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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]
A prismatic 5-polytope is constructed by a Cartesian product of two lower-dimensional polytopes. A prismatic 5-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of a square and a cube), but is considered separately because it has symmetries other than those inherited from its factors.
The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some ... Five-dimensional space, 5-polytope and ...
Face, a 2-dimensional element; Cell, a 3-dimensional element; Hypercell or Teron, a 4-dimensional element; Facet, an (n-1)-dimensional element; Ridge, an (n-2)-dimensional element; Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.
An important uniform 5-polytope is the 5-demicube, h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex is the vertex figure of the A 5 lattice, . It and has a doubled symmetry from its symmetric Coxeter diagram.
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 .
This category contains polytopes of 5-space, and honeycombs of 4-space. Pages in category "5-polytopes" The following 67 pages are in this category, out of 67 total. ...
McMullen conjectures that this list is complete regarding the compact compounds. If any more compact compounds exist, they must involve {4,3,3,5} or {5,3,3,5} being inscribed in {5,3,3,3} (the only case not yet excluded). [8] In five dimensions, there is only one regular hyperbolic honeycomb whose vertices are not at infinity: {3,4,3,3,3}.