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.

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.

Regular and uniform 5-polytopes by fundamental Coxeter groups

Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.

Uniform 5-polytopes can be generated by fundamental finite Coxeter groups and represented by permutations of rings of the Coxeter-Dynkin diagrams.

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. Simplex family: 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 4-faces. All elements are simplexes.
2. Hypercube/orthoplex family: B₅ [4,3,3,3] -
   • 31 uniform 5-polytopes as permutations of rings in the group diagram, including two regular forms and one alternated regular form:
     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, 5-orthoplex or 5-orthoplex.
        • It has 10 vertices, 40 edges, 80 faces, 80 cells, and 32 hypercells. All elements are simplexes.
     3. h{4,3,3,3} — demipenteract or E5 polytope, .
        • 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.
3. Demihypercube D₅/E₅ family: [3^2,1,1 -
   • 23 uniform 5-polytopes as permutations of rings in the group diagram, including:
     1. {3^1,2,1}, 1_2,1 demipenteract -
     2. {3^2,1,1}, 2_1,1 pentacross -

Uniform prismatic forms

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

#	Coxeter groups		Coxeter graph
1.	A4 × A1	[3,3,3] × [ ]
2.	B4 × A1	[4,3,3] × [ ]
3.	F4 × A1	[3,4,3] × [ ]
4.	H4 × A1	[5,3,3] × [ ]
5.	D4 × A1	[31,1,1 × [ ]
6.	A3 × I2p	[3,3] × [p]
7.	B3 × I2p	[4,3] × [p]
8.	H3 × I2p	[5,3] × [p]
9.	I2p × I2q × A1	[p] × [q] × [ ]

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.

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.

Although there is no agreed upon standard terminology for higher polytopes, for dimensional clarity George Olshevsky advocates borrowing from the SI prefix sequencing, which can covers up to 9-polytopes with 8-dimensional facets:

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

For specific polytopes, like the lower dimensional polytopes, they can be named by their number of facets. For example a 5-simplex, with 6 facets can explicitly be called a hexa-5-tope, representing a 6-faceted 5-polytope, and thus is named a hexateron.

See also

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

External links

• Polytope names, Guy Inchbald
• Polytopes of Various Dimensions, Jonathan Bowers
• Glossary for hyperspace, George Olshevsky
• Multi-dimensional Glossary, Garrett Jones
• polytope names, Wendy Krieger