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Polyhedral Painting with Group Averaging
Mathematics and Computer Science
  • Frank A. Farris, Santa Clara University
  • Ryan Tsao
Document Type
Publication Date
Taylor & Francis
The technique of group-averaging produces colorings of a sphere that have the symmetries of various polyhedra. The concepts are accessible at the undergraduate level, without being well-known in typical courses on algebra or geometry. The material makes an excellent discovery project, especially for students with some background in computer science; indeed, this is where the authors first worked through the material, as teacher and student, producing a previously unseen type of artistic image. The process uses a photograph as a palette, whose colors and textures appear in kaleidoscopic form on the surface of a sphere. We depict tetrahedral, octahedral, and icosahedral symmetries, with and without mirrors, along with the source photograph for comparison. We also describe a method to make images with color-reversing symmetry.

This is an Accepted Manuscript of an article published by Taylor & Francis in PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies in v. 26 no. 4 in April 2016, available online:

Citation Information
Farris, Frank A., and Ryan Tsao. "Polyhedral Painting with Group Averaging."Primus 26.4 (April 2016): 312-24.