Archimedean solid

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Truncated tetrahedron, cuboctahedron and truncated icosidodecahedron. The first and the last one can be described as the smallest and the largest Archimedean solid, respectively.
Rhombicuboctahedron and pseudo-rhombicuboctahedron

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five 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 each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. Branko Grünbaum (2009) observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

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 Archimedean 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[]

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. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)

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]

Classification[]

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, the snub cube and snub dodecahedron, 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).

Name/
(alternative name) 		Schläfli
Coxeter 		Transparent Solid Net Vertex
conf./fig. 		Faces Edges Vert. Volume
(unit edges) 		Point
group 		Sphericity
truncated tetrahedron t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 		Truncated tetrahedron   Cog-scripted-svg-blue.svg Polyhedron truncated 4a max.png Polyhedron truncated 4a net.svg 3.6.6
Polyhedron truncated 4a vertfig.png 		8 4 triangles
4 hexagons 		18 12 2.710576 Td 0.7754132
cuboctahedron
(rhombitetratetrahedron, triangular gyrobicupola)		 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   Cog-scripted-svg-blue.svg Polyhedron 6-8 max.png Polyhedron 6-8 net.svg 3.4.3.4
Polyhedron 6-8 vertfig.png 		14 8 triangles
6 squares 		24 12 2.357023 Oh 0.9049972
truncated cube t{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png 		Truncated hexahedron   Cog-scripted-svg-blue.svg Polyhedron truncated 6 max.png Polyhedron truncated 6 net.svg 3.8.8
Polyhedron truncated 6 vertfig.png 		14 8 triangles
6 octagons 		36 24 13.599663 Oh 0.8494937
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   Cog-scripted-svg-blue.svg Polyhedron truncated 8 max.png Polyhedron truncated 8 net.svg 4.6.6
Polyhedron truncated 8 vertfig.png 		14 6 squares
8 hexagons 		36 24 11.313709 Oh 0.9099178
rhombicuboctahedron
(small rhombicuboctahedron, elongated square orthobicupola)		 rr{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png 		Rhombicuboctahedron   Cog-scripted-svg-blue.svg Polyhedron small rhombi 6-8 max.png Polyhedron small rhombi 6-8 net.svg 3.4.4.4
Polyhedron small rhombi 6-8 vertfig.png 		26 8 triangles
18 squares 		48 24 8.714045 Oh 0.9540796
truncated cuboctahedron
(great rhombicuboctahedron)		 tr{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png 		Truncated cuboctahedron   Cog-scripted-svg-blue.svg Polyhedron great rhombi 6-8 max.png Polyhedron great rhombi 6-8 net.svg 4.6.8
Polyhedron great rhombi 6-8 vertfig light.png 		26 12 squares
8 hexagons
6 octagons 		72 48 41.798990 Oh 0.9431657
snub cube
(snub cuboctahedron)		 sr{4,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png 		Snub hexahedron (Ccw)   Cog-scripted-svg-blue.svg Polyhedron snub 6-8 left max.png Polyhedron snub 6-8 left net.svg 3.3.3.3.4
Polyhedron snub 6-8 left vertfig.png 		38 32 triangles
6 squares 		60 24 7.889295 O 0.9651814
icosidodecahedron
(pentagonal gyrobirotunda)		 r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png 		Icosidodecahedron   Cog-scripted-svg-blue.svg Polyhedron 12-20 max.png Polyhedron 12-20 net.svg 3.5.3.5
Polyhedron 12-20 vertfig.png 		32 20 triangles
12 pentagons 		60 30 13.835526 Ih 0.9510243
truncated dodecahedron t{5,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png 		Truncated dodecahedron   Cog-scripted-svg-blue.svg Polyhedron truncated 12 max.png Polyhedron truncated 12 net.svg 3.10.10
Polyhedron truncated 12 vertfig.png 		32 20 triangles
12 decagons 		90 60 85.039665 Ih 0.9260125
truncated icosahedron t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png 		Truncated icosahedron   Cog-scripted-svg-blue.svg Polyhedron truncated 20 max.png Polyhedron truncated 20 net.svg 5.6.6
Polyhedron truncated 20 vertfig.png 		32 12 pentagons
20 hexagons 		90 60 55.287731 Ih 0.9666219
rhombicosidodecahedron
(small rhombicosidodecahedron)		 rr{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png 		Rhombicosidodecahedron   Cog-scripted-svg-blue.svg Polyhedron small rhombi 12-20 max.png Polyhedron small rhombi 12-20 net.svg 3.4.5.4
Polyhedron small rhombi 12-20 vertfig.png 		62 20 triangles
30 squares
12 pentagons 		120 60 41.615324 Ih 0.9792370
truncated icosidodecahedron
(great rhombicosidodecahedron)		 tr{5,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png 		Truncated icosidodecahedron   Cog-scripted-svg-blue.svg Polyhedron great rhombi 12-20 max.png Polyhedron great rhombi 12-20 net.svg 4.6.10
Polyhedron great rhombi 12-20 vertfig light.png 		62 30 squares
20 hexagons
12 decagons 		180 120 206.803399 Ih 0.9703127
snub dodecahedron
(snub icosidodecahedron)		 sr{5,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png 		Snub dodecahedron (Cw)   Cog-scripted-svg-blue.svg Polyhedron snub 12-20 left max.png Polyhedron snub 12-20 left net.svg 3.3.3.3.5
Polyhedron snub 12-20 left vertfig.png 		92 80 triangles
12 pentagons 		150 60 37.616650 I 0.9820114

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

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.

