### The Platonic solids

The five Platonic solids have been known since antiquity. They are each constructed from multiple copies of five different, completely symmetric vertex stars. However, the vertices can be labeled so as to remove various degrees of their symmetries, and these labeled vertex stars can also be used to construct isogonal polyhedra. Below we list the different labeled versions of the five Platonic solids, along with VRML models of them and their vertex stars.

An interesting note is that the dodecahedron cannot be constructed using a completely asymmetric vertex star (that is, having vertex symbol  a+ b+ c+). Some degree of symmetry is required. Another observation is that a cube is a rectangular prism with all edges the same length. Lengthening the edges effectively marks them, so that the possible labelings for a rectangular prism are simply a subset of those for a cube, according to the symmetries of the prism.

#### The 5 Platonic solids and their isogonal labelings

• Tetrahedron   (3.3.3)
1. [a a a;  a]
2. [a+ a+ a+;  a+]
3. [a b+ b-;  a b+]
4. [a+ b+ c+;  a+ b+ c+]
5. [a+ b+ c+;  a+ c- b-]

• Cube   (4.4.4)
1. [a a a;  a]
2. [a+ a+ a+;  a+]
3. [a+ a+ a+;  a-]
4. [a b+ b-;  a b-]
5. [a+ b+ c+;  a+ b- c-]
6. [a+ b+ c+;  a- b- c-]
7. [a+ b+ c+;  a+ c+ b+]
8. [a+ b+ c+;  a- c+ b+]

• Octahedron   (3.3.3.3)
1. [a a a a;  a]
2. [a+ a+ a+ a+;  a+]
3. [a+ a- a+ a-;  a-]
4. [a b a b;  b a]
5. [a+ b+ a+ b+;  b+ a+]
6. [a+ b+ b- a-;  a+ b-]
7. [a+ b+ c+ d+;  a+ c+ b+ d+]
8. [a+ b+ c+ d+;  d- c+ b+ a-]

• Icosahedron   (3.3.3.3.3)
1. [a a a a a;  a]
2. [a+ a+ a+ a+ a+;  a+]
3. [a b+ c+ c- b-;  a c+ b+]
4. [a+ b+ c+ d+ e+;  a+ c+ b+ e+ d+]

• Dodecahedron   (5.5.5)
1. [a a a;  a]
2. [a+ a+ a+;  a+]