# Cayley Diagrams drawn on Platonic Solids

This page gives the Cayley diagrams of a very few infinite groups. Their presentations are also given. Above each presentation is a short list of elements in bold type: the colours used in the list correspond to the colours of the lines in the diagram.

Solid Order Group Presentation Cayley diagram

### The Five Regular Platonic Polyhedra

Tetrahedron 4 C2 × C2

C22

r g b
< r,g,b | r2, g2, b2, rg=b, (rg)2 >
Octahedron 6 D6 ≅ S3 k r g
< k,r,g | k2, r2, g2, rg=k, (rg)3 >
Cube 8 C2 × C2 × C2

C32

r g b
< r,g,b | r2, g2, b2, (rg)2, (gb)2, (br)2 >
D4 k r
< k,r | k4, r2, (kr)2 >
Icosahedron 12 A4 r g b
< r,g,b | r2, g3, b3, rbg >

If we add the rotations of the icosahedron (which form A5) to the above set of generators, we get the Mathieu group M12.
This is not a group. Unfortunately it this not the Cayley diagram of any group. Starting at the top of the diagram, rgb=1; but from most vertices, rgb has period 8.
Dodecahedron 20 This is not a group. Unfortunately it this not the Cayley diagram of any group. If it were, rg gb br rb bg and gr would all have period 10. Tracing out any of them on the diagram forms a Hamiltonian cycle.

r g b
< r,g,b | r2, g2, b2, (rg)10, (gb)10, (br)10 >

### Truncated Polyhedra

Truncated Tetrahedron 12 A4 k r
< r,g,b | k3, r2, (kr)3 >
This is not a group. This is a version of a diagram I found on the web. It purports to be a Cayley diagram for A4. It implies that you can generate A4 with three generators each of period 2: but A4 only has three elements of period 2, and they generate C2×C2.
Cuboctahedron 12 A4 r g
< r,g | r3, g3, (rg)2 >
Truncated Octahedron 24 S4 k r
< k,r | k4, r2, (kr)3 >
Truncated Octahedron 24 S4 b r g
< b,r,g | b2, r2, g2, (br)3, (rg)2, (gb)3 >
Truncated Cube 24 S4 k r
< k,r | k3, r2, (kr)4 >
Small Rhombicuboctahedron 24 S4 b r
< b,r | b4, r3, (br)2 >
Snub Cube 24 S4 r g b
< r,g,b | r2, g3, b2, rgb >
Icosidodecahedron 30 There is no group. The groups with 30 elements are not interesting. None of them could have a nice Cayley diagram such as this would have to be.
Truncated Cuboctahedron 48 S4 × C2 r g b
< r,g,b | r2, g2, b2, (rg)2, (gb)3, (br)4 >

The central element is (rgb)3.

Truncated Icosahedron 60 A5 k r
< k,r | k5, r2, (kr)3 >
Truncated Dodecahedron 60 A5 k r
< k,r | k3, r2, (kr)5 >
Small Rhombicosidodecahedron 60 A5 r b
< b,r | b5, r3, (br)2 >
Sunb Dodecahedron 60 A5 k g r
< k,g,r | k5, g3, r2, kgr >
Great Rhombicosidodecahedron 120 A5 × C2 k r g
< k,r,b | k2, r2, g2, (kr)2, (rg)3, (gk)5 >

### Less regular Polyhedra

Prism 2N DN k r
< k,r | k7, r2, krkr >
Antiprism 2N DN k r g
< k,r,g | k7, r2, g2, kgr >
Pyritohedron 20 This is not a group. This is equivalent to the dodecahedron, as can be seen by changing the angles until the pentagons are more nearly regular. Again it is not the Cayley diagram of any group.

Some more Cayley diagrams
and other pages on groups