In 4-dimensional

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...

. This 4-polytope has 16 cells (2

truncated octahedra In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...

connected by 6

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

s, 8

hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...

s.) It has 64 faces (48 squares and 16

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...

s), and 96 edges and 48 vertices. It has two symmetry constructions, one from the truncated

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...

, and one as an omnitruncation of the

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

. It is one of 18 uniform polyhedral prisms created by using uniform

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

s to connect pairs of parallel

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s and

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

Images

Alternative names

* Truncated octahedral dyadic prism ( Norman W. Johnson) * Truncated octahedral hyperprism * Tope (Jonathan Bowers: for truncated octahedral prism)

Related polytopes

The snub tetrahedral prism (also called an icosahedral prism), , sr×, is related to this polytope just like a snub tetrahedron (icosahedron), is the alternation of the

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...

in its tetrahedral symmetry . The ''snub tetrahedral prism'' has symmetry 3,3)⁺,2 order 24, although as an icosahedral prism, its full symmetry is ,3,2 order 240. Also related, the full snub tetrahedral antiprism or omnisnub tetrahedral antiprism is defined as an alternation of an omnitruncated tetrahedral prism, represented by

s\left\

= ht_0,1,2,3, or , although it cannot be constructed as a uniform 4-polytope. It can also be seen as an alternated truncated octahedral prism or pyritohedral icosahedral antiprism, . It has 2

icosahedra In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetri ...

connected by 6

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

and 8

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ...

, with 24 irregular

in the alternated gaps. In total it has 40 cells, 112 triangular faces, 96 edges, and 24 vertices. It has ,(3,2)⁺symmetry, order 48, and also ,3,2sup>+ symmetry, order 24. A construction exists with two regular icosahedra in snub positions with two edge lengths in a ratio of around 0.831 : 1. Omnisnub tetrahedral antiprism vertex figure

Omnisnub tetrahedral antiprism vertex figure

Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

for the omnisnub tetrahedral antiprism

External links

* * 4-polytopes {{4-polytopes-stub

Images

Alternative names

Related polytopes

See also

External links