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.
"Pinecones from White Bark Vista," by Frank A. Farris, Santa Clara University, CA. Courtesy of Princeton University Press
Digital 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