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Archimedean Solid
The 13 Archimedean solids are theconvex polyhedrathat have a similar arrangement of nonintersecting
regularconvex polygonsof two or more different types arranged in the same way about each vertexwith
all sides the same length (Cromwell 1997, pp. 91-92).
The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging
to adihedral groupof symmetries (e.g., the two infinite families of regular prisms and antiprisms), as well
as theelongated square gyrobicupola(because that surface's symmetry-breaking twist allows vertices
"near the equator" and those "in the polar regions" to be distinguished; Cromwell 1997, p. 92). The
Archimedean solids are sometimes also referred to as thesemiregular polyhedra.
http://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.html7/29/2019 Archimedean Solid2stdyust
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The Archimedean solids are illustrated above.
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Nets of the Archimedean solids are illustrated above.
The following table lists the uniform, Schlfli, Wythoff, and Cundy and Rollett symbols for the
Archimedean solids (Wenninger 1989, p. 9).
soliduniform
polyhedron
Schlfli
symbol
Wythoff
symbol
Cundy and Rollett
symbol
1 cuboctahedron
2great
rhombicosidodecahedron t4.6.10
3 great rhombicuboctahedron t
4.6.8
4 icosidodecahedron
5small
rhombicosidodecahedron r3.4.5.4
6 small rhombicuboctahedron r
http://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.html7/29/2019 Archimedean Solid2stdyust
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7 snub cubes
8 snub dodecahedron s
9 truncated cube t
10 truncated dodecahedron t
11 truncated icosahedron t
12 truncated octahedron t
13 truncated tetrahedron t
The following table gives the number of vertices , edges , and faces , together with the number of -
gonal faces for the Archimedean solids. The sorted numbers of edges are 18, 24, 36, 36, 48, 60, 60,
72, 90, 90, 120, 150, 180 (Sloane'sA092536), numbers of faces are 8, 14, 14, 14, 26, 26, 32, 32, 32, 38,
62, 62, 92 (Sloane'sA092537), and numbers of vertices are 12, 12, 24, 24, 24, 24, 30, 48, 60, 60, 60, 60,
120 (Sloane'sA092538).
solid
1 cuboctahedron 12 24 14 8 6
2 great rhombicosidodecahedron 120 180 62 30 20 12
3 great rhombicuboctahedron 48 72 26 12 8 6
4 icosidodecahedron 30 60 32 20 12
5 small rhombicosidodecahedron 60 120 62 20 30 12
6 small rhombicuboctahedron 24 48 26 8 18
7 snub cube 24 60 38 32 6
8 snub dodecahedron 60 150 92 80 12
9 truncated cube 24 36 14 8 6
10 truncated dodecahedron 60 90 32 20 12
http://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092536http://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.html7/29/2019 Archimedean Solid2stdyust
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11 truncated icosahedron 60 90 32 12 20
12 truncated octahedron 24 36 14 6 8
13 truncated tetrahedron 12 18 8 4 4
Seven of the 13 Archimedean solids (thecuboctahedron,icosidodecahedron,truncated cube,truncated
dodecahedron,truncated octahedron,truncated icosahedron, andtruncated tetrahedron) can be obtained
bytruncationof aPlatonic solid. The three truncation series producing these seven Archimedean solids
are illustrated above.
Two additional solids (thesmall rhombicosidodecahedronandsmall rhombicuboctahedron) can be
obtained byexpansionof aPlatonic solid, and two further solids (thegreat rhombicosidodecahedronand
great rhombicuboctahedron) can be obtained byexpansionof one of the previous 9 Archimedean solids
(Stott 1910; Ball and Coxeter 1987, pp. 139-140). It is sometimes stated (e.g., Wells 1991, p. 8) that
these four solids can be obtained by truncation of other solids. The confusion originated with Kepler
himself, who used the terms "truncated icosidodecahedron" and "truncated cuboctahedron" for thegreat
rhombicosidodecahedronandgreat rhombicuboctahedron, respectively. However, truncation alone is not
capable of producing these solids, but must be combined with distorting to turn the resulting rectangles
into squares (Ball and Coxeter 1987, pp. 137-138; Cromwell 1997, p. 81).
