Archimedean solid

In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search was completed around 1619 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot solids.

Classification

There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately). Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

The number of vertices is 720° divided by the vertex angle defect.

Name	picture	Faces		Edges	Vertices	Vertex configuration	Symmetry group
truncated tetrahedron	Truncated tetrahedron (Video)	8	4 triangles 4 hexagons	18	12	3,6,6	Td
cuboctahedron	Cuboctahedron (Video)	14	8 triangles 6 squares	24	12	3,4,3,4	Oh
truncated cube or truncated hexahedron	Truncated hexahedron (Video)	14	8 triangles 6 octagons	36	24	3,8,8	Oh
truncated octahedron	Truncated octahedron (Video)	14	6 squares 8 hexagons	36	24	4,6,6	Oh
rhombicuboctahedron or small rhombicuboctahedron	Rhombicuboctahedron (Video)	26	8 triangles 18 squares	48	24	3,4,4,4	Oh
truncated cuboctahedron or great rhombicuboctahedron	Truncated cuboctahedron (Video)	26	12 squares 8 hexagons 6 octagons	72	48	4,6,8	Oh
snub cube or snub cuboctahedron (2 chiral forms)	Snub hexahedron (Ccw) (Video) Snub hexahedron (Cw) (Video)	38	32 triangles 6 squares	60	24	3,3,3,3,4	O
icosidodecahedron	Icosidodecahedron (Video)	32	20 triangles 12 pentagons	60	30	3,5,3,5	Ih
truncated dodecahedron	Truncated dodecahedron (Video)	32	20 triangles 12 decagons	90	60	3,10,10	Ih
truncated icosahedron or buckyball or football/soccer ball	Truncated icosahedron (Video)	32	12 pentagons 20 hexagons	90	60	5,6,6	Ih
rhombicosidodecahedron or small rhombicosidodecahedron	Rhombicosidodecahedron (Video)	62	20 triangles 30 squares 12 pentagons	120	60	3,4,5,4	Ih
truncated icosidodecahedron or great rhombicosidodecahedron	Truncated icosidodecahedron (Video)	62	30 squares 20 hexagons 12 decagons	180	120	4,6,10	Ih
snub dodecahedron or snub icosidodecahedron (2 chiral forms)	Snub dodecahedron (Ccw) (Video) Snub dodecahedron (Cw) (Video)	92	80 triangles 12 pentagons	150	60	3,3,3,3,5	I

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of chemical compounds).

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

External links

• Paper models of Archimedean solids
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Penultimate Modular Origami
• Interactive 3D polyhedra in Java