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Polyhedra in which all vertices are the same
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons and are vertex-transitive,[citation needed] although they
Archimedean_solid
13 polyhedra; duals of the Archimedean solids
The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and
Catalan_solid
Convex polyhedron with regular faces
the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms. The solids are named after Norman Johnson and Victor Zalgaller
Johnson_solid
Greek mathematician and physicist (c. 287 – 212 BC)
deriving an approximation of pi (π), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very
Archimedes
Polyhedron resembling a soccerball
pioneered, are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron. The truncated icosahedron can be
Truncated_icosahedron
Any of the five regular polyhedra
the cube as {4,3}, and the octahedron as {3,4}. Archimedean solid Catalan solid Deltahedron Johnson solid Goldberg polyhedron Kepler-Poinsot polyhedron
Platonic_solid
Graph with an Archimedean solid as its skeleton
of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of
Archimedean_graph
of the Archimedean solids. Archimedean solids: Catalan solids: Construction instructions for the following solids are available: Platonic solids, Archimedean
Solids with icosahedral symmetry
Solids_with_icosahedral_symmetry
Archimedean solid with 62 faces
geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular
Rhombicosidodecahedron
Archimedean solid with 32 faces
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges
Truncated_dodecahedron
Topics referred to by the same term
Archimedean point Archimedean property Archimedean solid Archimedean spiral Archimedean tiling Archimedean screw Claw of Archimedes The Archimedeans, the mathematical
Archimedean
uniform polyhedra include Platonic and Archimedean solids as well as prisms and antiprisms. The Johnson solids are named after American mathematician
List_of_Johnson_solids
Archimedean solid with 26 faces
an Archimedean solid, and its dual is a Catalan solid, the deltoidal icositetrahedron. The elongated square gyrobicupola, the 37th Johnson solid, is
Rhombicuboctahedron
Archimedean solid with 14 faces
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and
Truncated_cube
Flat-sided three-dimensional shape
definitions overlap with the quasi-regular class. The Archimedean solids once had fourteenth solid known as the pseudorhombicuboctahedron, a mistaken construction
Polyhedron
Variously-defined concept in geometry
include: The thirteen Archimedean solids. The elongated square gyrobicupola (also called a pseudo-rhombicuboctahedron), a Johnson solid, has identical vertex
Semiregular_polyhedron
Archimedean solid with 92 faces
dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular
Snub_dodecahedron
Archimedean solid with 14 faces
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six equilateral square pyramids, one
Truncated_octahedron
Catalan solid with 24 faces
tetrakis cube, and kiscube) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be called a disdyakis hexahedron or
Tetrakis_hexahedron
Archimedean solid with 62 faces
dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular
Truncated_icosidodecahedron
Archimedean solid with 8 faces
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges
Truncated_tetrahedron
Archimedean solid with 32 faces
separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. One way to construct
Icosidodecahedron
Polyhedron related to sphere packing
polyhedron are obtained. Some Waterman polyhedra create Platonic solids and Archimedean solids. For this comparison of Waterman polyhedra they are normalized
Waterman_polyhedron
pyramid, Trapezo-rhombic dodecahedron, Rhombo-hexagonal dodecahedron Archimedean solid Truncated tetrahedron, Cuboctahedron, Truncated cube, Truncated octahedron
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Archimedean solid with 26 faces
geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square
Truncated_cuboctahedron
Natural number, composite number
Catalan solid that can tessellate space. The truncated octahedron contains 14 faces, is the permutohedron of order four, and the only Archimedean solid to
14_(number)
Archimedean solid with 38 faces
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and
Snub_cube
37th Johnson solid (26 faces)
be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. However
Elongated_square_gyrobicupola
Natural number
Officer Problem". MathWorld. Retrieved 2020-08-21. Weisstein, Eric W. "Archimedean Solid". MathWorld. Retrieved 2020-08-21. Weisstein, Eric W. "Domino Tiling"
36_(number)
Natural number
space, the most uniform solids are: the five Platonic solids (with one type of regular face) the fifteen Archimedean solids (counting enantimorphs, all
37_(number)
Isogonal polyhedron with regular faces
prisms and antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 quasiregular and 11 semiregular— the non-convex star polyhedra
Uniform_polyhedron
Convex polyhedron whose faces are almost regular polygons
Geodesic polyhedron Goldberg polyhedron Johnson solid Platonic solid Semiregular polyhedron Archimedean solid Prism Antiprism By definition, each face is
Near-miss_Johnson_solid
Catalan solid with 60 faces
hexacontahedron) is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian
Deltoidal_hexecontahedron
Catalan solid with 30 faces
Being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the
Rhombic_triacontahedron
