for each of the magic squares begins with the small black circle and ends with the larger one. The geometric path is shown in the remaining four squares, colored to correspond to the central magic square. --- Margaret Kepner

20 x 30 cm, digital print, 2016. In the wonderful, mysterious, and complex realms of chaos and strange attractors a seeker can find delicate, beautiful, and sometimes even very heartfelt phenomena... My artwork is dedicated to our two amazing friends, Dr. Raymond M. Smullyan (1919 – 2017), and Dr. Reza Sarhangi (1952 – 2016) who will live in our hearts forever. --- Teja Krasek

Honorable Mention - 2018 Mathematical Art Exhibition

52 x 52 x 5 cm, paper, 2017. Origami tessellations are complex geometrical 3-d structures. These surfaces are made using origami technique, which means only one sheet of paper is folded without stretching, cutting or gluing. This piece represents the result of continuous isometric mapping of the flat surface to a 3-dimensional surface. It's hard to believe, but it can be stretched back to a flat sheet at any time. --- Ekaterina Lukasheva

22 x 32 cm, computer art, 2016. Mathematics is a way to see through nature; it is a way to think of nature. We can see nature and derive math, but also we can do math and derive nature's shapes. This "parametric still life" (still life = "natura morta" in Italian), where each image is defined by a parametric equation, may show that mathematics through art is not only still living, but strongly living! --- Maria Mannone

28 x 43 x 2 cm, computer generated image printed on glossy paper, 2014. This artwork came out of my dissertation which involved proving theorems related to the Hilbert Metric. The Hilbert metric is a special way to define the distance between points. The Hilbert metric applied to the interiors of these two triangles results in a hyperbolic non-Euclidean geometry. With respect to this hyperbolic geometry the lines coming out of these vertices are parallel (non-intersecting). To our eyes, those lines do not look parallel, unless perhaps we see them as being parallel lines going off into the distance, drawn in perspective. As we look at the various patterns formed by the lines and the coloring, our mind tries to make sense of the complexity; our perception shifts between seeing hexagons (which are circles with respect to the Hilbert metric), parallel lines, and the occasional parallel-piped. --- Chris McCarthy

60 x 40 x 40 cm, yarn and shaped line, 2017. This piece started as an ellipsoid. I then crocheted a band spiraling around it from top to bottom. Eventually this band became hyperbolic. Solutions to differential equations can look like this. Or the object could look like a seed pod. --- Gabriele Meyer

41 x 51 cm, digital print onto aluminum panel, 2017. This image is a manipulation of a photograph of Jupiter's red spot, taken by the Juno spacecraft in July 2017. The image was taken at perijove, the point in the orbit when the spacecraft is closest to the surface of the planet, which gives the image its title. The manipulation was achieved using an iterated discrete form of a complex nonlinear partial differential equation. The form of the equation was chosen for aesthetics, yet the resulting structures suggest the intricate fluid dynamics present in the planet's giant storm. --- Kerry Mitchell

9 x 9 x 8 cm, 3D printed nylon plastic, 2015. A spine of a three-dimensional manifold with boundary is a two-dimensional complex that the manifold deformation retracts to. Here, we show the trefoil knot, together with a spine of its complement in the three-sphere, stereographically projected to euclidean space. The windows form a distorted rectangular grid, with all angles 90 degrees. In one grid direction the windows lie along semicircles, each with both ends on the vertical axis. In the other grid direction, the windows trace out trefoil knots. The only exception is the windows meeting the dual circle to the vertical axis. This design was suggested to us by Dylan Thurston. --- Saul Schleimer and Henry Segerman

24 x 24 x 24 cm, 3D-print, PLA plastic, 2017. In this sculpture, 24 of the "4-stub Dyck funnels" have been aligned with the 24 edges of a rhombic dodecahedron, and their stubs have been connected with 48 tunnels. This yields a surface of genus 50 -- the equivalent of the connected sum of 25 Klein bottles with 24 punctures, exhibiting the 24-fold symmetry of the oriented cube. It took 132 hours to build this model on a LULZBOT 3D-printer. Support removal took several more hours. --- Carlo Séquin

One of our ideas concerns transitions from order to disorder. Going linearly from one to the other can seem jarring, so we use modifying functions that are 0 for some interval, then sweep up to 1 for an equal interval. This is the form of all the modifying functions in this image. The orientations of the squares start out upright and the positions are regular, then they become more random on the right. Similarly, the edges start out straight then become curved. There are two sets of paths through 3-dimensional RGB color space, the paths of one set start on the left at white and the other at black. Then they proceed toward random mid-range colors, so that at the right the color distributions of the two sets become the same. --- John Shier and Doug Dunham

18 x 18 x 5 cm, wooden beads on copper wire, 2017. Wooden beads are made into 32 pentagons: 8 in the inner ring and 24 in the outer ring. Four pentagons meet at every vertex creating the folds in this {5, 4} hyperbolic surface. The undulating surface shares an organic beauty with the variations of color in the wooden beads. While the copper wire's ability to hold the beads in this irregular space adds a sense of motion to the piece. The visual simplicity of the 8 pentagons in the inner ring contrasts with the density of the outer ring, which has 3 times as many pentagons. --- Stacy Speyer

14 x 14 x 14 cm, 3cm glass bugle beads, 2014. An octet-truss lattice structure with face centered cubic (fcc) crystal symmetry is a truss-like space frame based on the stacking of tetrahedra and octahedra in a ratio of 2:1. The octet-truss space frame is strong and lightweight because of the inherent rigidity of their fundamental building motifs. The bugle bead model presented represents a finite skeletal structure according to the fcc close packing and consists of a central smaller cuboctahedron (brown bugle beads) and an outer layer of frequency-two cuboctahedron (yellow bugle beads). --- Chia-Chin Tsoo and Bih-Yaw Jin

50 x 50 cm, silver gelatin photographic print from 4" x 5" negative, 2017. Light painting of an LED attached to a Blackburn pendulum strung to produce orthogonal periods of motion in a 4-to-3 ratio. Musically, this would represent a major fourth interval. Exposure was made looking up in total darkness using a 4 x 5 inch sheet film camera. Total exposure time was 10 minutes 48 seconds, during which time the camera was rotated 90 degrees on a very slow turntable. Because the formula for the period of motion of a pendulum contains an angular dependence, the Blackburn pendulum goes in and out of resonance as the amplitudes decrease, resulting in interesting patterns that, to me, are quite beautiful. --- Paul Wainwright

6 x 6 x 6 cm, flame worked borosilicate glass, 2016. Art is a good excuse to play. Exploring patterns and symmetry through abstraction presents an opportunity to look outside of a regular pattern of seeing. "Möbius Shorts" is similar, but distinct from the Möbius strip. Here, "Möbius Shorts" is outlined with glass rods that hold an invisible surface that is topologically equivalent to a klein bottle with a hole in it. This one-sided surface seems to hold rings that float in glass thus revealing its simple and elegant shape. --- Anduriel Widmark