The rhombicosidodecahedron, one of the Archimedean solids

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.

"Identical vertices" are usually taken to mean that for any two vertices, there must be an isometry of the entire solid that takes one vertex to the other. Sometimes it is instead only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. This difference in definitions controls whether the Elongated square gyrobicupola is considered an Archimedean solid or a Johnson solid.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, despite meeting the above definition. With this restriction, there are only finitely many Archimedean solids. They can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Origin of name [edit]

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,^[1] who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

Classification [edit]

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).

Name (Vertex configuration)	Transparent	Solid	Net	Faces		Edges	Vertices	Point group
truncated tetrahedron (3.6.6)	(Animation)			8	4 triangles 4 hexagons	18	12	Td
cuboctahedron (3.4.3.4)	(Animation)			14	8 triangles 6 squares	24	12	Oh
truncated cube or truncated hexahedron (3.8.8)	(Animation)			14	8 triangles 6 octagons	36	24	Oh
truncated octahedron (4.6.6)	(Animation)			14	6 squares 8 hexagons	36	24	Oh
rhombicuboctahedron or small rhombicuboctahedron (3.4.4.4 )	(Animation)			26	8 triangles 18 squares	48	24	Oh
truncated cuboctahedron or great rhombicuboctahedron (4.6.8)	(Animation)			26	12 squares 8 hexagons 6 octagons	72	48	Oh
snub cube or snub hexahedron or snub cuboctahedron (2 chiral forms) (3.3.3.3.4)	(Animation) (Animation)			38	32 triangles 6 squares	60	24	O
icosidodecahedron (3.5.3.5)	(Animation)			32	20 triangles 12 pentagons	60	30	Ih
truncated dodecahedron (3.10.10)	(Animation)			32	20 triangles 12 decagons	90	60	Ih
Truncated icosahedron (5.6.6 )	(Animation)			32	12 pentagons 20 hexagons	90	60	Ih
rhombicosidodecahedron or small rhombicosidodecahedron (3.4.5.4)	(Animation)			62	20 triangles 30 squares 12 pentagons	120	60	Ih
truncated icosidodecahedron or great rhombicosidodecahedron (4.6.10)	(Animation)			62	30 squares 20 hexagons 12 decagons	180	120	Ih
snub dodecahedron or snub icosidodecahedron (2 chiral forms) (3.3.3.3.5)	(Animation) (Animation)			92	80 triangles 12 pentagons	150	60	I

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".^[2]

Properties [edit]

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 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.

Chirality [edit]

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).

See also [edit]

• Aperiodic tiling
• Catalan solid
• List of uniform polyhedra
• Platonic solid
• Johnson solid
• Toroidal polyhedron
• Quasicrystal
• semiregular polyhedron
• regular polyhedron
• uniform polyhedron

Notes [edit]

References [edit]

• Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette 89 (514): 76–81. 
• 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)
• Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G., Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92 .

External links [edit]

• Weisstein, Eric W., "Archimedean solid", MathWorld.
• Archemedian Solids by Eric W. Weisstein, Wolfram Demonstrations Project.
• Paper models of Archimedean Solids and Catalan Solids
• Free 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
• Solid Body Viewer 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