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 cuboctahedral prism or great rhombicuboctahedral prism is a convex

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

polychoron In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...

(four-dimensional

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

). It is one of 18 convex 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

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 edges c ...

s or

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 ...

s in parallel

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

Net

Alternative names

* Truncated-cuboctahedral dyadic prism (

Norman W. Johnson Norman Woodason Johnson () was a mathematician at Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper. Johnson earned his unde ...

) * Gircope (Jonathan Bowers: for great rhombicuboctahedral prism/hyperprism) * Great rhombicuboctahedral prism/hyperprism

Related polytopes

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, or , although it cannot be constructed as a uniform polychoron. It has 76 cells: 2

snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 square (geometry), squares and 32 equilateral triangles. It has 60 edge (geometry), edges and 24 vertex (geometry), vertices. It is a chiral polytope, ...

s connected by 12

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 o ...

s, 6

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular, it is a sem ...

s, and 8

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 ea ...

s, with 48

s in the alternated gaps. There are 48 vertices, 192 edges, and 220 faces (12 squares, and 16+192 triangles). It has ,3,2sup>+ 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. Omnisnub cubic 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

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 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 only r ...

but with ''D_2d'' symmetry), 8 octahedra (as triangular antiprisms), 24

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...

s (as ''C_s''-symmetry wedges) filling the gaps, and 48 vertices. It has ,(3,2)⁺symmetry, order 48. Its vertex figure is a

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

hexahedron A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There ...

topologically identical to the tetragonal antiwedge. Bialternatosnub octahedral hosochoron vertex figure

for the bialternatosnub octahedral hosochoron

External links

* * 4-polytopes {{Polyhedron-stub