A cube has 12 edges; an octahedron has 12 edges.
A cube has 8 vertexes; an octahedron has 8 faces.
A cube has 6 faces; an octahedron has 6 vertexes.
Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

So, cubes and octahedrons have something to do with each other.

Do tetrahedrons or icosahedrons have a similar pair?

Originally posted by talzamir A cube has 12 edges; an octahedron has 12 edges.
A cube has 8 vertexes; an octahedron has 8 faces.
A cube has 6 faces; an octahedron has 6 vertexes.
Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

So, cubes and octahedrons have something to do with each other.

Do tetrahedrons or icosahedrons have a similar pair?

So in the case of the tetrahedron, are we looking for a corresponding polyhedron whose edges are 6, faces are 4, vertexes are 4, sides of each face are 3, and each face has 3 sides?

Indeed we are. The case for a tetrahedron is pretty obvious.. the icosahedron perhaps a bit less so.

A tetrahedron has 6 edges; a ? has 12 edges.
A tetrahedron has 4 vertexes; a ? has 8 faces.
A tetrahedron has 4 faces; a ? has 4 vertexes.
Each face of a tetraedron has 3 sides; each vertex of a ? touches 3 edges.
Each vertex of a tetrahedron touches 3 edges; each face of a ? has 3 sides.

Oops. That's what I get from posting in a hurry. The above should read,

A tetrahedron has 6 edges; a ? has 6 edges.
A tetrahedron has 4 vertexes; a ? has 4 faces.
A tetrahedron has 4 faces; a ? has 4 vertexes.
Each face of a tetraedron has 3 sides; each vertex of a ? touches 3 edges.
Each vertex of a tetrahedron touches 3 edges; each face of a ? has 3 sides.

Originally posted by talzamir A cube has 12 edges; an octahedron has 12 edges.
A cube has 8 vertexes; an octahedron has 8 faces.
A cube has 6 faces; an octahedron has 6 vertexes.
Each face of a cube has 4 sides; each vertex of an octahedron touches 4 edges.
Each vertex of a cube touches 3 edges; each face of an octahedron has 3 sides.

So, cubes and octahedrons have something to do with each other.

Do tetrahedrons or icosahedrons have a similar pair?

Shades of Escher!

Although I don't think he used the icosahedron case - but he did use the tetrahedron one to great effect.

Indeed he did. An art gallery here was once filled with the works of Escher - a real delight. =)

As for the puzzle.. a tetrahedron (a four-sided die or D4) flips into another tetrahedron. A cube and an octahedron (d6 and d8) convert into each other. An icosahedron (D12) flips into a dodecahedron (D20). When I saw that I was rather stunned about it, and wondered why it works so. There is a clear reason for it.

Some gamers also use a ten-side die (D10) that looks like two pyramids of a base and five sides each, with the bases glued together. It has seven vertexes, ten faces, and 15 edges. It converts into a cylinder with a pentagon as base. 15 edges, 7 faces, 10 vertexes. In that, not every face looks the same, but then, the d10 is not one of Plato's perfect five.

The reason.. if you take the center point each face of a polyhedron and assign that as a vertex, and make an edge where two faces are adjacent, you get a face where the vertexes were. That projection converts
* vertexes into faces and vice versa
* maintains the number of edges
* replaces the number of sides of each face by edges from a vertex and vice versa

So a tetrahedron becomes a tetrahedron, a cube an octahedron, a dodecahedron an icosahedron, and vice versa.