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Archimedian solids

Feb 13, 2012, 3:48 AM 0

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons 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.

"Identical vertices" are usually taken to mean that for any two vertices, there must be an isometry of the entire solid that takes one vertex to the other. Sometimes it is instead only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. This difference in definitions controls whether the pseudorhombicuboctahedron is considered an Archimedean solid or a Johnson solid.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, despite meeting the above definition. With this restriction, there are only finitely many Archimedean solids. They can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.


The rhombicosidodecahedron, one of the Archimedean solids


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 1620 by Johannes Kepler,[1] who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

[edit] 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).

(Vertex configuration)
TransparentSolidNetFacesEdgesVerticesPoint group
truncated tetrahedron
Truncated tetrahedron
Truncated tetrahedron.png Truncated tetrahedron flat.svg 8 4 triangles
4 hexagons
18 12 Td
Cuboctahedron.png Cuboctahedron flat.svg  14  8 triangles
6 squares
24 12 Oh
truncated cube
or truncated hexahedron
Truncated hexahedron
Truncated hexahedron.png Truncated hexahedron flat.svg 14 8 triangles
6 octagons
36 24 Oh
truncated octahedron
Truncated octahedron


Truncated octahedron.png Truncated octahedron flat.png 14 6 squares
8 hexagons
36 24 Oh
or small rhombicuboctahedron
( )
Small rhombicuboctahedron.png Rhombicuboctahedron flat.png 26 8 triangles
18 squares
48 24 Oh
truncated cuboctahedron
or great rhombicuboctahedron
Truncated cuboctahedron
Great rhombicuboctahedron.png Truncated cuboctahedron flat.svg 26 12 squares
8 hexagons
6 octagons
72 48 Oh
snub cube
or snub hexahedron
or snub cuboctahedron
(2 chiral forms)
Snub hexahedron (Ccw)
Snub hexahedron (Cw)
Snub hexahedron.png Snub cube flat.png 38 32 triangles
6 squares
60 24 O
Icosidodecahedron.png Icosidodecahedron flat.png 32 20 triangles
12 pentagons
60 30 Ih
truncated dodecahedron
Truncated dodecahedron
Truncated dodecahedron.png Truncated dodecahedron flat.png 32 20 triangles
12 decagons
90 60 Ih
Truncated icosahedron
(5.6.6 )
Truncated icosahedron
Truncated icosahedron.png Truncated icosahedron flat.png 32 12 pentagons
20 hexagons
90 60 Ih
or small rhombicosidodecahedron
Small rhombicosidodecahedron.png Rhombicosidodecahedron flat.png 62 20 triangles
30 squares
12 pentagons
120 60 Ih
truncated icosidodecahedron
or great rhombicosidodecahedron
Truncated icosidodecahedron
Great rhombicosidodecahedron.png Truncated icosidodecahedron flat.png 62 30 squares
20 hexagons
12 decagons
180 120 Ih
snub dodecahedron
or snub icosidodecahedron
(2 chiral forms)
Snub dodecahedron (Ccw)
Snub dodecahedron (Cw)
Snub dodecahedron ccw.png Snub dodecahedron flat.svg 92 80 triangles
12 pentagons
150 60 I

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".[2]

[edit] Properties

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

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

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.

[edit] Chirality

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 certain chemical compounds).

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