Nonplanar expansions of polyhedral edges in Platonic and Archimedean solids

David A. Reimann
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)
Pages 143–150 Regular Papers


The process of replacing each edge of a regular polyhedron with a square results in the creation of a new object, similar to the process of Stott expansion. However, following the edge to square transformation, the resulting object's surface no longer has genus zero. In some cases, the object also contains bumps or craters to accommodate the additional length of material. This process can be generalized to any polyhedral form having equal length edges, such as Platonic solids, Archimedean solids, prisms, and anti-prisms. Examples are shown for these particular classes of polyhedra using a variety of materials and symmetries.