Archimedean solid

The truncated icosidodecahedron, with 120 vertices, is the largest Archimedean solid.	The pseudo-rhombicuboctahedron has a single vertex figure, 3.4.4.4, but with a twist on one square cupola. Unlike the (untwisted) rhombicuboctahedron it is not vertex transitive.

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 elongated square gyrobicupola (pseudo-rhombicuboctahedron) is considered an Archimedean solid or a Johnson solid: it 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 every vertex to every other vertex. Based on its existence, Branko Grünbaum (2009) has suggested a terminological distinction in which an Archimedean solid is defined as having the same vertex figure at each vertex (including the elongated square gyrobicupola) while a uniform polyhedron is defined as having each vertex symmetric to each other vertex (excluding the gyrobicupola).

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. All but 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 and rediscovered all of these forms. This search was almost entirely completed around 1620 by Johannes Kepler,^[2] who defined prisms, antiprisms, and the non-convex solids known as the 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]

Classification

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).

Name (Alternative name)	Schläfli Coxeter	Transparent	Vertex figure	Faces		Edges	Vert.	Point group
truncated tetrahedron	{3,3}	(Animation)	3.6.6	8	4 triangles 4 hexagons	18	12	T_d
cuboctahedron (rhombitetratetrahedron)	r{4,3} or rr{3,3} or	(Animation)	3.4.3.4	14	8 triangles 6 squares	24	12	O_h
truncated cube	t{4,3}	(Animation)	3.8.8	14	8 triangles 6 octagons	36	24	O_h
truncated octahedron (truncated tetratetrahedron)	t{3,4} or tr{3,3} or	(Animation)	4.6.6	14	6 squares 8 hexagons	36	24	O_h
rhombicuboctahedron (small rhombicuboctahedron)	rr{4,3}	(Animation)	3.4.4.4	26	8 triangles 18 squares	48	24	O_h
truncated cuboctahedron (great rhombicuboctahedron)	tr{4,3}	(Animation)	4.6.8	26	12 squares 8 hexagons 6 octagons	72	48	O_h
snub cube (snub cuboctahedron)	sr{4,3}	(Animation)	3.3.3.3.4	38	32 triangles 6 squares	60	24	O
icosidodecahedron	r{5,3}	(Animation)	3.5.3.5	32	20 triangles 12 pentagons	60	30	I_h
truncated dodecahedron	t{5,3}	(Animation)	3.10.10	32	20 triangles 12 decagons	90	60	I_h
truncated icosahedron	t{3,5}	(Animation)	5.6.6	32	12 pentagons 20 hexagons	90	60	I_h
rhombicosidodecahedron (small rhombicosidodecahedron)	rr{5,3}	(Animation)	3.4.5.4	62	20 triangles 30 squares 12 pentagons	120	60	I_h
truncated icosidodecahedron (great rhombicosidodecahedron)	tr{5,3}	(Animation)	4.6.10	62	30 squares 20 hexagons 12 decagons	180	120	I_h
snub dodecahedron (snub icosidodecahedron)	sr{5,3}	(Animation)	3.3.3.3.5	92	80 triangles 12 pentagons	150	60	I

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 (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

Further information: Uniform polyhedron and Conway polyhedron notation

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

The different Archimedean and Platonic solids can be related to each other using just several 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	Octahedral		Icosahedral
Starting solid Operation	Symbol {p,q}	Tetrahedron {3,3}	Cube {4,3}	Octahedron {3,4}	Dodecahedron {5,3}	Icosahedron {3,5}
Truncation (t)	t{p,q}	truncated tetrahedron	truncated cube	truncated octahedron	truncated dodecahedron	truncated icosahedron
Rectification (r) Ambo (a)	r{p,q}	tetratetrahedron	cuboctahedron		icosidodecahedron
Bitruncation (2t) Dual kis (dk)	2t{p,q}	truncated tetrahedron	truncated octahedron	truncated cube	truncated icosahedron	truncated dodecahedron
Birectification (2r) Dual (d)	2r{p,q}	tetrahedron	octahedron	cube	icosahedron	dodecahedron
cantellation (rr) Expansion (e)	rr{p,q}	rhombitetratetrahedron	rhombicuboctahedron		rhombicosidodecahedron
Snub rectified (sr) Snub (s)	sr{p,q}	snub tetratetrahedron	snub cuboctahedron		snub icosidodecahedron
Cantitruncation (tr) Bevel (b)	tr{p,q}	truncated tetratetrahedron	truncated cuboctahedron		truncated icosidodecahedron

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, in part due to self-duality of the tetrahedron, only one Archimedean solid has only tetrahedral symmetry.

Notes

↑ 1.0 1.1 Grünbaum (2009).
↑ 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
↑ Malkevitch (1988), p. 85

References

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