Media Highlights

Abstract

Frank Farris is a well-known expositor on symmetry and art, and author of Creating Symmetry, The Artful Mathematics of Wallpaper Patterns (Princeton University Press, 2015). Farris describes his journey that began with an Illustrating Mathematics semester at the Institute for Computational and Experimental Research in Mathematics, where he learned to use Rhino, a program designed for architects, and Grasshopper, a plug-in to create shapes from formulas. He first designed the planar profile of a knot in the complex plane, then used Grasshopper to create a two-dimensional array of identical knots with their centers at the Eisenstein integers. These are complex numbers of the form a + bω, where ω = −1/2 + √3i/2 and a and b are integers. To design the 7-Color Torus, a digital print shown on the cover of the Math Horizons issue, Farris exploited the property that every Eisenstein integer yields one of 7 remainders when divided by 2 − ω. This allowed him to color the knots with 7 colors, according to the congruence classes of their centers. Four of the nearest centers with a specified color form a parallelogram, and a torus is the quotient of the plane by vectors that span the parallelogram. As each knot is surrounded by 6 knots of different colors, this construction shows that at least 7 colors must be used to color a map on a torus. The final figure shows an attractive scarf designed by mathematical artist Ellie Baker, based on the 7-color pattern in the article. PR