primarily come from the families of platonic solids, Archimedean solids, Catalan solids, and Johnson solids, as well as dihedral symmetry families including
List of small polyhedra by vertex count
List_of_small_polyhedra_by_vertex_count
Solid with six equal square faces
vertices), resulting in a trirectangular tetrahedron. The snub cube is an Archimedean solid that can be constructed by separating the cube's faces, and filling
Cube
Branko Grünbaum calls the vertex-uniform tilings Archimedean, in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Subdivision of the plane into polygons that are all regular
Platonic and Archimedean solids". Math. Commun. 16 (2): 491–507. Pellicer, Daniel; Williams, Gordon (2012). "Minimal Covers of the Archimedean Tilings, Part
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
Catalan solid with 12 faces
is 120°. The rhombic dodecahedron is a Catalan solid, meaning the dual polyhedron of an Archimedean solid, the cuboctahedron; they share the same symmetry
Rhombic_dodecahedron
Solid with twenty equal triangular faces
Johnson solid is a polyhedron whose faces are all regular but which is not uniform. In other words, they do not include the Archimedean solids, the Catalan
Regular_icosahedron
Polyhedron with 8 triangles and 6 squares
from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is
Cuboctahedron
Polyhedron with 12 faces
dodecahedron is a Catalan solid with twelve rhombic faces and octahedral symmetry. It is dual to the cuboctahedron, an Archimedean solid, and occurs in nature
Dodecahedron
Overview of and topical guide to geometry
Parallelepiped Tetrahedron Heronian tetrahedron Platonic solid Archimedean solid Kepler-Poinsot polyhedra Johnson solid Uniform polyhedron Polyhedral compound Hilbert's
Outline_of_geometry
Shape with six sides
Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some
Hexagon
Polyhedron with eight triangular faces
a uniform polyhedron that is not a prism or antiprism, this is an Archimedean solid. Gyrobifastigium: Two uniform triangular prisms glued over one of
Octahedron
Natural number
62 is a semiprime. It is also the number of faces of two of the Archimedean solids, the rhombicosidodecahedron and truncated icosidodecahedron. 62 is
62_(number)
Topics referred to by the same term
Mecon may refer to: Truncated octahedron or mecon, an Archimedean solid Master of Economics (M.Econ.), a postgraduate master's degree MECON (company)
Mecon
Hydraulic machine
fountain. Archimedean spiral Screw-propelled vehicle Screw (simple machine) Spiral pump Toroidal propeller Vitruvius Also known as the Archimedean screw,[citation
Archimedes'_screw
Solid with 12 equal pentagonal faces
the regular dodecahedron by truncating its two axial vertices. Two Archimedean solids: truncated dodecahedron and snub dodecahedron. Respectively, these
Regular_dodecahedron
Catalan solid with 24 kite faces
the rhombicuboctahedron, an Archimedean solid. Deltoidal icositetrahedron and deltoidal hexecontahedron are two Catalan solids with kite faces only. The
Deltoidal_icositetrahedron
Catalan solid with 60 faces
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron
Triakis_icosahedron
Operation that cuts polytope vertices, creating a new facet in place of each vertex
place of each vertex. The term originates from Kepler's names for the Archimedean solids. A different truncation is defined for otherwise infinite surfaces
Truncation_(geometry)
Notation for a polyhedron's vertex figure
It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models. For uniform polyhedra, there
Vertex_configuration
6th Johnson solid (17 faces)
half of an icosidodecahedron, an Archimedean solid, or as half of a pentagonal orthobirotunda, another Johnson solid. Both polyhedra are constructed by
Pentagonal_rotunda
Edge-joined polygons which fold into a polyhedron
Zyrkel und Rychtscheyd) included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by
Net_(polyhedron)
65th Johnson solid (14 faces)
truncated tetrahedron. It is an example of a Johnson solid. Out of 19 modified Archimedean solids, it is the only one created by augmenting the truncated
Augmented truncated tetrahedron
Augmented_truncated_tetrahedron
Method of describing higher-order polyhedra
basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids. Some basic operations can be made as composites
Conway_polyhedron_notation
43rd Johnson solid (42 faces)
Johnson solids (J43). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimedean solids)
Elongated pentagonal gyrobirotunda
Elongated_pentagonal_gyrobirotunda
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book
List of Wenninger polyhedron models
List_of_Wenninger_polyhedron_models
72nd Johnson solid (62 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Gyrate_rhombicosidodecahedron
68th Johnson solid (42 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Augmented truncated dodecahedron
Augmented_truncated_dodecahedron
69th Johnson solid (52 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Parabiaugmented truncated dodecahedron
Parabiaugmented_truncated_dodecahedron
3D symmetry group
Platonic solid tetrahedron 4 6 4 Archimedean solid truncated tetrahedron 8 18 12 Catalan solid triakis tetrahedron 12 18 8 Near-miss Johnson solid Truncated
Tetrahedral_symmetry
73rd Johnson solid (62 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Parabigyrate rhombicosidodecahedron
Parabigyrate_rhombicosidodecahedron
83rd Johnson solid (32 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Tridiminished rhombicosidodecahedron
Tridiminished_rhombicosidodecahedron
62nd Johnson solid (12 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Metabidiminished_icosahedron
