In six-dimensional geometry , a SIX-DIMENSIONAL POLYTOPE or
6-POLYTOPE is a polytope , bounded by
**5-polytope** facets .

CONTENTS

* 1 Definition
* 2 Characteristics
* 3 Classification
* 4 Regular 6-polytopes
* 5 Uniform 6-polytopes
* 6 References
* 7 External links

DEFINITION

A
**6-polytope**

6-polytope is a closed six-dimensional figure with vertices , edges
, faces , cells (3-faces), 4-faces, and 5-faces. A vertex is a point
where six or more edges meet. An edge is a line segment where four or
more faces meet, and a face is a polygon where three or more cells
meet. A cell is a polyhedron . A 4-face is a polychoron , and a 5-face
is a
**5-polytope** . Furthermore, the following requirements must be met:

* Each 4-face must join exactly two 5-faces (facets).
* Adjacent facets are not in the same five-dimensional hyperplane .
* The figure is not a compound of other figures which meet the
requirements.

CHARACTERISTICS

The topology of any given
**6-polytope**

6-polytope is defined by its Betti numbers
and torsion coefficients .

The value of the
**Euler characteristic** used to characterise polyhedra
does not generalize usefully to higher dimensions, and is zero for all
6-polytopes, whatever their underlying topology. This inadequacy of
the
**Euler characteristic** to reliably distinguish between different
topologies in higher dimensions led to the discovery of the more
sophisticated Betti numbers.

Similarly, the notion of orientability of a polyhedron is
insufficient to characterise the surface twistings of toroidal
polytopes, and this led to the use of torsion coefficients.

CLASSIFICATION

6-polytopes may be classified by properties like "convexity " and
"symmetry ".

* A
**6-polytope**

6-polytope is convex if its boundary (including its 5-faces,
4-faces, cells, faces and edges) does not intersect itself and the
line segment joining any two points of the
**6-polytope**

6-polytope is contained in
the
**6-polytope**

6-polytope or its interior; otherwise, it is non-convex.
Self-intersecting
**6-polytope**

6-polytope are also known as star 6-polytopes , from
analogy with the star-like shapes of the non-convex Kepler-Poinsot
polyhedra .
* A REGULAR 6-POLYTOPE has all identical regular
**5-polytope** facets.
All regular
**6-polytope**

6-polytope are convex.

Main article: List of regular polytopes § Convex_5

* A SEMI-REGULAR 6-POLYTOPE contains two or more types of regular
4-polytope facets. There is only one such figure, called 221 .
* A UNIFORM 6-POLYTOPE has a symmetry group under which all vertices
are equivalent, and its facets are uniform 5-polytopes . The faces of
a uniform polytope must be regular .

Main article:
**Uniform 6-polytope**

* A PRISMATIC 6-POLYTOPE is constructed by the
**Cartesian product** of
two lower-dimensional polytopes. A prismatic
**6-polytope**

6-polytope is uniform if
its factors are uniform. The
**6-cube**

6-cube is prismatic (product of a squares
and a cube ), but is considered separately because it has symmetries
other than those inherited from its factors.
* A 5-space tessellation is the division of five-dimensional
**Euclidean space** into a regular grid of
**5-polytope** facets. Strictly
speaking, tessellations are not 6-polytopes as they do not bound a
"6D" volume, but we include them here for the sake of completeness
because they are similar in many ways to 6-polytope. A uniform 5-space
tessellation is one whose vertices are related by a space group and
whose facets are uniform 5-polytopes .

REGULAR 6-POLYTOPES

Regular 6-polytopes can be generated from Coxeter groups represented
by the
**Schläfli symbol** {p,q,r,s,t} with T {p,q,r,s}
**5-polytope** facets
around each cell .

There are only three such convex regular 6-polytopes :

* {3,3,3,3,3} -
**6-simplex**

6-simplex
* {4,3,3,3,3} -
**6-cube**

6-cube
* {3,3,3,3,4} -
**6-orthoplex**

There are no nonconvex regular polytopes of 5 or more dimensions.

For the 3 convex regular 6-polytopes , their elements are:

NAME
Schläfli
symbol Coxeter
diagram VERTICES
EDGES
FACES
CELLS
4-FACES
5-FACES
SYMMETRY (ORDER )

**6-simplex**

6-simplex
{3,3,3,3,3}

7
21
35
35
21
7
A6 (720)

**6-orthoplex**
{3,3,3,3,4}

12
60
160
240
192
64
B6 (46080)