Pyritification

Pyritification denotes a non-symmetric relationship which is seen between some pairs of regular maps. Some regular maps can be "pyritified" to create larger regular maps in the same genus.

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, S{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, with new vertices created on the existing edges. Some possible pyritifications are listed below. Many are obvious; those shown on a yellow background, perhaps less so.

Euler numberdiagramconvertsVertices, Faces, Edgescomments
Positive {2,3} → {4,3} V' = V+2E
F' = 2F
E' = 3E+F
Application restricted to 3-hosohedroncube.

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-hosohedronoctahedron and hemi-4-hosohedronhemioctahedron.

The dual process converts the square dihedron to the cube.

{3,3} → {5,3} V' = V+2E+F
F' = 3F
E' = 3E+3F
Application restricted to tetrahedrondodecahedron.
{4,3} → {5,3} V' = V+E
F' = 2F
E' = 2E+F
Application restricted to cubedodecahedron and hemicubehemidodecahedron.

The dual process converts the octahedron to the icosahedron.

0 {4,4} → {4,4} V' = V+E
F' = 2F
E' = 2E+F
Can be applied to any {4,4}, doubling the number of squares.
{6,3} → {6,3} V' = V
F' = 2F
E' = E+F
Can be applied to any {6,6}, tripling the number of hexagons.
{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 {8,3} → {7,3} V' = V+E/2
F' = 2F
E' = E+3F
{14,3} → {7,3} V' = V+E+7F
F' = 8F
E' = 2E+14F

The term "Pyritify"

Iron pyrites, also known as fool's gold, is a mineral (FeS2) which often forms cubic crystals. It can also form crystals with 12 pentagonal faces, but only tetrahedral symmetry. The form of these crystals in known as a pyritohedron.

As the pyritohedron is intermediate between the cube and the dodecahedron in shape, I use the term "pyritification" for the process of converting a cube to a dodecahedron by a process that goes via the pyritohedron.

Other relationships between regular maps
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