Take an icosahedron. Label opposing faces with 0 to 9. The resulting object is a Petersen graph. There are a few dice-makers that have sorta made these, but they always seem to switch the 6 and 9.
At Rolling Icosahedron, I give two solutions for rolling an icosahedron on a triangular grid so that all 120 orientations appear. These have not been proved minimal, and I found them by hand. They could likely be improved. The underlying graph is the Foster120B.
For the rolling Petersen graph on an icosahedron, there are 60 orientations producing The Foster060A graph.
What is the smallest triangular grid that has all 60 orientations of the rolling Petersen graph or hemi-icosahedron?
This grid works. The path must be repeated 5 times.
Here's a low quality GIF (2mb limit) showing the path. It should give you a rough idea of what's going on. The red dots keep track of orientations achieved (120 in all).
better quality YouTube video
Intuition:
Each "orbit" around a central vertex changes which icosahedron edge coincides with the central edge of the graph.