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.

"Wisdom," by Stephen RenThis image is centered on -1.5798, and is magnified 6500x from the original fractal. It is generated with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen RenNov 22, 2017

"Rolling Flames," by Stephen RenThis image is centered on 0.9717 + 2.21i, and depicts an endless sequence of “compound-star” figures. This image was produced with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren Nov 22, 2017

"Pyramid," by Stephen RenThis image is centered on -4.125, magnified 40x from the original fractal. The overall shape of this figure is just barely visible in the complete depiction, but, as shown in this image, it is heavily laced with detail. This figure is particularly interesting because it is set apart from the rest of the “main” fractal body, and further so far I had not found it repeated elsewhere in the fractal. This image was created with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren

Nov 22, 2017

"Negative Flower," by Stephen RenThis image is centered on -2.5, and is focused on the somewhat hexagonal figure found primarily to the left of the imaginary axis. This is magnified 3x from the overall image. This image was generated with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren Nov 22, 2017

"Heart of the Fractal," by Stephen RenThis image is centered on 0.170938545, and is magnified 6000x from the original fractal. This figure can be found repeated in many areas along the real axis, though it is often deeply blurred in a larger figure. This image is generated with 400 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren Nov 22, 2017

"Compound Star," by Stephen RenThis image is centered on 0.9073 + 2.354i, and is magnified 50x from the original fractal. This figure shows up repeatedly within the fractal, not only in isolation, but often within in other patterns and figures. This particular pattern bears resemblance to the Kneser’s Chi-Star function. This image was created with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren Nov 22, 2017

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