In five-dimensional geometry , a FIVE-DIMENSIONAL POLYTOPE or 5-POLYTOPE is a 5-dimensional polytope , bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets. CONTENTS * 1 Definition * 2 Characteristics * 3 Classification * 4 Regular 5-polytopes * 5 Uniform 5-polytopes * 6 Pyramids * 7 See also * 8 References * 9 External links DEFINITION A
* 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. 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 5-polytopes may be classified based on properties like "convexity " and "symmetry ". * A
Main article:
* A SEMI-REGULAR 5-POLYTOPE contains two or more types of regular
Main article: List_of_regular_polytopes § Convex_4 * A PRISMATIC 5-POLYTOPE is constructed by a
REGULAR 5-POLYTOPES Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with S {p,q,r} polychoral facets around each face . There are exactly three such convex regular 5-polytopes : * {3,3,3,3} -
For the 3 convex regular 5-polytopes and three semiregular 5-polytope, their elements are: NAME Schläfli symbol(s) Coxeter diagram(s) VERTICES EDGES FACES CELLS 4-FACES SYMMETRY (ORDER )
6 15 20 15 6 A5, (120)
32 80 80 40 10 BC5, (3820)
UNIFORM 5-POLYTOPES Main article:
For three of the semiregular 5-polytope, their elements are: NAME Schläfli symbol(s) Coxeter diagram(s) VERTICES EDGES FACES CELLS 4-FACES SYMMETRY (ORDER ) Expanded
30 120 210 180 162 2×A5, (240)
Rectified
The expanded
PYRAMIDS Pyramidal 5-polytopes, or 5-PYRAMIDS, can be generated by a
SEE ALSO * List of regular polytopes#Five-dimensional regular polytopes and higher 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, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 * (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. "5D uniform polytopes (polytera)". |