## Table of Contents

46 relations: Alternation (geometry), Blend word, Cartesian coordinate system, Convex polytope, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, Cube, Demihypercube, Dimension, Dual polyhedron, Edge (geometry), Face (geometry), Facet (geometry), Geometry, Greek language, Hanner polytope, Harold Scott MacDonald Coxeter, Hexadecagon, Hypercube, Norman Johnson (mathematician), Pascal's triangle, Petrie polygon, Polyhedral combinatorics, Projection (linear algebra), Regular Polytopes (book), Schläfli symbol, Square, Tesseract, Tetrahedron, Triangle, Uniform 8-polytope, Uniform polytope, Vertex (geometry), Vertex figure, Wythoff construction, 5-cell, 5-cube, 5-simplex, 6-cube, 6-simplex, 7-cube, 7-demicube, 7-simplex, 8-demicube, 8-orthoplex.

- 8-polytopes

## Alternation (geometry)

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

See 8-cube and Alternation (geometry)

## Blend word

In linguistics, a blend—also known as a blend word, lexical blend, or portmanteau—is a word formed, usually intentionally, by combining the sounds and meanings of two or more words.

## Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

See 8-cube and Cartesian coordinate system

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.

See 8-cube and Convex polytope

## Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).

## Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

See 8-cube and Coxeter–Dynkin diagram

## Cross-polytope

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces.

See 8-cube and Cube

## Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Dual polyhedron

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

See 8-cube and Dual polyhedron

## Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

See 8-cube and Edge (geometry)

## Face (geometry)

In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.

See 8-cube and Face (geometry)

## Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

See 8-cube and Facet (geometry)

## Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

## Greek language

Greek (Elliniká,; Hellēnikḗ) is an independent branch of the Indo-European family of languages, native to Greece, Cyprus, Italy (in Calabria and Salento), southern Albania, and other regions of the Balkans, the Black Sea coast, Asia Minor, and the Eastern Mediterranean.

## Hanner polytope

In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations.

See 8-cube and Hanner polytope

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

See 8-cube and Harold Scott MacDonald Coxeter

## Hexadecagon

In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon.

## Hypercube

In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.

## Norman Johnson (mathematician)

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

See 8-cube and Norman Johnson (mathematician)

## Pascal's triangle

In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.

See 8-cube and Pascal's triangle

## Petrie polygon

In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) belongs to one of the facets.

## Polyhedral combinatorics

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

See 8-cube and Polyhedral combinatorics

## Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P.

See 8-cube and Projection (linear algebra)

## Regular Polytopes (book)

Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter.

See 8-cube and Regular Polytopes (book)

## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.

See 8-cube and Schläfli symbol

## Square

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles (90-degree angles, π/2 radian angles, or right angles).

## Tesseract

In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube.

## Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

## Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

## Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. 8-cube and Uniform 8-polytope are 8-polytopes.

See 8-cube and Uniform 8-polytope

## Uniform polytope

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

See 8-cube and Uniform polytope

## Vertex (geometry)

In geometry, a vertex (vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect.

See 8-cube and Vertex (geometry)

## Vertex figure

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

## Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

See 8-cube and Wythoff construction

## 5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.

## 5-cube

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

## 6-cube

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

## 6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

## 7-cube

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

## 7-demicube

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.

## 7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

## 8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. 8-cube and 8-demicube are 8-polytopes.

## 8-orthoplex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces. 8-cube and 8-orthoplex are 8-polytopes.

## See also

### 8-polytopes

- 1 33 honeycomb
- 1 42 polytope
- 2 41 polytope
- 3 31 honeycomb
- 4 21 polytope
- 7-cubic honeycomb
- 7-demicubic honeycomb
- 7-simplex honeycomb
- 8-cube
- 8-demicube
- 8-orthoplex
- 8-simplex
- A8 polytope
- B8 polytope
- Cantellated 8-simplexes
- Cantic 8-cube
- Cyclotruncated 7-simplex honeycomb
- D8 polytope
- E8 polytope
- Heptellated 8-simplexes
- Hexicated 8-simplexes
- Omnitruncated 7-simplex honeycomb
- Pentellated 8-simplexes
- Quarter 7-cubic honeycomb
- Rectified 8-cubes
- Rectified 8-orthoplexes
- Rectified 8-simplexes
- Runcinated 8-simplexes
- Stericated 8-simplexes
- Truncated 8-cubes
- Truncated 8-orthoplexes
- Truncated 8-simplexes
- Uniform 8-polytope

## References

Also known as 8 cube, 8-hypercube, Octeract.