Transformations of Vertices, Edges and Faces to Derive Polyhedra

Robert McDermott
Bridges London: Mathematics, Music, Art, Architecture, Culture (2006)
Pages 359–364


Three geometric transformations produced a large number of polyhedra, each originating from an initial polyhedron. In the first transformation, vertices were slid along edges and across faces producing nested polyhedra. A second transformation produced dual polyhedra, whereby edges of the initial polyhedron were rotated and scaled and the end points of these edges derived the dual polyhedra. In a third transformation, faces of an initial polyhedron were rotated and scaled producing snub polyhedra. The vertices of these rotated and scaled faces were used to derive other polyhedra. This geometric approach which derives new vertices from previous vertices, edges and faces, produced precise results. A CD-ROM accompanying this paper contains three animations and data for all the derived polyhedra. This CD-ROM can be obtained by sending me email.