48th Johnson solid (52 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Gyroelongated pentagonal birotunda
Gyroelongated_pentagonal_birotunda
15th-century book on polyhedra
corporibus regularibus includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural
De quinque corporibus regularibus
De_quinque_corporibus_regularibus
80th Johnson solid (42 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Parabidiminished rhombicosidodecahedron
Parabidiminished_rhombicosidodecahedron
3D symmetry group
octahedron by x + y + z = 1 (or the corresponding inequalities, to get the solid instead of the surface). ax + by + cz = 1 gives a polyhedron with 48 faces
Octahedral_symmetry
55th Johnson solid (14 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Parabiaugmented hexagonal prism
Parabiaugmented_hexagonal_prism
85th Johnson solid (26 faces)
not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it is a relative of the icosahedron that has fourfold symmetry
Snub_square_antiprism
Hypothetical "God's-eye view" of the world
An Archimedean point (Latin: Punctum Archimedis) is a hypothetical viewpoint from which certain objective truths can perfectly be perceived (also known
Archimedean_point
70th Johnson solid (52 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Metabiaugmented truncated dodecahedron
Metabiaugmented_truncated_dodecahedron
Type of map projection
is a polyhedral globe. Often the polyhedron used is a Platonic solid or Archimedean solid. However, other polyhedra can be used: the AuthaGraph projection
Polyhedral_map_projection
59th Johnson solid (20 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Parabiaugmented_dodecahedron
33rd Johnson solid (27 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Pentagonal_gyrocupolarotunda
87th Johnson solid (17 faces)
components are not all prisms, antiprisms or sections of Platonic or Archimedean solids. The augmented sphenocorona is constructed by attaching equilateral
Augmented_sphenocorona
81st Johnson solid (42 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Metabidiminished rhombicosidodecahedron
Metabidiminished_rhombicosidodecahedron
46th Johnson solid (42 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Gyroelongated pentagonal bicupola
Gyroelongated_pentagonal_bicupola
Geometric operation which truncates the edges of polyhedra
569–591. Chamfered Tetrahedron Chamfered Solids Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra
Chamfer_(geometry)
57th Johnson solid (17 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Triaugmented_hexagonal_prism
75th Johnson solid (62 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Trigyrate rhombicosidodecahedron
Trigyrate_rhombicosidodecahedron
created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes. small rhombicuboctahedral prism (Small)
Rhombicuboctahedral_prism
Generalisation of dice with identical faces
Catalan solids, the Platonic Solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, Platonic
Isohedral_figure
Problems which attempt to find the most efficient way to pack objects into containers
shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra, tripods (unions of cubes along three positive
Packing_problems
76th Johnson solid (52 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Diminished rhombicosidodecahedron
Diminished_rhombicosidodecahedron
polyhedron) Hemicube Hemi-octahedron Hemi-dodecahedron Hemi-icosahedron Archimedean solid Truncated tetrahedron Cuboctahedron Truncated cube Truncated octahedron
List_of_mathematical_shapes
Type of polyhedron with many holes
proportione in three phases: drawing Platonic solids and Archimedean solids; replacing the edges of those solids by struts, forming a convex polygon, and this
Leonardo_polyhedron
66th Johnson solid (22 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Augmented_truncated_cube
Value determined from a polyhedron
Platonic solids, implying that the other solids cannot tile space and that they cannot be dissected into a cube. All of the Archimedean solids have Dehn
Dehn_invariant
24th Johnson solid (32 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Gyroelongated pentagonal cupola
Gyroelongated_pentagonal_cupola
Catalan solid with 60 faces
60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices
Pentagonal_hexecontahedron
71st Johnson solid (62 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Triaugmented truncated dodecahedron
Triaugmented_truncated_dodecahedron
47th Johnson solid (47 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Gyroelongated pentagonal cupolarotunda
Gyroelongated_pentagonal_cupolarotunda
Prism with a 3-sided base
Johnson solid is a convex polyhedron with regular faces, and this definition sometimes omits uniform polyhedra such as Archimedean solids, Catalan solids, prisms
Triangular_prism
Uniform 4-polytope
using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids. Tetrahedral dyadic prism (Norman W. Johnson) Tepe (Jonathan
Tetrahedral_prism
21st Johnson solid (27 faces)
but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who
Elongated_pentagonal_rotunda
Ball used in the sport of football
design is often referenced when describing the truncated icosahedron Archimedean solid, carbon buckyballs, or the root structure of geodesic domes. Along
Football_(ball)
34th Johnson solid (32 faces)
similar to the icosidodecahedron (or pentagonal gyrobirotunda), an Archimedean solid: the difference is that one of its rotundas is twisted around 36°
Pentagonal_orthobirotunda
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