Archimedean solid

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Archimedean solid

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
an Archimedean solid is a highly symmetric, semi-regular convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
composed of two or more types of regular polygon

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s meeting in identical vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. They are distinct from the Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s, whose regular polygonal faces do not meet in identical vertices. The symmetry of the Archimedean solids excludes the members of the dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
, the prisms

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
and antiprism

Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s.

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Encyclopedia

In geometry

Geometry

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
polyhedron

Polyhedron

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s meeting in identical vertices

Vertex (geometry)

Platonic solid

Johnson solid

Dihedral group

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
and antiprism

Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s. The Archimedean solids can all be made via Wythoff construction

Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
s from the Platonic solids with tetrahedral

Tetrahedral symmetry

A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
, octahedral

Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
and icosahedral symmetry

Icosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
. See Convex uniform polyhedron

Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
.

Origin of name

The Archimedean solids take their name from Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
, who discussed them in a now-lost work. During the Renaissance

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe....
, artist

Artist

The definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art....
s and mathematician

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s valued pure forms and rediscovered all of these forms. This search was completed around 1620 by Johannes Kepler

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
, who defined prisms

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

Classification

There are 13 Archimedean solids (15 if the mirror image

Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
s of two enantiomorphs

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
, 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

Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
of (4,6,8) means that a square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
, hexagon

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
, and octagon

Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
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

Defect (geometry)

In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle....
.

Name (Vertex configuration Vertex configuration In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex.... )	Transparent	Faces	Faces (By type)	Edges	Vertices	Symmetry group List of spherical symmetry groups List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
truncated tetrahedron Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges.... (3.6.6)	(Animation)	8	4 triangles 4 hexagon Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol .... s	18	12	T_d
cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square.... (3.4.3.4)	(Animation)	14	8 triangles 6 squares Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	24	12	O_h
truncated cube Truncated cube The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges.... or truncated hexahedron (3.8.8)	(Animation)	14	8 triangles 6 octagon Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol .... s	36	24	O_h
truncated octahedron Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... (4.6.6)	(Animation)	14	6 squares 8 hexagons	36	24	O_h
rhombicuboctahedron Rhombicuboctahedron The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each.... or small rhombicuboctahedron (3.4.4.4 )	(Animation)	26	8 triangles 18 squares	48	24	O_h
truncated cuboctahedron Truncated cuboctahedron The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges.... or great rhombicuboctahedron (4.6.8)	(Animation)	26	12 squares 8 hexagons 6 octagons	72	48	O_h
snub cube Snub cube The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles.... or snub hexahedron or snub cuboctahedron (2 chiral Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone.... forms) (3.3.3.3.4)	(Animation) (Animation)	38	32 triangles 6 squares	60	24	O
icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon.... (3.5.3.5)	(Animation)	32	20 triangles 12 pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?.... s	60	30	I_h
truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges.... (3.10.10)	(Animation)	32	20 triangles 12 decagon Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 .... s	90	60	I_h
truncated icosahedron Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges.... or buckyball Fullerene Fullerene are a family of carbon Allotropy, molecules composed entirely of carbon, in the form of a hollow sphere, ellipsoid, cylinder , or plane.... or football Football (ball) A football is a ball used to play one of the various sports known as football.In the distant past, crude balls such as inflated pigs' bladders were used, but balls are now designed by teams of engineers to exacting specifications.... /soccer ball (5.6.6 )	(Animation)	32	12 pentagons 20 hexagons	90	60	I_h
rhombicosidodecahedron Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges.... or small rhombicosidodecahedron (3.4.5.4)	(Animation)	62	20 triangles 30 squares 12 pentagons	120	60	I_h
truncated icosidodecahedron Truncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges.... or great rhombicosidodecahedron (4.6.10)	(Animation)	62	30 squares 20 hexagons 12 decagons	180	120	I_h
snub dodecahedron Snub dodecahedron The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles.... or snub icosidodecahedron (2 chiral Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone.... forms) (3.3.3.3.5)	(Animation) (Animation)	92	80 triangles 12 pentagons	150	60	I

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular

Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
.

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

Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compound

Chemical compound

A chemical compound is a Chemical substance consisting of two or more different chemical element Chemical bond together in a fixed mass ratio that can be split into simpler substances....
s).

The duals

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
of the Archimedean solids are called the Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
s. Together with the bipyramid

Bipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
s and trapezohedra

Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
, these are the face-uniform solids with regular vertices.

External links

by Eric W. Weisstein
Eric W. Weisstein

Eric W. Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science . He currently works for Wolfram Research, Inc....
, Wolfram Demonstrations Project
Wolfram Demonstrations Project

The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
.
by Dr. R. Mäder
, The Encyclopedia of Polyhedra by George W. Hart
by James S. Plank
in Java
Designed by Tom Barber
: Software used to create many of the images on this page.

Archimedean solid

Origin of name

Classification

See also

External links