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A cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertex-transitive and edge-transitive. Image File history File links This is a lossless scalable vector image. ...
Spinning cuboctahedron, made by me using POV-Ray, see image:poly. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ...
Coxeter groups in the plane with equivalent diagrams. ...
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Image File history File links CDW_3. ...
Image File history File links CDW_dot. ...
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Image File history File links CDW_dot. ...
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// List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ...
The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ...
A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
H.S.M. Coxeter. ...
This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ...
// A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular. ...
Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links Cuboctahedron_vertfig. ...
In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...
Download high resolution version (849x754, 59 KB)Rhombic dodecahedron, made by me using POV-Ray, see image:poly. ...
The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ...
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...
Image File history File links This is a lossless scalable vector image. ...
Categories: Polyhedra | Stub ...
A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ...
// A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ...
In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. ...
Its dual polyhedron is the rhombic dodecahedron. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...
The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ...
Other Names
Richard Buckminster “Bucky†Fuller (July 12, 1895 – July 1, 1983)[1] was an American visionary, designer, architect, poet, author, and inventor. ...
The term dymaxion was used by inventor Buckminster Fuller for several of his projects, such as the Dymaxion car and Dymaxion House or the Dymaxion Map. ...
Unfolded Dymaxion map with nearly-contiguous land masses. ...
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ...
Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ...
A cantellated cube - Red faces are reduced. ...
In geometry, the triangular orthobicupola is one of the Johnson solids (J27). ...
Area and volume The area A and the volume V of the cuboctahedron of edge length a are:  
Geometric relations A cuboctahedron can be obtained by taking an appropriate cross section of a four-dimensional cross-polytope. A 3-D view of a beverage-can stove with a cross section in yellow. ...
In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...
A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...
A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
The cuboctahedron is a rectified cube and also a rectified octahedron. In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ...
A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
It is also a cantellated tetrahedron. With this construction it is given the Wythoff Symbol: 3 3 | 2.
 A cantellated cube - Red faces are reduced. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 246 KB, MIME type: image/png)See: Stella (software) Colors edited from cuboctahedron to show Chiral tetrahedral symmetry. ...
A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group. The symmetry group of an object (e. ...
The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola. A regular hexagon. ...
In geometry, the triangular cupola is one of the Johnson solids (J3). ...
The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ...
The gyrobifastigium (J26) can be considered a linear gyrobicupola. ...
In geometry, the gyrobifastigium is one of the Johnson solids (J26). ...
In geometry, the triangular orthobicupola is one of the Johnson solids (J27). ...
Both triangular bicupolae are important in sphere packing. The distance from the solid's centre to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's centre. In mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical spheres which fill a space. ...
A sphere is a perfectly symmetrical geometrical object. ...
A regular hexagon. ...
Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform polychora. In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ...
In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...
The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron; it is 5/3 √2 times the cube of the length of an edge.
Cartesian coordinates The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin are Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
- (±1,±1,0)
- (±1,0,±1)
- (0,±1,±1)
Related polyhedra Compare: Image File history File links Download high-resolution version (800x800, 14 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Cube I, the creator of this work, hereby release it into the public domain. ...
A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
Image File history File links Download high-resolution version (799x800, 16 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated cube I, the creator of this work, hereby release it into the public domain. ...
The truncated cube, or truncated hexahedron, is an Archimedean solid. ...
Image File history File links Download high-resolution version (800x800, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Cuboctahedron I, the creator of this work, hereby release it into the public domain. ...
Image File history File links Download high-resolution version (800x800, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated octahedron I, the creator of this work, hereby release it into the public domain. ...
The truncated octahedron is an Archimedean solid. ...
Image File history File links Download high-resolution version (800x800, 18 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks octahedron as dual to cube I, the creator of this work, hereby release it into the public domain. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
See also A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ...
A colored model Net (polyhedron) The truncated cuboctahedron is an Archimedean solid. ...
References - 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)
This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
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