Which triangulations of the sphere with maximum degree 6 can be realized as non-intersecting polyhedra with unit edges? Take a look below.

This is work in collaboration with Peter Doyle and Zili Wang, using my code and work on the realization of abstract simplicial complexes with specified edge lengths.

Triangulations of the sphere generated using Gunnar Brinkmann and Brendan McKay's plantri.

Visualization following Lee Stemkowski's wonderful examples.

4 vertices (1 shape)

5 vertices (1 shape)

6 vertices (2 shapes)

7 vertices (5 shapes)

8 vertices (10 shapes) manual proof of shape 3 by computer

9 vertices (15 shapes)

10 vertices (30 shapes)

11 vertices (44 shapes)

12 vertices (77 shapes) manual proof of shape 21 by computer

13 vertices (115 shapes) manual proof of shape 74 by computer

14 vertices (184 shapes) manual proofs of shapes 6 and 124

15 vertices (267 shapes) manual proofs of shapes 8, 194 and 248

16 vertices (420 shapes) manual proofs of shapes 6, 7, 8, 9, 11, 163, 311 and 328

17 vertices (595 shapes) manual proofs of shapes 8, 9, 10, 16 and 484

18 vertices (883 shapes) manual proofs of shapes 6, 23, 722 and 728

19 vertices (1242 shapes) manual proofs of shapes 38, 204 and 408

20 vertices (1783 shapes) manual proofs of shapes 6, 79, 250, 280 and 1028

21 vertices (2445 shapes) manual proofs of shapes 437, 1016, 1019 and 1848

22 vertices (3443 shapes) manual proofs of shapes 39, 759, 1649, 1652, 1653 and 2913

23 vertices (4622 shapes) manual proofs of shapes 133, 1468, 2004, 2657, 2658, 2752 and 4096