geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an Archimedean solid is one of the 13 solids first enumerated by

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

. They are the

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

composed of

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

s meeting in identical vertices, excluding the five

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s (which are composed of only one type of polygon), excluding the prisms and

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s, and excluding the

pseudorhombicuboctahedron In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same patte ...

. They are a subset of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s, whose regular polygonal faces do not need to meet in identical vertices. "Identical vertices" means that each two vertices are symmetric to each other: A global

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

of the entire solid takes one vertex to the other while laying the solid directly on its initial position. observed that a 14th polyhedron, the

elongated square gyrobicupola In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its Face (geometry), faces consist of regular polygons that meet ...

(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

s, whose

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

s are the

dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...

s, 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 construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

s from the Platonic solids with

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

octahedral In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

and

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...

Origin of name

The Archimedean solids take their name from

, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.. During the

Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...

artist An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating an art. The common usage in both everyday speech and academic discourse refers to a practitioner in the visual arts only. However, th ...

s and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s valued ''pure forms'' with high symmetry, and by around 1620

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

had completed the rediscovery of the 13 polyhedra, as well as defining the prisms,

antiprisms In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass ...

, and the non-convex solids known as Kepler-Poinsot polyhedra. (See for more information about the rediscovery of the Archimedean solids during the renaissance.) Kepler may have also found the

(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 Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...

Classification

There are 13 Archimedean solids (not counting the

; 15 if the

mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...

s 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 In geometry, a vertex configurationCrystallography ...

of 4.6.8 means that a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

, and

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

meet at a vertex (with the order taken to be clockwise around the vertex). Some definitions of

Semiregular polyhedron In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

include one more figure, the

Elongated square gyrobicupola In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its Face (geometry), faces consist of regular polygons that meet ...

or "pseudo-rhombicuboctahedron"., p. 85

Properties

The number of vertices is 720° divided by the vertex

angle defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the de ...

. The cuboctahedron and icosidodecahedron are

edge-uniform In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

and are called quasi-regular. The

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...

of the Archimedean solids are called the

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

s. Together with the

bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not ...

s and

trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

, these are the

face-uniform In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

solids with regular vertices.

Chirality

The snub cube and snub dodecahedron are known as ''

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

'', 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

, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain

chemical compound A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...

s.)

Construction of Archimedean solids

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid,

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

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 Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

, or

cantellation In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...

, 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 as the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification. 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

Citations

Works cited

*. Reprinted in . *.

General references

*. * Chapter 2 * (Section 3–9) *.

External links

*
Archimedean Solids
by

Eric W. Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

.
Paper models of Archimedean Solids and Catalan Solids

Free paper models(nets) of Archimedean solids

The Uniform Polyhedra
by Dr. R. Mäder

at Visual Polyhedra by David I. McCooey

''The Encyclopedia of Polyhedra'' by George W. Hart

by James S. Plank

in Java

is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
Stella: Polyhedron Navigator
Software used to create many of the images on this page.
Paper Models of Archimedean (and other) Polyhedra
{{DEFAULTSORT:Archimedean Solid