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 ands, 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.

This is an action model; pull his head and he fiddles his bass. --- Robert J. Lang Jul 05, 2017

"The Temple of the Peach," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 16", 2016

Who visits the Temple of the Peach? My first step in going there was to paste the Hyperbolic Peach pattern that I used to illustrate non-Euclidean geometry around the inside of a cylinder, the temple wall. The giant globe holder with its peach medallion holds a sphere decorated with another pattern made from the same peach photograph, which is propped on the floor. Photoshop allows me to tell that floor that it should be highly reflective. I had to let the ray-tracing software run for a day to produce this scene. --- Frank A. Farris Jul 05, 2017

"Pinecones from White Bark Vista," by Frank A. Farris, Santa Clara University, CA. Courtesy of Princeton University PressDigital print on aluminum, 20" x 20", 2013

Part of the enjoyment of wallpaper patterns is the way your mind knows how to continue the pattern outside the given frame, in both the left/right and up/down directions. (When mathematicians use the word wallpaper, we just mean any pattern with translational symmetry in two independent directions.) The undulating flip-and-slide symmetry in the up/down direction makes this pattern type one of my favorites. It is symmetric, but not overly so. Here we see alternating axes of glide reflections that are not related by
translation. This pattern type is called pg by the International Union of Crystallographers. --- Frank A. Farris Jul 05, 2017

"Puzzle Pieces from a Sierra Sunset," by Frank A. Farris, Santa Clara University, CA. Courtesy of Princeton University PressDigital print on aluminum, 20" x 16", 2012

Photographs with rather minimal variation (and artistic value) can turn into beautiful patterns. Can you find the color-reversing symmetry? If you turn the pink puzzle pieces 90°, they match the green ones with the same shapes but opposite colors. The inset at the lower right shows that my source photograph is actually a collage, combining the original sunset photo with its negative, rotated upside down. Again, the black band is considered a neutral color to separate positive and negative colors. --- Frank A. Farris Jul 05, 2017

"Mossy Frogs and Granite Bugs Spiral on a Globe," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 24" x 20", 2015

The pattern in the background of this image does not exactly have color-reversing symmetry. The source photograph of granite and moss is only vaguely color-reversing when you turn it upside down: the greens turn into grays. Still, when I used it with a formula that would yield color-reversing symmetry, it led to the two similar-but-different shapes: frogs and bugs. Then I wound the pattern onto a sphere in a spiral pattern. I added the purple haze by hand, using Photoshop. --- Frank A. Farris Jul 05, 2017

"Icosahedral Lampflower," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 16", 2016

This image tests the limits of ray-tracing, with a virtual lamp inside a partially transparent, icosahedrally painted globe placed atop a pumpkin stem in a crystal vase. Two opposing slightly curved reflective walls create a festival of reflections and projections. The source of the colors is my now-familiar stained glass window. --- Frank A. Farris Jul 05, 2017

"A Frieze Sampler," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 24", 2016

Friezes take their name from an architectural feature: a band of decoration typically along the top of a wall. In the theory of plane symmetry, the first famous classification result says that there exist exactly seven types of frieze patterns. Every pattern ever constructed by anyone, as long as it repeats exactly along one direction, can be classified as belonging to one of these seven types. Mathematicians, like many humans, like to collect exactly one artifact of each type. Here is my sampler of seven friezes, made from source photographs of California wildflowers. I especially like the fourth one down, which exemplifies a glide reflection symmetry—a flip-and-slide motion that leaves the pattern unchanged. --- Frank A. Farris Jul 05, 2017

"Autumn Moths from a Hike in the Hills," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 20", 2016

The big surprise about wallpaper patterns is that there are exactly 17 types. After making many examples of each type, I still return to the type of this pattern as a favorite; it’s called p31m. Notice the three-fold rotational symmetry, but also the alternation of mirror symmetry with three-way pinwheels. For me, this gives a beautiful balance of sameness and variation. It’s symmetric, yet organic. I took the source photograph on an autumn day, hiking in the golden hills above San Jose, CA. This piece shows that interesting patterns can arise from photographs with rather limited color palettes. --- Frank A. Farris Jul 05, 2017

"The Alchemist's Shelf," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 51 x 61 cm, 2017

The rich colors of the alchemist's globes come from scenes captured with a camera that records the entire visual sphere. They tell my story of leaving home in San Jose for an autumn sabbatical in Minnesota. At the top left, we see my stairwell at home, where a stained glass window made by Hans Schepker hangs next to a quilt. Moving to the right, we find the glorious Lakewood Chapel and St. John's Abbey. Bright fall colors at the bottom left give way to another view of St. John's Abbey, and then we return home. Texture maps on the spheres use quotients of harmonic polynomials with icosahedral or octahedral symmetry. The scene is composed in Photoshop, with about 14 hours of ray-tracing. --- Frank A. Farris Jul 05, 2017

"Torus," by Jiangmei Wu (Indiana University, Bloomington)Best textile, sculpture, or other medium - 2017 Mathematical Art Exhibition

45 x 45 x 20 cm, Hi-tec Kozo paper, 2014

"Torus" is folded from one single sheet of uncut paper. Gauss’s Theorema Egregium states that the Gaussian curvature of a surface doesn’t change if one bends the surface without stretching it. Therefore, the isometric embedding from a flat square or rectangle to a torus is impossible. The famous Hévéa Torus is the first computerized visualization of Nash Problem: isometric embedding of a flat square to a torus of C1 continuity without cutting and stretching. Interestingly, the solution presented in Hévéa Torus uses fractal hierarchy of corrugations that are similar to pleats in fabric and folds in origami. In my Torus, isometric embedding of a flat rectangle to a torus of C0 continuity is obtained by using periodic waterbomb tessellation. --- Jiangmei Wu

May 09, 2017

"Animal Heads - Wolf," by Chris Watson (Tessellation Art, Prague, Czech Republic)50 x 50 x 5 cm, digital print on canvas, 2016

"Animal Heads - Wolf" is made from hundreds of tessellating wolves. It uses a mosaic technique that I developed specifically to create mathematical artwork. As with traditional square mosaics, each tile is a different colour. In this case, the individual tiles are howling wolves. Look closely to see the detail on each, or take a step backwards and the wolf head is revealed. --- Chris Watson

May 09, 2017

"4-Wheel Decomposition," by Ally Stacey (Oregon State University, Corvallis)45 x 60 cm, Crayola crayon and Sharpie, 2015

Learning math inspires me to make art which in turn helps me better understand the math. This work is essentially a change-of-basis calculation going from the space of Closed Jacobi diagrams (trivalent graphs with an outer circle) to the space of Chord Diagrams, which are trivalent graphs with all vertices on the outer circle. These algebras are important to the field of Vassiliev Knot Invariants. This is done via what's the called the STU relation which is a way of resolving internal vertices. All algebraic steps are shown in the foreground. Making these helps me keep track of calculations in my research in an aesthetically pleasing way. --- Ally Stacey May 09, 2017