Visualizing Affine Regular, Area-Preserving Decompositions of Irregular 3D Pentagons and Heptagons

Douglas G. Burkholder
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 333–336 Short Papers


We demonstrate a simple, elegant, and visual method for decomposing irregular pentagons into a pair of affine images of the two distinct types of regular pentagons. Moreover, the sum of the area of the two affine images equals the area of the original irregular pentagon. Similarly, we decompose irregular heptagons into a triple of affine images of the three distinct types of regular heptagons. One can use these decompositions to design visually interesting sculptures reflecting these geometric relationships.