Truncated tetrahedron, cuboctahedron and truncated icosidodecahedron. The first can be described as the smallest Archimedean solid, the last as the largest.
Rhombicuboctahedron and pseudo-rhombicuboctahedron
In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids (each with only one type of polygon face), and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive.[1] An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
In these polyhedra, the vertices are identical, in the sense that a global isometry of the entire solid takes any one vertex to any other. Branko Grünbaum (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 parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of 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.
^Steckles, Katie. "The Unwanted Shape". YouTube. Retrieved 20 January 2022.
In geometry, an Archimedeansolid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They...
mathematics, a Catalan solid, or Archimedean dual, is a polyhedron that is dual to an Archimedeansolid. There are 13 Catalan solids. They are named after...
Platonic solid, Archimedeansolid, uniform prism, or uniform antiprism) before they refer to it as a "Johnson solid". As in any strictly convex solid, at least...
geometry, the rhombicosidodecahedron is an Archimedeansolid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular...
and the octahedron as {3,4}. Archimedeansolid Catalan solid Deltahedron Johnson solid Goldberg solid Kepler–Poinsot solids List of regular polytopes Prince...
Archimedean point Archimedean property ArchimedeansolidArchimedean spiral Archimedean tiling Archimedean screw Claw of Archimedes The Archimedeans, the mathematical...
Platonic solids - regular polyhedra (all faces of the same type) Archimedeansolids - polyhedra with more than one polygon face type. Catalan solids - duals...
= 1). Truncated figures The truncated tetrahedron is the simplest Archimedeansolid, made of four triangles and four hexagons, the hexagonal prism, which...
from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedeansolid that is not only vertex-transitive but also edge-transitive. It is...
the same length, ensuring the triangles are equilateral), it is an Archimedeansolid. The polyhedron has octahedral symmetry, like the cube and octahedron...
include: The thirteen Archimedeansolids. The elongated square gyrobicupola, also called a pseudo-rhombicuboctahedron, a Johnson solid, has identical vertex...
geometry, the truncated icosahedron is an Archimedeansolid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular...
each separating a triangle from a pentagon. As such it is one of the Archimedeansolids and more particularly, a quasiregular polyhedron. An icosidodecahedron...
dodecahedron or omnitruncated icosahedron is an Archimedeansolid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular...
In geometry, the truncated dodecahedron is an Archimedeansolid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges...
highly symmetrical Platonic solids, the Archimedeansolids and their duals the Catalan solids, and the regular-faced Johnson solids. Many of the most studied...
In geometry, the snub cube, or snub cuboctahedron, is an Archimedeansolid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and...
polyhedron are obtained. Some Waterman polyhedra create Platonic solids and Archimedeansolids. For this comparison of Waterman polyhedra they are normalized...
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedeansolids, prisms, or antiprisms). They were named by Norman Johnson, who...
be an Archimedeansolid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedeansolids. It is...
dodecahedron, or snub icosidodecahedron, is an Archimedeansolid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular...
In geometry, the truncated octahedron is the Archimedeansolid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's...
Platonic and Archimedeansolids". Math. Commun. 16 (2): 491–507. Pellicer, Daniel; Williams, Gordon (2012). "Minimal Covers of the Archimedean Tilings, Part...
five regular Platonic solids: the tetrahedron, octahedron, cube, dodecahedron and icosahedron; the two quasiregular Archimedeansolids: the cuboctahedron...
In geometry, the truncated cube, or truncated hexahedron, is an Archimedeansolid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and...
faces of non-prismatic uniform polyhedra such as the Platonic solids or the Archimedeansolids: the triangle, the square, the pentagon, the hexagon, the octagon...
An Archimedean point (Latin: Punctum Archimedis) is a hypothetical viewpoint from which certain objective truths can perfectly be perceived (also known...
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