A Computer Aided Geometric Model of a Ten-Plane Polyhedral Transformation

Robert McDermott and Will Hawkins
Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture (2008)
Pages 171–178


A computer aided geometric model for a polyhedral transformation was written in C++, OpenGL and GLUT. This geometric model allows a user to interactively change the shape of a closed convex polyhedron derived from 10 face planes in a fundamental region. Given different inputs, a different shaped polyhedron was output. For some inputs, the output was a Platonic or Archimedean polyhedron. For other inputs a more complex polyhedron was output. This transformation is based on a fundamental region that is formed by 3 symmetry planes intersecting at the centroid of a polyhedron. This fundamental region is combined with 3 different sets of transformations to produce a polyhedron for either a tetrahedral, an octahedral, or an icosahedral family of polyhedra. Using the interactive software, a polyhedron transforms smoothly and continuously from a regular polyhedron, to a truncate polyhedron, to a dual polyhedron, to a stellate polyhedron along with many transitional polyhedra.