In response to Peter Beck's thought-provoking idea about connecting
Rubikoid puzzles to plate tectonics, I have two slight spoilers.
First, plate tectonics involves spreading zones, which are places where
new crust is created, and subduction zones, where crust is destroyed.
In any permutation group, the things being permuted are not allowed to
appear or disappear. So it seems unlikely that group theory can be
directly applied to tectonics.
Second, just because a puzzle is Rubikoid does not mean it has parity
constraints. Consider the "Magic Octohedron", which has eight
triangular faces. You can grab any pyramidal cluster of four faces and
rotate it. This is really the 2x2x2 Cube in disguise. In this form, it
has no parity constraints, that is, all the 8-factorial different
configurations are achievable. So even if group theory could be applied
to tectonics, we couldn't assume parity constraints in the general case.