Incomplete Open Platonic Solids

TL;DR

This paper extends the enumeration of incomplete open solids from cubes to all Platonic solids, revealing the number of such structures for each solid type with significant computational effort.

Contribution

It provides the first comprehensive enumeration of incomplete open Platonic solids beyond cubes, including detailed counts for tetrahedra, octahedra, dodecahedra, and icosahedra.

Findings

122 incomplete open cubes, matching LeWitt's enumeration

6 incomplete open tetrahedra

185 incomplete open octahedra

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.