Archimedean solid

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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

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