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I've recently studied rolling polyhedra graphs. A polyhedron is rolled over its edges until it has reached all possible orientations. One more roll puts it back where it started. Here's the 120 states of the rolling dodecahedron in a closed path. Each vertex is the center of a pentagon.

rolling dodecahedron

What is a simple closed path that goes through all 120 orientations for a rolling dodecahedron? Failing that... a simpler path.

Some examples of simple closed rolling polyhedra paths are below. The first path gives all orientations for both the icosahedron and the gyroelongated square dipyramid. The next two work for the triaugmented triangular prism and snub disphenoid.

rolling paths

One solving method would be to make a simple grid of pentagons, possibly overlapping, and analyze random paths, eliminating those that repeat an orientation. And if the grid has no solutions, trying a new grid. For the cube, a 2x4 grid of squares works. I don't know if the below grid would work.

pentgrid

Here's what the rolling octahedron graph looks like with triangles. No closed path exists that returns to the starting point. This is equivalent to the Nauru graph.

Nauru graph

Here's the rolling icosahedron. The connectivity graph is Foster 120B

icosahedron

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  • $\begingroup$ Can you elaborate on the meaning of a rolling polyhedral graph? Your link explains this in terms of tiles on a board, but not in terms of a concatenation of straight line segments as depicted in this question. $\endgroup$ – Lee Mosher Nov 24 '19 at 17:53
  • $\begingroup$ Also, have you considered the possibility that no such simple closed path exists? To the extent that I understand the intuitive idea, it seems clear to me that none such exists for a regular tetrahedron. $\endgroup$ – Lee Mosher Nov 24 '19 at 17:54
  • $\begingroup$ No simple closed path exists for the octahedron either. I did find a complicated path for the dodecahedron, so I'm fairly certain there's a closed path at least simpler than the one I found. $\endgroup$ – Ed Pegg Nov 24 '19 at 18:11
  • $\begingroup$ What software (if any) are you using to generate the path? $\endgroup$ – user3257842 Nov 26 '19 at 21:06

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