The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

"Parametric Natura Morta," by Maria Mannone (University of Minnesota, Minneapolis)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 Apr 27, 2018

"Energy-D," by Ekaterina Lukasheva (San Jose, CA)52 x 52 x 5 cm, paper, 2017

Honorable Mention - 2018 Mathematical Art Exhibition

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 Apr 27, 2018

"A Heart for Raymond and Reza," by Teja Krasek (Ljubljana, Slovenia)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 Apr 27, 2018

"The Five Faces of Jaenisch," by Margaret Kepner (Washington, DC)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

Apr 27, 2018

"Dodecahedral 11-Hole Torus," by David Honda (Marshall Middle School, San Diego, CA) 34 x 37 x 37 cm, cardstock paper, 2016

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

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 Apr 27, 2018

"Pythagoras' Rainbow Quilt," by Natalie Hobson (Sonoma State University, Rohnert Park, CA)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 Apr 27, 2018

"Fibonacci Trivet," by Andrea Heald (University of Washington, Seattle)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 Apr 27, 2018

"Kleine Fröschlein," by Zdeňka Guadarrama (Rockhurst University, Kansas City, MO)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 Apr 27, 2018

"Harnessing Chaos #5-9-53023," by Gary Greenfield (professor emeritus, University of Richmond, VA)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 Apr 27, 2018

"Serpentine Symmetries," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City)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 Apr 27, 2018

"Series of Rhombic Polyhedra," by Faye E Goldman (Ardmore, PA)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 Apr 27, 2018

"The Maximum Genus of the Complete Graph on Ten Vertices is Eighteen," by Robert Franzosa (University of Maine, Orono)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 Apr 27, 2018

"Flatworm Form No. 3," by Robert Fathauer (Tessellations Company, Phoenix, AZ)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 Apr 27, 2018

"A Gooseberry/Fibonacci Spiral," by Frank A Farris (Santa Clara University, Santa Clara, CA) 51 x 51 cm, digital print on aluminum, 2017

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

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 Apr 27, 2018

"Spiky ball," by Mircea Draghicescu (Portland, OR)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 Apr 27, 2018