Pyritification of Regular Maps

To the left is a diagram of a cube, the regular map S⁰{4,3}.

We can divide each face of the cube into two faces, as shown by the red lines, to the right. Then, if we regard the cube as a wire frame in 3-space, we can shorten the new edges, putting a slight angle in each each black line where the red line joins it. With the right amount of shortening, we can arrange for the resulting shape to be the same as a crystal of iron pyrites. This shape is called a pyritohedron. It has lost some of the S4 rotational symmetry of the cube, its rotational symmetry group is A4.

If we shorten the new edges a bit more, we can arrange for it to become a regular dodecahedron, S⁰{5,3}. Now its rotational symmetry group is A5.

I use the term pyritification for such a process of converting a regular map to a larger regular map by dividing up each of its faces in the same way. Some possible pyritifications are listed below. Many are obvious; those shown on a yellow bacground, perhaps less so.

Euler number	diagram	-converts-	Vertices, Faces, Edges	comments
Positive		{2,3} → {4,3}	V' = V+2E F' = 2F E' = 3E+F	Application restricted to 3-hosohedron→cube. The dual process {3,2}→{3,4} converts the triangular dihedron to the octahedron.
		{2,4} → {3,4}	V' = V+E F' = 2F E' = 2E+F	Application restricted to 4-hosohedron→octahedron. The dual process converts the square dihedron to the tetrahedron.
		{3,3} → {5,3}	V' = V+2E+F F' = 3F E' = 3E+3F	Application restricted to tetrahedron→dodecahedron.
		{4,3} → {5,3}	V' = V+E F' = 2F E' = 2E+F	Application restricted to cube→dodecahedron and hemicube→hemidodecahedron. The dual process converts the octahedron to the icosahedron.
0		{3,6} → {3,6}	V' = V+E F' = 4F E' = 2E+3F	Can be applied to any {3,6}, quadrupling the number of triangles.
		{4,4} → {4,4}	V' = V+E F' = 2F E' = 2E+F	Can be applied to any {4,4}, doubling the number of squares.
		{4,4} → {3,6}	V' = V+E+F F' = 4F E' = 2E+4F	Can be applied to any {4,4}, converting each square to two triangles.
		{6,3} → {6,3}	V' = V F' = 2F E' = E+F	Can be applied to any {4,4}, tripling the number of hexagons.
		{6,3} → {3,6}	V' = V+F F' = 6F E' = E+6F	Can be applied to any {6,3}, converting each hexagon to three triangles.
		{6,3} → {6,3}	V' = V+2E+6F F' = 7F E' = 3E+12F	Can be applied to any {6,3}, increasing the number of hexagons by a factor of seven.
Negative		{4,6} → {3,9}	V' = V F' = 2F E' = E+F	It does not preserve regularity. It generates maps which lack 9-fold symmetry.
		{6,4} → {4,6}	V' = V F' = 4F E' = E+3F	Not yet tested
		{6,4} → {3,8}	V' = V+F F' = 3F E' = E+3F	It does not preserve regularity.
		{8,4} → {5,4}	V' = 2V+F F' = 4F E' = E+V+4F	This is not a simple pyritification. See pyritification of {8,4} for details. A related process converts {8,4}→{4,5} It does not preserve regularity. It generates maps which lack 5-fold symmetry.
		{14,3} → {7,3}	V' = V+E+7F F' = 8F E' = 2E+14F