More Langton's Ants on polyhedra. The rule used here is a minimal adaptation of the basic binary scheme: turn right on 0 and left on 1, and flip the cell value when moving on. The turn angles are the smallest nonzero ones, i.e. pi/3 or 60° for 6-fold vertices of the geodesic polyhedron.
I first tested this rule in a 2D hex lattice about a year ago, and it made a distinctive laterally symmetric pattern. Here the symmetry is broken by the occasional 5-fold vertices. It is also worth noting that in 5-fold vertices, there is no way to continue straight forward, so the minimal left/right turning angles are actually smaller, i.e. pi/5.
Depending on the random-seeded initial location and direction, I sometimes got these nice hexagonal details. To ensure I'd always get them, I switched from the icosahedral geodesic to a Goldberg polyhedron, shown in the second part. Now the laterally symmetric pattern is clear, at least for some time; the overall topology is still different from the 2D lattice.
Since only one cell is being updated at a time, the overall progress is relatively slow, and here I'm doing ten updates per video frame.
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