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Truncated square antiprism | |
---|---|

Type | Truncated antiprism |

Schläfli symbol | ts{2,8} tsr{4,2} or |

Conway notation | tA4 |

Faces | 18: 2 {8}, 8 {6}, 8 {4} |

Edges | 48 |

Vertices | 32 |

Symmetry group | D_{4d}, [2^{+},8], (2*4), order 16 |

Rotation group | D_{4}, [2,4]^{+}, (224), order 8 |

Dual polyhedron | |

Properties | convex, zonohedron |

The **truncated square antiprism** one in an infinite series of truncated antiprisms, constructed as a truncated square antiprism. It has 18 faces, 2 octagons, 8 hexagons, and 8 squares.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, the **square antiprism** is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an *anticube*.

If the hexagons are folded, it can be constructed by regular polygons. Or each folded hexagon can be replaced by two triamonds, adding 8 edges (56), and 4 faces (32). This form is called a *gyroelongated triamond square bicupola*.^{ [1] }

Symmetry | D_{2d}, [2^{+},4], (2*2) | D_{3d}, [2^{+},6], (2*3) | D_{4d}, [2^{+},8], (2*4) | D_{5d}, [2^{+},10], (2*5) |
---|---|---|---|---|

Antiprisms | s{2,4} (v:4; e:8; f:6) | s{2,6} (v:6; e:12; f:8) | s{2,8} (v:8; e:16; f:10) | s{2,10} (v:10; e:20; f:12) |

Truncated antiprisms | ts{2,4} (v:16;e:24;f:10) | ts{2,6} (v:24; e:36; f:14) | ts{2,8} (v:32; e:48; f:18) | ts{2,10} (v:40; e:60; f:22) |

Although it can't be made by all regular planar faces, its alternation is the Johnson solid, the snub square antiprism.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a **Johnson solid** is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (*J*_{1}); it has 1 square face and 4 triangular faces.

In geometry, the **snub square antiprism** is one of the Johnson solids (*J*_{85}). A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966.

In geometry, an *n*-sided **antiprism** is a polyhedron composed of two parallel copies of some particular *n*-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a (degenerate) type of snub polyhedra.

In geometry, a **dodecahedron** is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, a **dodecagon** or 12-gon is any twelve-sided polygon.

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

The **cubic honeycomb** or **cubic cellulation** is the only 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 calls this honeycomb a **cubille**.

A **snub polyhedron** is a polyhedron obtained by alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some but not all authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces.

In geometry, a **near-miss Johnson solid** is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.

In geometry, a **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms, with clockwise or counterclockwise orientations. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron, with the faces moved apart, and twists on their centers, adding new polygons centered on the original vertices, and pairs of triangles fitting between the original edges.

A **tetradecahedron** is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.

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 **truncated cuboctahedral prism** or **great rhombicuboctahedral prism** is a convex uniform polychoron.

In geometry, an **edge-contracted icosahedron** is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C_{2v} symmetry, order 4.

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}, has three square tilings, {4,4} around each edge, and 6 square tilings around each vertex in a cubic {4,3} vertex figure.

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