# Archimedean solid

The **Archimedean solids** are the 13 polyhedra that are convex, isogonal (vertex-transitive), having only regular faces, and are not Platonic solids, prisms, or antiprisms. They were first described and fully enumerated by their namesake Archimedes, but his work on them is now lost; they were "rediscovered" during the Renaissance due to mathematicians' and artists' fascination with pure forms.

The Archimedean solids formed the basis of the more general concept of uniform polyhedra, which are not required to be convex, and were only characterized in the 20th century.

The elongated square gyrobicupola, or pseudorhombicuboctahedron, is not traditionally considered an Archimedean solid as it does not fit the definition of vertex transitivity, only congruence of vertex figures. Branko Grünbaum has extensively documented how mathematicians throughout history, including Johannes Kepler himself, incorrectly defined the Archimedean solids based on the "local criterion" of having a single vertex figure, rather than the "global criterion" of full vertex transitivity.^{[1]}

The duals of the Archimedean solids are known as the Catalan solids.

## List of the Archimedean solids[edit | edit source]

Name; Coxeter diagram | Image | Faces | Edges | Vertices | Vertex figure | Dual |
---|---|---|---|---|---|---|

Truncated tetrahedron |
4 triangles 4 hexagons |
6+12 | 12 | Isosceles triangle edge lengths 1 (base) and √3 (sides) |
Triakis tetrahedron | |

Truncated cube |
8 triangles 6 octagons |
12+24 | 24 | Isosceles triangle edge lengths 1 (base) and √2+√2 (sides) |
Triakis octahedron | |

Cuboctahedron |
8 triangles 6 squares |
24 | 12 | Rectangle edge lengths 1 and √2 |
Rhombic dodecahedron | |

Truncated octahedron |
8 hexagons 6 squares |
12+24 | 24 | Isosceles triangle edge lengths √2 (base) and √3 (sides) |
Tetrakis hexahedron | |

Small rhombicuboctahedron |
8 triangles 6+12 squares |
24+24 | 24 | Isosceles trapezoid edge lengths 1 (top, sides) and √2 (bottom) |
Deltoidal icositetrahedron | |

Great rhombicuboctahedron |
12 squares 8 hexagons 6 octagons |
24+24+24 | 48 | Scalene triangle edge lengths √2, √3, and √2+√2 |
Disdyakis dodecahedron | |

Snub cube |
8+24 triangles 6 squares |
12+24+24 | 24 | Floret pentagon edge lengths 1, 1, 1, 1, √2 |
Pentagonal icositetrahedron | |

Truncated dodecahedron |
20 triangles 12 decagons |
30+60 | 60 | Isosceles triangle edge lengths 1 (base) and √(5+√5)/2 (sides) |
Triakis icosahedron | |

Icosidodecahedron |
20 triangles 12 pentagons |
60 | 30 | Rectangle edge lengths 1 and (1+√5)/2 |
Rhombic triacontahedron | |

Truncated icosahedron |
20 hexagons 12 pentagons |
30+60 | 60 | Isosceles triangle edge lengths (1+√5)/2 (base) and √3 (sides) |
Pentakis dodecahedron | |

Small rhombicosidodecahedron |
30 squares 20 triangles 12 pentagons |
60+60 | 60 | Isosceles trapezoid edge lengths 1 (top), √2 (sides), and (1+√5)/2 (bottom) |
Deltoidal hexecontahedron | |

Great rhombicosidodecahedron |
30 squares 20 hexagons 12 decagons |
60+60+60 | 120 | Scalene triangle edge lengths √2, √3, and √(5+√5)/2 |
Disdyakis triacontahedron | |

Snub dodecahedron |
20+60 triangles 12 pentagons |
30+60+60 | 60 | Floret pentagon edge lengths 1, 1, 1, 1, (1+√5)/2 |
Pentagonal hexecontahedron |

## External links[edit | edit source]

- Wikipedia contributors. "Archimedean solid".

## References[edit | edit source]

- ↑ Grunbaum, Branko. "An enduring error."

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