A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 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 that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.
In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron 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 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid [sic?].
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.
In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.
In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
In geometry, a truncated 24-cell is a uniform 4-polytope formed as the truncation of the regular 24-cell.
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.
The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.
In 4-dimensional geometry, the tetrahedral cupola is a polychoron bounded by one tetrahedron, a parallel cuboctahedron, connected by 10 triangular prisms, and 4 triangular pyramids.
In 4-dimensional geometry, the cubic cupola is a 4-polytope bounded by a rhombicuboctahedron, a parallel cube, connected by 6 square prisms, 12 triangular prisms, 8 triangular pyramids.
In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids can be made with regular faces by computing the appropriate height.
In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
