5-polytope

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In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. Each polyhedral cell being shared by exactly two polychoron facets.

A proposed name polyteron (plural: polytera) has been advocated , from the Greek root poly- meaning "many", a shortened tetra- meaning "four", and suffix -on. "Four" refers to the dimension of the 5-polytope facets.

Contents 1 Definition 2 Regular and uniform polytera by fundamental Coxeter groups 3 Uniform prismatic forms 4 Pyramids 5 A note on generality of terms for n-polytopes and elements 6 See also 7 References 8 External links

[edit] Definition

A 5-polytope, or polyteron, is a closed five-dimensional figure with vertices, edges, faces, and cells, and hypercells. 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 hypercell is a polychoron. Furthermore, the following requirements must be met:

1. Each cell must join exactly two hypercells.
2. Adjacent hypercells are not in the same four-dimensional hyperplane.
3. The figure is not a compound of other figures which meet the requirements.

[edit] Regular and uniform polytera by fundamental Coxeter groups

Regular polytera can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.

Uniform polytera can be generated by fundamental finite Coxeter groups.

There are three regular and many other uniform 6-polytopes, enumerated by Coxeter groups, two having linear graphs and one having a bifurcated graph.

1. A₅ [3,3,3,3] -
   • 19 uniform polytera as permutations of rings in the group diagram, including one regular:
     1. {3,3,3,3} - Hexateron, hexa-5-tope or 5-simplex
        • It has 6 vertices, 15 edges, 20 faces, 15 cells, and 6 hypercells. All elements are simplexes.
2. C₅ [4,3,3,3] -
   • 31 uniform polytera as permutations of rings in the group diagram, including two regulars:
     1. {4,3,3,3} - penteract, deca-5-tope, or 5-hypercube
        • It has 32 vertices, 80 edges, 80 faces, 40 cells, and 10 hypercells. All elements are hypercubes.
     2. {3,3,3,4} - pentacross, triacontadi-5-tope, or 5-cross-polytope
        • It has 10 vertices, 40 edges, 80 faces, 80 cells, and 32 hypercells. All elements are simplexes.
3. B₅ [3^2,1,1] -
   • 23 uniform polytera as permutations of rings in the group diagram, including one semiregular from the demihypercube family:
     1. {3^2,1,1}, 1_2,1 demipenteract - ; also as h{4,3,3,3} .
        • It has 16 vertices, 80 edges, 160 faces, 120 cells, and 26 4-faces. The regular facets are 10 16-cells, and 16 5-cells.

[edit] Uniform prismatic forms

There are 9 categorical uniform prismatic forms based on Cartesian products of lower dimensional regular polytopes:

• 1. A₄xA₁: [3,3,3]x[ ]
  2. C₄xA₁: [4,3,3]x[ ] -
  3. F₄xA₁: [3,4,3]x[ ] -
  4. G₄xA₁: [5,3,3]x[ ] -
  5. B₄xA₁: [3^1,1,1]x[ ] -
  6. A₃xD₂^p: [3,3]x[p] -
  7. C₃xD₂^p: [4,3]x[p] -
  8. G₃xD₂^p: [5,3]x[p] -
  9. D₂^pxD₂^qxA₁: [p]x[q]x[ ] -

[edit] Pyramids

Pyramidal polyterons, or 5-pyramids, can be generated by a polychoron base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.

[edit] A note on generality of terms for n-polytopes and elements

A 5-polytope, or polyteron, follows from the lower dimensional polytopes: 2: polygon, 3: polyhedron, and 4: polychoron.

In more generality, although there is no agreed upon standard for higher polytopes, following a SI prefix sequencing, a proposed sequence of higher polytopes may be called:

• Polyteron as a name for 5-polytope (tera for 4D faceted polytope), and terons for 4-face element.
• Polypeton as a name for 6-polytope (peta for 5D faceted polytope), and petons for 5-face elements.
• Polyexon as a name for 7-polytope (exa for 6D faceted polytope), and exons for 6-face elements.
• Polyzetton as a name for 8-polytope (zetta for 7D faceted polytope), and zettons for 7-face elements.
• Polyyotton as a name for 9-polytope (yotta for 8D faceted polytope), and yottons for 8-face elements.

[edit] See also

• List of regular polytopes#Five Dimensions
• Polygon - 2-polytopes
• Polyhedron - 3-polytopes
• Polychoron - 4-polytopes
• Regular polygon
• Uniform polyhedron
• Uniform polychoron

[edit] References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• 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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

[edit] External links

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
• Glossary for hyperspace