http://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.html7/29/2019 Archimedean Solid2stdyust
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The remaining two solids, thesnub cubeandsnub dodecahedron, can be obtained by moving the faces
of acubeanddodecahedronoutward while giving each face a twist. The resulting spaces are then filled
with ribbons ofequilateral triangles(Wells 1991, p. 8).
Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular
tetrahedronso that four of their faces lie on the faces of thattetrahedron.
The Archimedean solids satisfy
(1)
where is the sum of face-angles at a vertex and is the number of vertices (Steinitz and Rademacher
1934, Ball and Coxeter 1987).
Let the cyclic sequence represent the degrees of the faces surrounding a vertex (i.e.,
is a list of the number of sides of all polygons surrounding any vertex). Then the definition of an
Archimedean solid requires that the sequence must be the same for each vertex to withinrotationand
reflection. Walsh (1972) demonstrates that represents the degrees of the faces surrounding each vertex
of a semiregular convex polyhedron ortessellationof the planeiff
1. and every member of is at least 3,
2. , with equality in the case of a planetessellation, and
3. for everyodd number , contains a subsequence ( , , ).
Condition (1) simply says that the figure consists of two or more polygons, each having at least three
sides. Condition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for
the figure to lie in the plane, and less than a full rotation for a solid figure to be convex.
The usual way of enumerating the semiregular polyhedra is to eliminate solutions of conditions (1) and (2)
using several classes of arguments and then prove that the solutions left are, in fact, semiregular (Kepler
1864, pp. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940, p. 394; Coxeteret al. 1954; Lines 1965,
pp. 202-203; Walsh 1972). The following table gives all possible regular and semiregular polyhedra and
tessellations. In the table, 'P' denotesPlatonic solid, 'M' denotes aprismorantiprism, 'A' denotes an
Archimedean solid, and 'T' a plane tessellation.
http://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.html7/29/2019 Archimedean Solid2stdyust
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fg. solid Schlfli symbol
(3, 3, 3) P tetrahedron
(3, 4, 4) M triangular prism t
(3, 6, 6) A truncated tetrahedron t
(3, 8, 8) A truncated cube t
(3, 10, 10) A truncated dodecahedron t
(3, 12, 12) T tessellation t
(4, 4, ) M -gonalprism t
(4, 4, 4) P cube
(4, 6, 6) A truncated octahedron t
(4, 6, 8) A great rhombicuboctahedron t
(4, 6, 10) A great rhombicosidodecahedron t
(4, 6, 12) T tessellation t
(4, 8, 8) T tessellation t
(5, 5, 5) P dodecahedron
(5, 6, 6) A truncated icosahedron t
(6, 6, 6) T tessellation
(3, 3, 3, ) M -gonalantiprisms
(3, 3, 3, 3) P octahedron
(3, 4, 3, 4) A cuboctahedron
(3, 5, 3, 5) A icosidodecahedron
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(3, 6, 3, 6) T tessellation
(3, 4, 4, 4) A small rhombicuboctahedronr
(3, 4, 5, 4) A small rhombicosidodecahedron r
(3, 4, 6, 4) T tessellation r
(4, 4, 4, 4) T tessellation
(3, 3, 3, 3, 3) P icosahedron
(3, 3, 3, 3, 4) A snub cube
s
(3, 3, 3, 3, 5) A snub dodecahedron s
(3, 3, 3, 3, 6) T tessellation s
(3, 3, 3, 4, 4) T tessellation --
(3, 3, 4, 3, 4) T tessellation s
(3, 3, 3, 3, 3, 3) T tessellation
As shown in the above table, there are exactly 13 Archimedean solids (Walsh 1972, Ball and Coxeter
1987). They are called thecuboctahedron,great rhombicosidodecahedron,great rhombicuboctahedron,
icosidodecahedron,small rhombicosidodecahedron,small rhombicuboctahedron,snub cube,snub
dodecahedron,truncated cube,truncated dodecahedron,truncated icosahedron(soccer ball),truncated
octahedron, andtruncated tetrahedron.
