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.
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.
| truncated tetrahedron|
|8|| 4 triangles|
|14|| 8 triangles|
| truncated cube|
or truncated hexahedron
|14|| 8 triangles|
| truncated octahedron|
|14|| 6 squares|
or small rhombicuboctahedron
| truncated cuboctahedron|
or great rhombicuboctahedron
| snub cube|
or snub hexahedron
or snub cuboctahedron
(2 chiral forms)
|32|| 20 triangles|
| truncated dodecahedron|
| truncated icosahedron|
or football/soccer ball
|32|| 12 pentagons|
or small rhombicosidodecahedron
|62|| 20 triangles|
| truncated icosidodecahedron|
or great rhombicosidodecahedron
| snub dodecahedron|
or snub icosidodecahedron
(2 chiral forms)
|92|| 80 triangles|
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).
- 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)
- Weisstein, Eric W., "Archimedean solid" from MathWorld.
- Archemedian Solids by Eric W. Weisstein, Wolfram Demonstrations Project.
- Paper models of Archimedean Solids and Catalan Solids
- Paper models(nets) of Archimedean solids
- The Uniform Polyhedra by Dr. R. Mäder
- Virtual Reality Polyhedra, The Encyclopedia of Polyhedra by George W. Hart
- Penultimate Modular Origami by James S. Plank
- Interactive 3D polyhedra in Java
- Contemporary Archimedean Solid Surfaces Designed by Tom Barber
- Stella: Polyhedron Navigator: Software used to create many of the images on this page.
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