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Concept
6-polytope
Summary
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
A 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.
The topology of any given 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.
6-polytopes may be classified by properties like "convexity" and "symmetry".
A 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 is contained in the 6-polytope or its interior; otherwise, it is non-convex. Self-intersecting 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 are convex.
List of regular polytopes#Convex_5
A semi-regular 6-polytope contains two or more types of regular 4-polytope facets.
Official source
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