Archimedean solid

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The truncated icosidodecahedron is the largest Archimedean solid by volume (when all are drawn with equal edge lengths). It also has the most vertices and edges.
The pseudo-rhombicuboctahedron has a single vertex figure,, but with a twist on one square cupola. Unlike the (untwisted) rhombicuboctahedron it is not vertex transitive. This is a Johnson solid; most authors do not consider it to be Archimedean.

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

"Identical vertices" means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Branko Grünbaum (2009) pointed out a widespread error in the literature on Archimedan solids: some authors only require that the faces that meet at one vertex are related by a local isometry to the faces that meet at any other vertex, and incorrectly claim that there are 13 solids satisfying this definition. There are 14, because the elongated square gyrobicupola (pseudo-rhombicuboctahedron) is the unique convex polyhedron that has regular polygons meeting in the same way at each vertex, but that does not have a global symmetry taking any vertex to any other vertex. Because of this Grünbaum suggested that the elongated square gyrobicupola should be counted as an Archimedean solid, which would give 14 Archimedean solids, but most authors (including Archimedes himself) do not include it in their lists of 13 Archimedean solids.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedan solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Origin of name[edit]

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.[1] During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,[2] as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra.

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.[1]


There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 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).

(Alternative name)
Transparent Solid Net Vertex
Faces Edges Vert. Volume
(unit edges)
truncated tetrahedron t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated tetrahedron
Truncated tetrahedron.png Truncated tetrahedron flat.svg 3.6.6
Truncated tetrahedron vertfig.png
8 4 triangles
4 hexagons
18 12 2.710576 Td
r{4,3} or rr{3,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cuboctahedron.png Cuboctahedron flat.svg
Cuboctahedron vertfig.png
14 8 triangles
6 squares
24 12 2.357023 Oh
truncated cube t{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated hexahedron
Truncated hexahedron.png Truncated hexahedron flat.svg 3.8.8
Truncated cube vertfig.png
14 8 triangles
6 octagons
36 24 13.599663 Oh
truncated octahedron
(truncated tetratetrahedron)
t{3,4} or tr{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated octahedron


Truncated octahedron.png Truncated octahedron flat.png 4.6.6
Truncated octahedron vertfig.png
14 6 squares
8 hexagons
36 24 11.313709 Oh
(small rhombicuboctahedron)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Small rhombicuboctahedron.png Rhombicuboctahedron flat.png
Small rhombicuboctahedron vertfig.png
26 8 triangles
18 squares
48 24 8.714045 Oh
truncated cuboctahedron
(great rhombicuboctahedron)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated cuboctahedron
Great rhombicuboctahedron.png Truncated cuboctahedron flat.svg 4.6.8
Great rhombicuboctahedron vertfig.png
26 12 squares
8 hexagons
6 octagons
72 48 41.798990 Oh
snub cube
(snub cuboctahedron)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub hexahedron (Ccw)
Snub hexahedron.png Snub cube flat.svg
Snub cube vertfig.png
38 32 triangles
6 squares
60 24 7.889295 O
icosidodecahedron r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png Icosidodecahedron flat.svg
Icosidodecahedron vertfig.png
32 20 triangles
12 pentagons
60 30 13.835526 Ih
truncated dodecahedron t{5,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated dodecahedron
Truncated dodecahedron.png Truncated dodecahedron flat.png 3.10.10
Truncated dodecahedron vertfig.png
32 20 triangles
12 decagons
90 60 85.039665 Ih
truncated icosahedron t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Truncated icosahedron
Truncated icosahedron.png Truncated icosahedron flat-2.svg 5.6.6
Truncated icosahedron vertfig.png
32 12 pentagons
20 hexagons
90 60 55.287731 Ih
(small rhombicosidodecahedron)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Small rhombicosidodecahedron.png Rhombicosidodecahedron flat.png
Small rhombicosidodecahedron vertfig.png
62 20 triangles
30 squares
12 pentagons
120 60 41.615324 Ih
truncated icosidodecahedron
(great rhombicosidodecahedron)
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated icosidodecahedron
Great rhombicosidodecahedron.png Truncated icosidodecahedron flat.svg 4.6.10
Great rhombicosidodecahedron vertfig.png
62 30 squares
20 hexagons
12 decagons
180 120 206.803399 Ih
snub dodecahedron
(snub icosidodecahedron)
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub dodecahedron (Ccw)
Snub dodecahedron ccw.png Snub dodecahedron flat.svg
Snub dodecahedron vertfig.png
92 80 triangles
12 pentagons
150 60 37.616650 I

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".[3]


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.


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).

Construction of Archimedean solids[edit]

The Archimedean solids can be constructed as generator positions in a kaleidoscope.

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. Expansion or cantellation involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges but with a particular ratio between corner and edge truncation.

Construction of Archimedean Solids
Symmetry Tetrahedral
Tetrahedral reflection domains.png
Octahedral reflection domains.png
Icosahedral reflection domains.png
Starting solid
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Uniform polyhedron-33-t0.png
Uniform polyhedron-43-t0.png
Uniform polyhedron-43-t2.png
Uniform polyhedron-53-t0.png
Uniform polyhedron-53-t2.png
Truncation (t) t{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
truncated tetrahedron
Uniform polyhedron-33-t01.png
truncated cube
Uniform polyhedron-43-t01.png
truncated octahedron
Uniform polyhedron-43-t12.png
truncated dodecahedron
Uniform polyhedron-53-t01.png
truncated icosahedron
Uniform polyhedron-53-t12.png
Rectification (r)
Ambo (a)
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t1.png
Uniform polyhedron-53-t1.png
Bitruncation (2t)
Dual kis (dk)
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetrahedron
Uniform polyhedron-33-t12.png
truncated octahedron
Uniform polyhedron-43-t12.png
truncated cube
Uniform polyhedron-43-t01.png
truncated icosahedron
Uniform polyhedron-53-t12.png
truncated dodecahedron
Uniform polyhedron-53-t01.png
Birectification (2r)
Dual (d)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Uniform polyhedron-33-t2.png
Uniform polyhedron-43-t2.png
Uniform polyhedron-43-t0.png
Uniform polyhedron-53-t2.png
Uniform polyhedron-53-t0.png
cantellation (rr)
Expansion (e)
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t02.png
Uniform polyhedron-53-t02.png
Snub rectified (sr)
Snub (s)
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
snub tetratetrahedron
Uniform polyhedron-33-s012.png
snub cuboctahedron
Uniform polyhedron-43-s012.png
snub icosidodecahedron
Uniform polyhedron-53-s012.png
Cantitruncation (tr)
Bevel (b)
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetratetrahedron
Uniform polyhedron-33-t012.png
truncated cuboctahedron
Uniform polyhedron-43-t012.png
truncated icosidodecahedron
Uniform polyhedron-53-t012.png

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid has only tetrahedral symmetry.

See also[edit]


  1. ^ a b Grünbaum (2009).
  2. ^ Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227
  3. ^ Malkevitch (1988), p. 85


  • Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81. 
  • Grünbaum, Branko (2009), "An enduring error", Elemente der Mathematik, 64 (3): 89–101, doi:10.4171/EM/120, MR 2520469 . Reprinted in Pitici, Mircea, ed. (2011), The Best Writing on Mathematics 2010, Princeton University Press, pp. 18–31 .
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G., Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92 .
  • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  Chapter 2

External links[edit]