Incomplete Open Platonic Solids

• Published in 2026
• Added on 2026-02-26

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.

Links

• https://arxiv.org/abs/2602.20425v1
• https://arxiv.org/pdf/2602.20425v1
• https://incomplete-open.mikael.johanssons.org/

Other information

BibTeX entry

@article{IncompleteOpenPlatonicSolids,
	key = {IncompleteOpenPlatonicSolids},
	type = {article},
	title = {Incomplete Open Platonic Solids},
	author = {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.},
	comment = {},
	date_added = {2026-02-26},
	date_published = {2026-02-26},
	urls = {https://arxiv.org/abs/2602.20425v1,https://arxiv.org/pdf/2602.20425v1,https://incomplete-open.mikael.johanssons.org/},
	collections = {combinatorics,lists-and-catalogues,things-to-make-and-do},
	url = {https://arxiv.org/abs/2602.20425v1 https://arxiv.org/pdf/2602.20425v1 https://incomplete-open.mikael.johanssons.org/},
	urldate = {2026-02-26},
	year = 2026,
	archivePrefix = {arXiv},
	eprint = {2602.20425},
	primaryClass = {math.CO}
}