Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.
History of discovery
- Regular polytopes:
- *1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
- Convex semiregular polytopes:
- *1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
- Convex uniform polytopes:
- *1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III.
- *1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, The Theory of Uniform Polytopes and Honeycombs, University of Toronto
Regular 5-polytopes
- - 5-simplex
- - 5-cube
- - 5-orthoplex
Convex uniform 5-polytopes
There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.Symmetry of uniform 5-polytopes in four dimensions
The 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the regular forms in the B5 family. The bifurcating graph of the D5 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. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form , have an extended symmetry,
If all mirrors of a given color are unringed in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed, an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
;Fundamental families
;Uniform prisms
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms ××.
;Uniform duoprisms
There are 3 categorical uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedra and regular polygons: ×.
Enumerating the convex uniform 5-polytopes
- Simplex family: A5
- * 19 uniform 5-polytopes
- Hypercube/Orthoplex family: BC5
- * 31 uniform 5-polytopes
- Demihypercube D5/E5 family:
- * 23 uniform 5-polytopes
- Prisms and duoprisms:
- * 56 uniform 5-polytope constructions based on prismatic families: ×, ×, ×, ×.
- * One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.
In addition there are:
- Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: ××.
- Infinitely many uniform 5-polytope constructions based on duoprismatic families: ×, ×, ×.
The A5 family
They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex.
The A5 family has symmetry of order 720. 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.
The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector.
The B5 family
The B5 family has symmetry of order 3840.This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram.
For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.
The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.
The D5 family
The D5 family has symmetry of order 1920.This family has 23 Wythoffian uniform polyhedra, from 3x8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 are repeated from the B5 family and 8 are unique to this family.
Uniform prismatic forms
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes:A4 × A1
This prismatic family has 9 forms:The A1 x A4 family has symmetry of order 240.
B4 × A1
This prismatic family has 16 forms.The A1×B4 family has symmetry of order 768.
F4 × A1
This prismatic family has 10 forms.The A1 x F4 family has symmetry of order 2304. Three polytopes 85, 86 and 89 have double symmetry 3,4,3],2], order 4608. The last one, snub 24-cell prism, has symmetry, order 1152.
H4 × A1
This prismatic family has 15 forms:The A1 x H4 family has symmetry of order 28800.
Grand antiprism prism
The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces, 1360 cells, and 322 hypercells.Regular and uniform honeycombs
There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.There are three regular honeycombs of Euclidean 4-space:
- tesseractic honeycomb, with symbols, =. There are 19 uniform honeycombs in this family.
- 24-cell honeycomb, with symbols,. There are 31 reflective uniform honeycombs in this family, and one alternated form.
- * Truncated 24-cell honeycomb with symbols t,
- * Snub 24-cell honeycomb, with symbols s, and constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
- 16-cell honeycomb, with symbols,
- There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h it is geometrically identical to the 16-cell honeycomb, =
- There are 7 uniquely ringed forms from the, family, all new, including:
- * 4-simplex honeycomb
- * Truncated 4-simplex honeycomb
- * Omnitruncated 4-simplex honeycomb
- There are 9 uniquely ringed forms in the : family, two new ones, including the quarter tesseractic honeycomb, =, and the bitruncated tesseractic honeycomb, =.
Compact regular tessellations of hyperbolic 4-space
There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H4 space:| Honeycomb name | Schläfli Symbol | Coxeter diagram | Facet type | Cell type | Face type | Face figure | Edge figure | Vertex figure | Dual |
| Order-5 5-cell | |||||||||
| Order-3 120-cell | |||||||||
| Order-5 tesseractic | |||||||||
| Order-4 120-cell | |||||||||
| Order-5 120-cell | Self-dual |
There are four regular star-honeycombs in H4 space:
| Honeycomb name | Schläfli Symbol | Coxeter diagram | Facet type | Cell type | Face type | Face figure | Edge figure | Vertex figure | Dual |
| Order-3 small stellated 120-cell | |||||||||
| Order-5/2 600-cell | |||||||||
| Order-5 icosahedral 120-cell | |||||||||
| Order-3 great 120-cell |
Regular and uniform hyperbolic honeycombs
There are 5 compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams. There are also 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite facets or vertex figures.= : | = : | = : = : = : |