# 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

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