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David,

Interesting thoughts.

It seems to me that the icosahedron is a more attractive physical target than the

dodecahedron. It has 12 shear planes. A move would be a 72o rotation of a

five-triangle subsection. My intuition is that it is buildable more or less along

the line of the Rubik cube, except that the fixed axes point at vertices instead of

center faces.

The dual of the 2x2x2 cube seems to be an octahedron with shear planes along its

edges.

More generally slice a sphere a buncha times. The cuts are shear planes. Don't

let it fall apart (there is the trick!). Now paint the surface pieces. Now twist to

mix.

The actual shape does not seem to matter much. It is the interaction of the shear

planes that makes the puzzle.

Happy sphereing,

Bill