In six-dimensional geometry , a SIX-DIMENSIONAL POLYTOPE or 6-POLYTOPE is a polytope , bounded by 5-polytope facets .
* 1 Definition * 2 Characteristics * 3 Classification * 4 Regular 6-polytopes * 5 Uniform 6-polytopes * 6 References * 7 External links
* 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.
The topology of any given
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.
6-polytopes may be classified by properties like "convexity " and "symmetry ".
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
There are only three such convex regular 6-polytopes :
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 )
7 21 35 35 21 7 A6 (720)
12 60 160 240 192 64 B6 (46080)