Let be theinradiusof the dual polyhedron (corresponding to theinsphere, which touches the faces of
the dual solid), be themidradiusof both the polyhedron and its dual (corresponding to the
midsphere, which touches the edges of both the polyhedron and its duals), thecircumradius
(corresponding to thecircumsphereof the solid which touches the vertices of the solid) of the
Archimedean solid, and the edge length of the solid Since thecircumsphereandinsphereare dual to
each other, they obey the relationship
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Archimedean Solid2stdyust
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(2)
(Cundy and Rollett 1989, Table II following p. 144). In addition,
(3)
(4)
(5)
(6)
(7)
(8)
The following tables give the analytic and numerical values of , , and for the Archimedean solids with
polyhedron edgesof unit length (Coxeteret al. 1954; Cundy and Rollett 1989, Table II following p. 144).
Hume (1986) gives approximate expressions for thedihedral anglesof the Archimedean solid (and exact
expressions for their duals).
solid
1 cuboctahedron 1
2
great
rhombicosidodecahedro
n
3great
rhombicuboctahedron
4 icosidodecahedron
5
small
rhombicosidodecahedro
n
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6small
rhombicuboctahedron
7 snub cube * * *
8 snub dodecahedron * * *
9 truncated cube
1
0
truncated
dodecahedron
1
1truncated icosahedron
1
2truncated octahedron
1
3truncated tetrahedron
*The complicated analytic expressions for thecircumradiiof these solids are given in the entries for the
snub cubeandsnub dodecahedron.
solid
1 cuboctahedron 0.75 0.86603 1
2 great rhombicosidodecahedron 3.73665 3.76938 3.80239
3 great rhombicuboctahedron 2.20974 2.26303 2.31761
4 icosidodecahedron 1.46353 1.53884 1.61803
5 small rhombicosidodecahedron 2.12099 2.17625 2.23295
6 small rhombicuboctahedron 1.22026 1.30656 1.39897
7 snub cube 1.15763 1.24719 1.34371
8 snub dodecahedron 2.03969 2.09688 2.15583
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9 truncated cube 1.63828 1.70711 1.77882
10 truncated dodecahedron 2.88526 2.92705 2.96945
11 truncated icosahedron 2.37713 2.42705 2.47802
12 truncated octahedron 1.42302 1.5 1.58114
13 truncated tetrahedron 0.95940 1.06066 1.17260
The Archimedean solids and theirdualsare allcanonical polyhedra. Since the Archimedean solids are
convex, theconvex hullof each Archimedean solid is the solid itself.
SEE ALSO:Archimedean Solid Stellations,Catalan Solid,Deltahedron,Isohedron,Johnson Solid,Kepler-
Poinsot Solid,Platonic Solid,Quasiregular Polyhedron,Semiregular Polyhedron,Uniform Polyhedron,
Uniform Tessellation
REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M.Mathematical Recreations and Essays, 13th ed.New York: Dover, p. 136, 1987.
Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.).Fundamentals of Mathematics, Vol. 2: Geometry.Cambridge, MA:
MIT Press, pp. 269-286, 1974.
Catalan, E. "Mmoire sur la Thorie des Polydres." J. l'cole Polytechnique (Paris)41, 1-71, 1865.
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc.24, 1-9,
1928.
Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z.46, 380-407, 1940.
Coxeter, H. S. M. 2.9 inRegular Polytopes, 3rd ed."Historical Remarks." New York: Dover, pp. 30-32, 1973.
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A
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