Truncated cuboctahedral prism | |
---|---|

Schlegel diagram | |

Type | Prismatic uniform polychoron |

Uniform index | 55 |

Schläfli symbol | t_{0,1,2,3}{4,3,2} or tr{4,3}×{} |

Coxeter-Dynkin | |

Cells | 28 total: 2 4.6.8 12 4.4.4 8 4.4.6 6 4.4.8 |

Faces | 124 total: 96 {4} 16 {6} 12 {8} |

Edges | 192 |

Vertices | 96 |

Vertex figure | Irregular tetrahedron |

Symmetry group | [4,3,2], order 96 |

Properties | convex |

In geometry, a **truncated cuboctahedral prism** or **great rhombicuboctahedral prism** is a convex uniform polychoron (four-dimensional polytope).

It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.

- Truncated-cuboctahedral dyadic prism (Norman W. Johnson)
- Gircope (Jonathan Bowers: for great rhombicuboctahedral prism/hyperprism)
- Great rhombicuboctahedral prism/hyperprism

A **full snub cubic antiprism** or **omnisnub cubic antiprism** can be defined as an alternation of an truncated cuboctahedral prism, represented by ht_{0,1,2,3}{4,3,2}, or , although it cannot be constructed as a uniform polychoron. It has 76 cells: 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedrons in the alternated gaps. There are 48 vertices, 192 edges, and 220 faces (12 squares, and 16+192 triangles). It has [4,3,2]^{+} symmetry, order 48.

A construction exists with two uniform snub cubes in snub positions with two edge lengths in a ratio of around 1 : 1.138.

Vertex figure for the **omnisnub cubic antiprism**

Also related is the **bialternatosnub octahedral hosochoron**, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has 40 cells: 2 rhombicuboctahedra (with *T _{h}* symmetry), 6 rectangular trapezoprisms (topologically equivalent to a cube but with

Vertex figure for the **bialternatosnub octahedral hosochoron**

- 6. Convex uniform prismatic polychora - Model 55 , George Olshevsky.
- Klitzing, Richard. "4D uniform polytopes (polychora) x3x4x x - gircope".

In geometry, a **uniform 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a **double prism** or **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

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, the **rectified 5-cell** is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a **cubille**.

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-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, the **icosahedral honeycomb** is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

In geometry, a **truncated 24-cell** is a uniform 4-polytope formed as the truncation of the regular 24-cell.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In geometry, the **rectified 24-cell** or **rectified icositetrachoron** is a uniform 4-dimensional polytope, which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.

In four-dimensional geometry, a **cantellated 5-cell** is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In four-dimensional geometry, a **cantellated 24-cell** is a convex uniform 4-polytope, being a cantellation of the regular 24-cell.

In four-dimensional geometry, a **runcinated 24-cell** is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In 4-dimensional geometry, a **truncated octahedral prism** or **omnitruncated tetrahedral prism** is a convex uniform 4-polytope. This 4-polytope has 16 cells It has 64 faces, and 96 edges and 48 vertices.

In geometry, a **rhombicuboctahedral prism** is a convex uniform polychoron.

In geometry, a **truncated icosidodecahedral prism** or **great rhombicosidodecahedral prism** is a convex uniform 4-polytope.

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.

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