Truncated cuboctahedral prism

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Truncated cuboctahedral prism
Truncated cuboctahedral prism.png
Schlegel diagram
Type Prismatic uniform polychoron
Uniform index55
Schläfli symbol t0,1,2,3{4,3,2} or tr{4,3}×{}
Coxeter-Dynkin CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Cells28 total:
2 Great rhombicuboctahedron.png 4.6.8
12 Hexahedron.png 4.4.4
8 Hexagonal prism.png 4.4.6
6 Octagonal prism.png 4.4.8
Faces124 total:
96 {4}
16 {6}
12 {8}
Edges192
Vertices96
Vertex figure Truncated cuboctahedral prism verf.png
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).

Geometry Branch of mathematics that studies the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

Contents

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.

Prism (geometry) geometric shape, a polyhedron with an n-sided polygonal base

In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent, regular, polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:

Archimedean solid one of the 13 solids (semi-regular convex polyhedrons composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms)

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedra composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

Great rhombicuboctahedral prism net.png
Net

Alternative names

Norman Johnson (mathematician) American mathematician

Norman Woodason Johnson was a mathematician at Wheaton College, Norton, Massachusetts.

A full snub cubic antiprism or omnisnub cubic antiprism can be defined as an alternation of an truncated cuboctahedral prism, represented by ht0,1,2,3{4,3,2}, or CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, 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.

Alternation (geometry) operation on a polyhedron or tiling that removes alternate vertices

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

Snub cube Archimedean solid

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.

Tetrahedron Polyhedron with 4 faces

In geometry, a tetrahedron, 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 ordinary convex polyhedra and the only one that has fewer than 5 faces.

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.png
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 Th symmetry), 6 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 8 octahedra (as triangular antiprisms), 24 triangular prisms (as Cs-symmetry wedges) filling the gaps, and 48 vertices. It has [4,(3,2)+] symmetry, order 48. Its vertex figure is a chiral hexahedron topologically identical to the tetragonal antiwedge.

Rhombicuboctahedron Archimedean solid with eight triangular and eighteen square faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

Cube A geometric shape with 6 square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Triangular prism three-sided 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 uniform triangular prism is a right triangular prism with equilateral bases, and square sides.

Bialternatosnub octahedral hosochoron vertex figure.png
Vertex figure for the bialternatosnub octahedral hosochoron

Vertex figure figure exposed when a corner of a polyhedron or polytope is sliced off

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.


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Truncated 24-cells

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

Truncated 5-cell

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

Rectified 24-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.

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

Snub cubic prism

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