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 [a,b,b,a], have an extended symmetry,
a,b,b,a, like [3,3,3,3], doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry. If all mirrors of a given color are unringed (inactive) 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 (active), 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 {p}×{q}×{ }. ;Uniform duoprisms There are 3 categorical
uniform duoprismatic families of polytopes based on
Cartesian products of the
uniform polyhedra and
regular polygons: {
q,
r}×{
p}.
Enumerating the convex uniform 5-polytopes •
Simplex family: A5 [34] • 19 uniform 5-polytopes •
Hypercube/
Orthoplex family: B5 [4,33] • 31 uniform 5-polytopes •
Demihypercube D5/E5 family: [32,1,1] • 23 uniform 5-polytopes (8 unique) • Polychoral prisms: • 56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [31,1,1]×[ ]. • 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. That brings the tally to: 19+31+8+45+1=104 In addition there are: • Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [
p]×[
q]×[ ]. • Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[
p], [4,3]×[
p], [5,3]×[
p].
The A5 family There are 19 forms based on all permutations of the
Coxeter diagrams with one or more rings. (16+4-1 cases) They are named by
Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron). The
A5 family has symmetry of order 720 (6
factorial). 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 (1,1,1,1,1,1).
The B5 family The
B5 family has symmetry of order 3840 (5!×25). This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the
Coxeter diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.) 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 (5! x 24). This family has 23 Wythoffian uniform polytopes, from
3×8-1 permutations of the D5
Coxeter diagram with one or more rings. 15 (2×8-1) are repeated from the B5 family and 8 are unique to this family, though even those 8 duplicate the alternations from the B5 family. In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5-polytopes 51 through 58 above.
Uniform prismatic forms There are 5 finite categorical
uniform prismatic families of polytopes based on the nonprismatic uniform
4-polytopes. For simplicity, most alternations are not shown.
A4 × A1 This prismatic family has
9 forms: The
A1 x A4 family has symmetry of order 240 (2*5!).
B4 × A1 This prismatic family has
16 forms. (Three are shared with [3,4,3]×[ ] family) The
A1×B4 family has symmetry of order 768 (254!). The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.
F4 × A1 This prismatic family has
10 forms. The
A1 x F4 family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry 3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3+,4,3,2] symmetry, order 1152.
H4 × A1 This prismatic family has
15 forms: The
A1 x H4 family has symmetry of order 28800 (2*14400).
Duoprism prisms Uniform duoprism prisms, {
p}×{
q}×{ }, form an infinite class for all integers
p,
q>2. {4}×{4}×{ } makes a lower symmetry form of the
5-cube. The extended
f-vector of {
p}×{
q}×{ } is computed as (
p,
p,
1)*(
q,
q,
1)*(2,
1) = (2
pq,5
pq,4
pq+2
p+2
q,3
pq+3
p+3
q,
p+
q+2,
1).
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 (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600
tetrahedra, 40
pentagonal antiprisms, 700
triangular prisms, 20
pentagonal prisms), and 322 hypercells (2
grand antiprisms , 20
pentagonal antiprism prisms , and 300
tetrahedral prisms ). == Notes on the Wythoff construction for the uniform 5-polytopes ==