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Incomplete Open Platonic Solids

Mikael Vejdemo-Johansson

Abstract

Sol LeWitt famously enumerated all the incomplete open cubes, finding 122 of these connected, non-planar subsets of the edges of the cube. Since then, while several projects have revisited the cube enumeration, no such enumeration has been published for any other interesting solid. In this paper we present work on enumerating all the incomplete open platonic solids, finding 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2\,423\,206 dodecahedra and 16\,096\,166 icosahedra.

Incomplete Open Platonic Solids

Abstract

Sol LeWitt famously enumerated all the incomplete open cubes, finding 122 of these connected, non-planar subsets of the edges of the cube. Since then, while several projects have revisited the cube enumeration, no such enumeration has been published for any other interesting solid. In this paper we present work on enumerating all the incomplete open platonic solids, finding 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2\,423\,206 dodecahedra and 16\,096\,166 icosahedra.
Paper Structure (10 sections, 2 figures, 1 table)

This paper contains 10 sections, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ruleset: what is an incomplete open polyhedron?
  3. Enumeration
  4. Subset encoding
  5. Symmetry groups and their actions on edges and subsets
  6. Putting everything together computationally
  7. Connected
  8. Planar
  9. All the incomplete open platonic solids
  10. Discussion

Figures (2)

  • Figure 1: Exhibition poster for Sol LeWitt's Variations on Incomplete Open Cubes from a show at the John Weber Gallery in 1974.
  • Figure 2: Number of different incomplete polyhedra by edge-count for each. The left graph shows the tetrahedron (), the cube () and the octahedron () and the right graph shows the dodecahedron () and the icosahedron ().