In six-dimensional geometry , a SIX-DIMENSIONAL POLYTOPE or
6-POLYTOPE is a polytope , bounded by
CONTENTS * 1 Definition * 2 Characteristics * 3 Classification * 4 Regular 6-polytopes * 5 Uniform 6-polytopes * 6 References * 7 External links DEFINITION A
* 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. CHARACTERISTICS The topology of any given
The value of the
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. CLASSIFICATION 6-polytopes may be classified by properties like "convexity " and "symmetry ". * A
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:
* A PRISMATIC 6-POLYTOPE is constructed by the
REGULAR 6-POLYTOPES Regular 6-polytopes can be generated from Coxeter groups represented
by the
There are only three such convex regular 6-polytopes : * {3,3,3,3,3} -
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)
64 192 240 160 60 12 B6 (46080) UNIFORM 6-POLYTOPES Main article:
Here are six simple uniform convex 6-polytopes, including the
NAME Schläfli symbol(s) Coxeter diagram(s) VERTICES EDGES FACES CELLS 4-FACES 5-FACES SYMMETRY (ORDER ) Expanded
42 210 490 630 434 126 2×A6 (1440)
12 60 160 240 192 64 D6 (23040)
Rectified
221 polytope {3,3,32,1} 27 216 720 1080 648 99 E6 (51840) 122 polytope {3,32,2} or 72 720 2160 2160 702 54 2×E6 (103680) The expanded
REFERENCES * ^ A B C Richeson, D.; Euler's Gem: The
* T. Gosset : On the Regular and Semi-Regular Figures in Space of n
Dimensions,
* H.S.M. Coxeter : * H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954 * H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 * KALEIDOSCOPES: SELECTED WRITINGS OF H.S.M. COXETER, edited by F.
Arthur Sherk,
* (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, * (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, * (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, * N.W. Johnson : The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 * Klitzing, Richard. "6D uniform polytopes (polypeta)". |