In
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 scientis ...
. They are the
convex uniform polyhedra composed of
regular polygons 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 e ...
s (which are composed of only one type of polygon), excluding the
prisms
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
and
antiprisms, and excluding the
pseudorhombicuboctahedron. 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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...
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'' ...
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 faces consist of regular polygons that meet in the same p ...
(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
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
and
antiprisms, 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 amb ...
s 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
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 scientis ...
, 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 ide ...
,
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
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
,
antiprisms, 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
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 faces consist of regular polygons that meet in the same p ...
(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.
Classification
There are 13 Archimedean solids (not counting 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 faces consist of regular polygons that meet in the same p ...
; 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 substance ...
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 of 4.6.8 means that a
square,
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
, and
octagon 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 faces consist of regular polygons that meet in the same p ...
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 def ...
.
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 tw ...
and are called
quasi-regular.
The
duals 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 so ...
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 ...
s and
trapezohedra, 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 Face (geometry), faces are the same. More specifically, all faces must be not ...
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
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 substance ...
, 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 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, 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
See also
*
Aperiodic tiling
*
Archimedean graph
*
Icosahedral twins
*
List of uniform polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are ...
*
Prince Rupert's cube#Generalizations
*
Quasicrystal
*
Regular polyhedron
*
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 ...
*
Toroidal polyhedron
*
Uniform polyhedron
Citations
Works cited
*. Reprinted in .
*.
General references
*.
* Chapter 2
* (Section 3–9)
*.
External links
*
Archimedean Solidsby
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.
Paper models of Archimedean Solids and Catalan SolidsFree paper models(nets) of Archimedean solidsThe Uniform Polyhedraby 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