Frank A Farris

Professor of Mathematics and Computer Science
Santa Clara University
San Jose, California, USA

Inspired by joining the Illustrating Mathematics semester at the Institute for Computational and Experimental Research in Mathematics (ICERM), I am interested in promoting the role of mathematical art in the broader community. Mathematical artists do more than reach out to non-mathematicians. We make important contributions to mathematical research, exposition, and education. Recent work involves creating shapes invariant under various group actions using Grasshopper in Rhino. The shapes are then staged in scenes with texture mapping and ray tracing, or printed as sculptures.

Three Manifestations of Polyhedral Symmetry
13 x 10 x 5 cm
3D printed steel with golden coating

Three small sculptures illustrate how various rotational polyhedral symmetries can be exhibited in real objects, without any straight lines or corners. The smallest and largest balls display rotational icosahedral symmetry. These are constructed from 10 bands, each created to have D3 symmetry (like a curvy triangle that is symmetric when you turn it over), with full twists in the bands of the larger sculpture. The medium-sized sculpture has rotational tetrahedral symmetry. Made from only 3 bands, each with D2 symmetry, it is basically a set of Borromean Rings with a full twist in each one. Some models are available at cost from Shapeways. An article explaining how you can make these will soon appear in the JMA.

Polyhedral Symmetries from Twisted Bands
41 x 32 cm
Digital print

Each of the three types of polyhedral rotational symmetry---icosahedral, octahedral, and tetrahedral---is illustrated with a complex of twisted bands. I created the shapes in Rhino, using the Grasshopper plug-in, then used Photoshop to apply wood grain texture maps. Ray tracing produced the three inset images, which I then arranged in a collage. An article explaining how you can make these will soon appear in the Journal of Mathematics and the Arts. Two of the shapes appear as 3D-prints in the photograph, "Three Manifestations of Polyhedral Symmetry."