### The Archimedean solids

Along with the Platonic solids, the Archimedean solids have also been known since ancient times. While the Platonic solids are constructed from vertex stars using all the same kind of regular polygon, the vertex stars for the Archimedean solids have mixed types of regular polygons. There are 13 such solids, though two of them have two chiral (twisting to the left or right) forms, so that it could be stated that there are 15 different solids. Or, put another way, there are 13 different mixed-polygon vertex stars that can construct 15 different convex, isogonal polyhedra:  11 that construct a single polyhedron and 2 that construct 2 polyhedra each.

These 13 vertex stars can also be labeled, each according to their various symmetries, and their multiple labeled versions determined. We list them all below. While the many symmetries of the Platonic solid vertex stars allow many different labeled versions, the Archimedean vertex stars have far fewer symmeteries (owing to their mixed polygon compositions) and thus they end up producing far fewer labeled versions. So, while the varied shapes of the Archimedean solids make very pleasant viewing, in terms of isogonality they are really quite dull!

Nevertheless, they do have some noteworthy points. The snub cube and snub dodecahedron, the two vertex stars that form chiral polyhedra, do not have symmetric symbols, even though their vertex stars do. Of course, this must be the case because if they could form symmetric incidence symbols then they would not have two chiral forms. (The truncated cuboctahedron and the truncated icosidodecahedron do not form symmetric incidence symbols because their vertex stars have no symmetry.) Similar to the (Platonic) dodecahedron, the icosidodecahedron cannot form a completely asymmetric incidence symbol:  some degree of symmetry in its vertex symbol is required. Unlike any of the other Platonic or Archimedean polyhedra, both of these latter polyhedra have more than one pentagon in their vertex stars.

#### The 13 Archimedean vertex stars and their labeled isogonal polyhedra

1. Truncated tetrahedron   (3.6.6)
• [a b+ b-;  a b-]
• [a+ b+ c+;  a+ c+ b+]

2. Truncated cube   (3.8.8)
• [a b+ b-;  a b-]
• [a+ b+ c+;  a+ c+ b+]
• [a+ b+ c+;  a- c+ b+]

3. Truncated octahedron   (4.6.6)
• [a b+ b-;  a b-]
• [a+ b+ c+;  a- b- c-]
• [a+ b+ c+;  a+ c+ b+]

4. Truncated icosahedron   (5.6.6)
• [a b+ b-;  a b-]
• [a+ b+ c+;  a+ c+ b+]

5. Truncated dodecahedron   (3.10.10)
• [a b+ b-;  a b-]
• [a+ b+ c+;  a+ c+ b+]

6. Cuboctahedron   (3.4.3.4)
• [a+ a- a+ a-;  a-]
• [a+ b+ a+ b+;  b+ a+]
• [a+ b+ b- a-;  a- b-]
• [a+ a- b+ b-;  b- b+ a- a+]
• [a+ b+ c+ d+;  d+ c+ b+ a+]

7. Truncated cuboctahedron   (4.6.8)
• [a+ b+ c+;  a- b- c-]

8. Snub cube (two forms)   (3.3.3.3.4)
9. Snub dodecahedron (two forms)   (3.3.3.3.5)
10. Rhombicuboctahedron   (3.4.4.4)
• [a+ b+ b- a-;  a- b-]
• [a+ b+ c+ d+;  d+ b- c- a+]
• [a+ b+ c+ d+;  d+ c+ b+ a+]

11. Rhombicosidodecahedron   (3.4.5.4)
• [a+ b+ b- a-;  a- b-]
• [a+ b+ c+ d+;  d+ c+ b+ a+]

12. Icosidodecahedron   (3.5.3.5)
• [a+ a- a+ a-;  a-]
• [a+ b+ a+ b+;  b+ a+]

13. Truncated icosidodecahedron   (4.6.10)
• [a+ b+ c+;  a- b- c-]

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Last updated: April 18, 2019