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Concept
5-polytope
Summary
In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five 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, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met:
Each cell must join exactly two 4-faces.
Adjacent 4-faces are not in the same four-dimensional hyperplane.
The figure is not a compound of other figures which meet the requirements.
The topology of any given 5-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, 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.
5-polytopes may be classified based on properties like "convexity" and "symmetry".
A 5-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 5-polytope is contained in the 5-polytope or its interior; otherwise, it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.
A uniform 5-polytope has a symmetry group under which all vertices are equivalent, and its facets are uniform 4-polytopes. The faces of a uniform polytope must be regular.
Uniform 5-polytope
A semi-regular 5-polytope contains two or more types of regular 4-polytope facets.
Official source
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