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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.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. [5] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. [6]
The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples. Polytope elements [ edit ]
The bifurcating graph of the D 5 family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube. Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction , represented by rings around permutations of nodes in a Coxeter diagram .
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
In 5-dimensional geometry, there are 23 uniform polytopes with D 5 symmetry, 8 are unique, and 15 are shared with the B 5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively. They can be visualized as symmetric orthographic projections in Coxeter planes of the D 6 Coxeter group, and other ...
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any ...
In 5-dimensional geometry, there are 19 uniform polytopes with A 5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices. Each can be visualized as symmetric orthographic projections in the Coxeter planes of the A 5 Coxeter group and other subgroups.