Archimedean solid

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. The symmetry of the Archimedean solids excludes the members of the dihedral group, the prisms and antiprisms. The Archimedean solids can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry. See Convex uniform polyhedron.

Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued "pure forms" and rediscovered all of these forms. This search was completed around 1620 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

Classification

There are 13 Archimedean solids (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).

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 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).

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.

See also

* semiregular polyhedron
* uniform polyhedron
* List of uniform polyhedra

References

* (Section 3-9)

External links

*
* [http://demonstrations.wolfram.com/ArchimedeanSolids/ Archemedian Solids] by Eric W. Weisstein, The Wolfram Demonstrations Project.
* [http://www.software3d.com/Archimedean.php Paper models of Archimedean Solids and Catalan Solids]
* [http://www.korthalsaltes.com/archimedean_solids_pictures.html Paper models(nets) of Archimedean solids]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] by Dr. R. Mäder
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] , "The Encyclopedia of Polyhedra" by George W. Hart
* [http://www.cs.utk.edu/~plank/plank/origami/penultimate/intro.html Penultimate Modular Origami] by James S. Plank
* [http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra] in Java
* [http://video.google.com/videoplay?docid=7084140981126344386&q=tom+barber&hl=en Contemporary Archimedean Solid Surfaces] Designed by Tom Barber
* [http://www.software3d.com/Stella.php Stella: Polyhedron Navigator] : Software used to create many of the images on this page.


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