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Archimedean solid

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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. The symmetry of the Archimedean solids excludes the members of the dihedral group, the prisms and antiprisms. The Archimedean solids can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry. See Convex uniform polyhedron.

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, who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

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
(Vertex configuration)
Transparent Solid Net Faces Faces
(By type)
Edges Vertices Symmetry group
truncated tetrahedron
(3.6.6)
Truncatedtetrahedron
(Animation)
Truncated tetrahedron Truncated tetrahedron flat 8 4 triangles
4 hexagons
18 12 Td
cuboctahedron
(3.4.3.4)
Cuboctahedron
(Animation)
Cuboctahedron Cuboctahedron flat  14  8 triangles
6 squares
24 12 Oh
truncated cube
or truncated hexahedron
(3.8.8)
Truncatedhexahedron
(Animation)
Truncated hexahedron Truncated hexahedron flat 14 8 triangles
6 octagons
36 24 Oh
truncated octahedron
(4.6.6)
Truncatedoctahedron
(Animation)
Truncated octahedron Truncated octahedron flat 14 6 squares
8 hexagons
36 24 Oh
rhombicuboctahedron
or small rhombicuboctahedron
(3.4.4.4 )
Rhombicuboctahedron
(Animation)
Small rhombicuboctahedron Rhombicuboctahedron flat 26 8 triangles
18 squares
48 24 Oh
truncated cuboctahedron
or great rhombicuboctahedron
(4.6.8)
Truncatedcuboctahedron
(Animation)
Great rhombicuboctahedron Truncated cuboctahedron flat 26 12 squares
8 hexagons
6 octagons
72 48 Oh
snub cube
or snub hexahedron
or snub cuboctahedron
(2 chiral forms)
(3.3.3.3.4)
Snubhexahedronccw
(Animation)
Snubhexahedroncw
(Animation)
Snub hexahedron Snub cube flat 38 32 triangles
6 squares
60 24 O
icosidodecahedron
(3.5.3.5)
Icosidodecahedron
(Animation)
Icosidodecahedron Icosidodecahedron flat 32 20 triangles
12 pentagons
60 30 Ih
truncated dodecahedron
(3.10.10)
Truncateddodecahedron
(Animation)
Truncated dodecahedron Truncated dodecahedron flat 32 20 triangles
12 decagons
90 60 Ih
truncated icosahedron
or buckyball
or football/soccer ball
(5.6.6 )
Truncatedicosahedron
(Animation)
Truncated icosahedron Truncated icosahedron flat 32 12 pentagons
20 hexagons
90 60 Ih
rhombicosidodecahedron
or small rhombicosidodecahedron
(3.4.5.4)
Rhombicosidodecahedron
(Animation)
Small rhombicosidodecahedron Rhombicosidodecahedron flat 62 20 triangles
30 squares
12 pentagons
120 60 Ih
truncated icosidodecahedron
or great rhombicosidodecahedron
(4.6.10)
Truncatedicosidodecahedron
(Animation)
Great rhombicosidodecahedron Truncated icosidodecahedron flat 62 30 squares
20 hexagons
12 decagons
180 120 Ih
snub dodecahedron
or snub icosidodecahedron
(2 chiral forms)
(3.3.3.3.5)
Snubdodecahedronccw
(Animation)
Snubdodecahedroncw
(Animation)
Snub dodecahedron ccw Snub dodecahedron flat 92 80 triangles
12 pentagons
150 60 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 certain 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.

See also

References

External links


bg:Архимедово тяло ca:Políedre arquimediàeo:Arĥimeda solidoit:Solido archimedeo he:פאון ארכימדי mk:Архимедови тела nl:Archimedisch lichaam pl:Wielościan półforemny pt:Sólidos de Arquimedesth:ทรงตันอาร์คิมิดีส

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