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.

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"Mazzocchio by Leonardo da Vinci," by Nedeljko and Milka Adžić (Novi Sad, Serbia)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

"Slinky Spheres," by Dan Bach (Oakland, CA)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

"Chaos Game Cut 6," by Tom Bates (Santa Barbara, CA)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

"Bounded by the Trefoil," by Elliott A. Best (University of California Riverside)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

"A Butterfly Through Time," by Linda Beverly (California State University East Bay, Hayward)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

"The Symmetric Four-Color Simple Imperfect Squared Square," by Regina Bittencourt (Santiago, Chile)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

"K(5,5)," by Ethan Bolker (professor emeritus, UMass Boston)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

"Temari Permutation Ball," by Debra K. Borkovitz (Wheelock College, Boston, MA)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

"Three Hamilton Cycles of the Complete Graph on 4096 Vertices," by Robert Bosch (Oberlin College, Oberlin, OH)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

"Jelly Fish," by Carol Branch (Carlsbad, CA)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

"Limit Set #170922," by Vladimir Bulatov (Shapeways, Corvallis, OR)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

"Stellated Penrose," by Douglas G. Burkholder (Lenoir-Rhyne University, Hickory, NC)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

"Extended Regularity," by Conan Chadbourne (San Antonio, TX)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

"Calculated Chaos," by Sandra DeLozier Coleman (Niceville, FL)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

"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