In geometry, an **Archimedean solid** is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon), excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron.^{ [1] } They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

- Origin of name
- Classification
- Properties
- Chirality
- Construction of Archimedean solids
- Stereographic projection
- See also
- Citations
- General references
- External links

"Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. BrankoGrünbaum ( 2009 ) observed that a 14th polyhedron, the elongated square gyrobicupola (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 antiprisms, whose symmetry groups 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.

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.^{ [2] } During the Renaissance, artists and mathematicians valued *pure forms* with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,^{ [3] } as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the elongated square gyrobicupola (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.^{ [2] }

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images 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, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

Name/ (alternative name) | Schläfli Coxeter | Transparent | Solid | Net | Vertex conf./fig. | Faces | Edges | Vert. | Volume (unit edges) | Point group | Sphericity | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Truncated tetrahedron | t{3,3} | 3.6.6 | 8 | 4 triangles 4 hexagons | 18 | 12 | 2.710576 | T_{d} | 0.7754132 | |||

Cuboctahedron (rhombitetratetrahedron, triangular gyrobicupola) | r{4,3} or rr{3,3} or | 3.4.3.4 | 14 | 8 triangles 6 squares | 24 | 12 | 2.357023 | O_{h} | 0.9049972 | |||

Truncated cube | t{4,3} | 3.8.8 | 14 | 8 triangles 6 octagons | 36 | 24 | 13.599663 | O_{h} | 0.8494937 | |||

Truncated octahedron (truncated tetratetrahedron) | t{3,4} or tr{3,3} or | 4.6.6 | 14 | 6 squares 8 hexagons | 36 | 24 | 11.313709 | O_{h} | 0.9099178 | |||

Rhombicuboctahedron (small rhombicuboctahedron, elongated square orthobicupola) | rr{4,3} | 3.4.4.4 | 26 | 8 triangles 18 squares | 48 | 24 | 8.714045 | O_{h} | 0.9540796 | |||

Truncated cuboctahedron (great rhombicuboctahedron) | tr{4,3} | 4.6.8 | 26 | 12 squares 8 hexagons 6 octagons | 72 | 48 | 41.798990 | O_{h} | 0.9431657 | |||

Snub cube (snub cuboctahedron) | sr{4,3} | 3.3.3.3.4 | 38 | 32 triangles 6 squares | 60 | 24 | 7.889295 | O | 0.9651814 | |||

Icosidodecahedron (pentagonal gyrobirotunda) | r{5,3} | 3.5.3.5 | 32 | 20 triangles 12 pentagons | 60 | 30 | 13.835526 | I_{h} | 0.9510243 | |||

Truncated dodecahedron | t{5,3} | 3.10.10 | 32 | 20 triangles 12 decagons | 90 | 60 | 85.039665 | I_{h} | 0.9260125 | |||

Truncated icosahedron | t{3,5} | 5.6.6 | 32 | 12 pentagons 20 hexagons | 90 | 60 | 55.287731 | I_{h} | 0.9666219 | |||

Rhombicosidodecahedron (small rhombicosidodecahedron) | rr{5,3} | 3.4.5.4 | 62 | 20 triangles 30 squares 12 pentagons | 120 | 60 | 41.615324 | I_{h} | 0.9792370 | |||

Truncated icosidodecahedron (great rhombicosidodecahedron) | tr{5,3} | 4.6.10 | 62 | 30 squares 20 hexagons 12 decagons | 180 | 120 | 206.803399 | I_{h} | 0.9703127 | |||

Snub dodecahedron (snub icosidodecahedron) | sr{5,3} | 3.3.3.3.5 | 92 | 80 triangles 12 pentagons | 150 | 60 | 37.616650 | I | 0.9820114 |

Some definitions of Semiregular polyhedron include one more figure, the Elongated square gyrobicupola or "pseudo-rhombicuboctahedron".^{ [4] }

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.

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

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, 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 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.)

truncated tetrahedron | truncated cube | truncated octahedron | truncated dodecahedron | truncated icosahedron | |||||
---|---|---|---|---|---|---|---|---|---|

triangle-centered | hexagon-centered | octagon-centered | triangle-centered | square-centered | hexagon-centered | Decagon-centered | Triangle-centered | pentagon-centered | hexagon-centered |

cuboctahedron | icosidodecahedron | rhombicuboctahedron | rhombicosidodecahedron | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

square-centered | triangle-centered | vertex-centered | pentagon-centered | triangle-centered | square-centered | square-centered | triangle-centered | Pentagon-centered | Triangle-centered | Square-centered |

truncated cuboctahedron | truncated icosidodecahedron | snub cube | ||||
---|---|---|---|---|---|---|

square-centered | hexagon-centered | octagon-centered | square-centered | |||

- ↑ Steckles, Katie. "The Unwanted Shape".
*YouTube*. Retrieved 20 January 2022. - 1 2 Grünbaum (2009).
- ↑ Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler,
*Archive for History of Exact Sciences*,**50**, 1997, 227 - ↑ Malkevitch (1988), p. 85

- Grünbaum, Branko (2009), "An enduring error",
*Elemente der Mathematik*,**64**(3): 89–101, doi: 10.4171/EM/120 , MR 2520469 . Reprinted in Pitici, Mircea, ed. (2011),*The Best Writing on Mathematics 2010*, Princeton University Press, pp. 18–31. - Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra".
*Mathematical Gazette*.**89**(514): 76–81. doi:10.1017/S0025557200176818. S2CID 125675814.. - Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G. (eds.),
*Shaping Space: A Polyhedral Approach*, Boston: Birkhäuser, pp. 80–92. - Pugh, Anthony (1976).
*Polyhedra: A visual approach*. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2 - 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) - Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (2008). "New light on the rediscovery of the Archimedean solids during the renaissance".
*Archive for History of Exact Sciences*.**62**(4): 457–467. Bibcode:2008AHES...62..457S. doi:10.1007/s00407-008-0024-z. ISSN 0003-9519. S2CID 122216140..

- Weisstein, Eric W. "Archimedean solid".
*MathWorld*. - Archimedean 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
- Archimedean Solids at Visual Polyhedra by David I. McCooey
- 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
^{[ permanent dead link ]}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

In geometry, a **dodecahedron** or **duodecahedron** is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

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 Johnson solid is the square-based pyramid with equilateral sides (*J*_{1}); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a “Johnson solid”.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as *poly-* + *-hedron*.

A **Platonic solid** is a convex regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, the **rhombicuboctahedron**, or **small rhombicuboctahedron**, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, the **truncated cuboctahedron** is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a **9**-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

In geometry, the **truncated icosidodecahedron** is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

In geometry, the **elongated square gyrobicupola** or **pseudo-rhombicuboctahedron** is one of the Johnson solids (*J*_{37}). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex. It strongly resembles, but should not be mistaken for, the small rhombicuboctahedron, which *is* an Archimedean solid. It is also a canonical polyhedron.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

A **uniform polyhedron** has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, the **great icosahedron** is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

In geometry, a **vertex configuration** is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, **Conway polyhedron notation**, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

A **pseudo-uniform polyhedron** is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.

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