Chirality[]

The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). 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[]

Further information: Uniform polyhedron and Conway polyhedron notation
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. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. An 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 splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.

Construction of Archimedean Solids
Symmetry Tetrahedral
Tetrahedral reflection domains.png 		Octahedral
Octahedral reflection domains.png 		Icosahedral
Icosahedral reflection domains.png
Starting solid
Operation		 Symbol
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png 		Tetrahedron
{3,3}
Uniform polyhedron-33-t0.png		 Cube
{4,3}
Uniform polyhedron-43-t0.svg		 Octahedron
{3,4}
Uniform polyhedron-43-t2.svg		 Dodecahedron
{5,3}
Uniform polyhedron-53-t0.svg		 Icosahedron
{3,5}
Uniform polyhedron-53-t2.svg
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.svg		 truncated octahedron
Uniform polyhedron-43-t12.svg		 truncated dodecahedron
Uniform polyhedron-53-t01.svg		 truncated icosahedron
Uniform polyhedron-53-t12.svg
Rectification (r)
Ambo (a)		 r{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png 		tetratetrahedron
(octahedron)
Uniform polyhedron-33-t1.png		 cuboctahedron
Uniform polyhedron-43-t1.svg		 icosidodecahedron
Uniform polyhedron-53-t1.svg
Bitruncation (2t)
Dual kis (dk)		 2t{p,q}
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.svg		 truncated icosahedron
Uniform polyhedron-53-t12.svg		 truncated dodecahedron
Uniform polyhedron-53-t01.svg
Birectification (2r)
Dual (d)		 2r{p,q}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png 		tetrahedron
Uniform polyhedron-33-t2.png		 octahedron
Uniform polyhedron-43-t2.svg		 cube
Uniform polyhedron-43-t0.svg		 icosahedron
Uniform polyhedron-53-t2.svg		 dodecahedron
Uniform polyhedron-53-t0.svg
cantellation (rr)
Expansion (e)		 rr{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png 		rhombitetratetrahedron
(cuboctahedron)
Uniform polyhedron-33-t02.png		 rhombicuboctahedron
Uniform polyhedron-43-t02.png		 rhombicosidodecahedron
Uniform polyhedron-53-t02.png
Snub rectified (sr)
Snub (s)		 sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png 		snub tetratetrahedron
(icosahedron)
Uniform polyhedron-33-s012.svg		 snub cuboctahedron
Uniform polyhedron-43-s012.png		 snub icosidodecahedron
Uniform polyhedron-53-s012.png
Cantitruncation (tr)
Bevel (b)		 tr{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png 		truncated tetratetrahedron
(truncated octahedron)
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 that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)

Stereographic projection[]

truncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron
Truncated tetrahedron stereographic projection triangle.png
triangle-centered 		Truncated tetrahedron stereographic projection hexagon.png
hexagon-centered 		Truncated cube stereographic projection octagon.png
octagon-centered 		Truncated cube stereographic projection triangle.png
triangle-centered 		Truncated octahedron stereographic projection square.png
square-centered 		Truncated octahedron stereographic projection hexagon.png
hexagon-centered 		Truncated dodecahedron stereographic projection decagon.png
Decagon-centered 		Truncated dodecahedron stereographic projection triangle.png
Triangle-centered 		Truncated icosahedron stereographic projection pentagon.png
pentagon-centered 		Truncated icosahedron stereographic projection hexagon.png
hexagon-centered
cuboctahedron icosidodecahedron rhombicuboctahedron rhombicosidodecahedron
Cuboctahedron stereographic projection square.png
square-centered 		Cuboctahedron stereographic projection triangle.png
triangle-centered 		Cuboctahedron stereographic projection vertex.png
vertex-centered 		Icosidodecahedron stereographic projection pentagon.png
pentagon-centered 		Icosidodecahedron stereographic projection triangle.png
triangle-centered 		Rhombicuboctahedron stereographic projection square.png
square-centered 		Rhombicuboctahedron stereographic projection square2.png
square-centered 		Rhombicuboctahedron stereographic projection triangle.png
triangle-centered 		Rhombicosidodecahedron stereographic projection pentagon'.png
Pentagon-centered 		Rhombicosidodecahedron stereographic projection triangle.png
Triangle-centered 		Rhombicosidodecahedron stereographic projection square.png
Square-centered
truncated cuboctahedron truncated icosidodecahedron snub cube
Truncated cuboctahedron stereographic projection square.png
square-centered 		Truncated cuboctahedron stereographic projection hexagon.png
hexagon-centered 		Truncated cuboctahedron stereographic projection octagon.png
octagon-centered 		Truncated icosidodecahedron stereographic projection decagon.png Truncated icosidodecahedron stereographic projection hexagon.png Truncated icosidodecahedron stereographic projection square.png Snub cube stereographic projection.png
square-centered

See also[]

Citations[]

  1. ^ Jump up to: 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

General references[]

  • 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.
  • Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81..
  • Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G. (eds.), Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92.
  • Pugh, Anthony (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2
  • 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)
  • Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (2008). "New light on the rediscovery of the Archimedean solids during the renaissance". Archive for History of Exact Sciences. 62 (4): 457–467. doi:10.1007/s00407-008-0024-z. ISSN 0003-9519..

External links[]

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