The Magic Octahedron (Octahedron Cube?) is out. A friend got one for me
over the weekend. Each face is 9 triangles; it works sort of like two
tetrahedrons base to base, except each tetra is really a 4-sided pyramid.
The solution is not too hard, and not too different from the tetrahedron.
Like the tetra, the corner and next-to-corner pieces don't travel-they
only twist. The remaining edges are isomorphic to the edges of a cube.
In fact, if you corner-center a Cubes coloring (ie each face has 4 colors,
each corner has one; see the Scientific American article), and then peel
off the labels from the corners, you have an exact isomorphism of the
Octahedron.
By the way, this version twists around the corners, using the same
type mechanism as the original Cube (I think). It should be possible to
use the mechanism of the Tetrahedron (with its 4 axes of rotation) to
build an Octahetron whose faces twist. Any mechanical engineers out there
to do so?
--- Stan
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