  Communication
Free Faster access than browser!

# Uniform 9-polytope

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. 

## Alicia Boole Stott

Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish-English mathematician.

## Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

## Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

## 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 the spatial relations between a collection of mirrors (or reflecting hyperplanes).

## Cross-polytope

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

## 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 h&gamma;n for being half of the hypercube family, &gamma;n.

## Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler&ndash;Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

## Face (geometry)

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

## 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.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

## Hypercube

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

## Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

## Messenger of Mathematics

The Messenger of Mathematics is a defunct mathematics journal.

## Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## Quarter 8-cubic honeycomb

In seven-dimensional Euclidean geometry, the quarter 8-cubic honeycomb is a uniform space-filling tessellation (or honeycomb).

## Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

## Rectified 9-cubes

In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube.

## Rectified 9-orthoplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.

## Rectified 9-simplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

## Schläfli symbol

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

## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

## Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 &ndash; December 1962) was an English lawyer and an amateur mathematician.

## Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets.

## Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.

## Uniform polytope

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

## Vertex figure

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

## 1 52 honeycomb

In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

## 2 51 honeycomb

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.

## 5 21 honeycomb

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

## 8-cubic honeycomb

The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space.

## 8-demicubic honeycomb

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space.

## 8-simplex honeycomb

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb).

## 9-cube

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

## 9-demicube

In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed.

## 9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

## 9-simplex

In geometry, a 9-simplex is a self-dual regular 9-polytope.

## References

Hey! We are on Facebook now! »