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Uniform 5-polytope
Five-dimensional geometric shape
Graphs of regular and uniform 5-polytopes.
5-simplexRectified 5-simplexTruncated 5-simplex
Cantellated 5-simplexRuncinated 5-simplexStericated 5-simplex
5-orthoplexTruncated 5-orthoplexRectified 5-orthoplex
Cantellated 5-orthoplexRuncinated 5-orthoplex
Cantellated 5-cubeRuncinated 5-cubeStericated 5-cube
5-cubeTruncated 5-cubeRectified 5-cube
5-demicubeTruncated 5-demicube
Cantellated 5-demicubeRuncinated 5-demicube

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.

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History of discovery

  • Regular polytopes: (convex faces)
    • 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: (Various definitions before Coxeter's uniform category)
    • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.1
  • 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
  • Non-convex uniform polytopes:
    • 1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.2
    • 2000-2024: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,3 with a current count of 1348 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.45

Regular 5-polytopes

Main article: List of regular polytopes § Five Dimensions

Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex:

There are no nonconvex regular polytopes in 5 dimensions or above.

Convex uniform 5-polytopes

Unsolved problem in mathematics What is the complete set of convex uniform 5-polytopes?6 More unsolved problems in mathematics

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 families7
GroupsymbolOrderCoxetergraphBracketnotationCommutatorsubgroupCoxeternumber(h)Reflectionsm=5/2 h8
A5720[3,3,3,3][3,3,3,3]+615
D51920[3,3,31,1][3,3,31,1]+820
B53840[4,3,3,3]105 20
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}×{ }.

CoxetergroupOrderCoxeterdiagramCoxeternotationCommutatorsubgroupReflections
A4A1120[3,3,3,2] = [3,3,3]×[ ][3,3,3]+10 1
D4A1384[31,1,1,2] = [31,1,1]×[ ][31,1,1]+12 1
B4A1768[4,3,3,2] = [4,3,3]×[ ]4 12 1
F4A12304[3,4,3,2] = [3,4,3]×[ ][3+,4,3+]12 12 1
H4A128800[5,3,3,2] = [3,4,3]×[ ][5,3,3]+60 1
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A18pq[p,2,q,2] = [p]×[q]×[ ][p+,2,q+]p q 1
I2(2p)I2(q)A116pq[2p,2,q,2] = [2p]×[q]×[ ]p p q 1
I2(2p)I2(2q)A132pq[2p,2,2q,2] = [2p]×[2q]×[ ]p p q q 1
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}.

CoxetergroupOrderCoxeterdiagramCoxeternotationCommutatorsubgroupReflections
Prismatic groups (use 2p for even)
A3I2(p)48p[3,3,2,p] = [3,3]×[p][(3,3)+,2,p+]6 p
A3I2(2p)96p[3,3,2,2p] = [3,3]×[2p]6 p p
B3I2(p)96p[4,3,2,p] = [4,3]×[p]3 6p
B3I2(2p)192p[4,3,2,2p] = [4,3]×[2p]3 6 p p
H3I2(p)240p[5,3,2,p] = [5,3]×[p][(5,3)+,2,p+]15 p
H3I2(2p)480p[5,3,2,2p] = [5,3]×[2p]15 p 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

Further information: A5 polytope

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).

#Base pointJohnson naming systemBowers name and (acronym)Coxeter diagramk-face element countsVertexfigureFacet counts by location: [3,3,3,3]
43210[3,3,3](6)[3,3,2](15)[3,2,3](20)[2,3,3](15)[3,3,3](6)Alt
1(0,0,0,0,0,1) or (0,1,1,1,1,1)5-simplexhexateron (hix)61520156{3,3,3}{3,3,3}----
2(0,0,0,0,1,1) or (0,0,1,1,1,1)Rectified 5-simplexrectified hexateron (rix)1245806015t{3,3}×{ }r{3,3,3}---{3,3,3}
3(0,0,0,0,1,2) or (0,1,2,2,2,2)Truncated 5-simplextruncated hexateron (tix)1245807530Tetrah.pyrt{3,3,3}---{3,3,3}
4(0,0,0,1,1,2) or (0,1,1,2,2,2)Cantellated 5-simplexsmall rhombated hexateron (sarx)2713529024060prism-wedgerr{3,3,3}--{ }×{3,3}r{3,3,3}
5(0,0,0,1,2,2) or (0,0,1,2,2,2)Bitruncated 5-simplex bitruncated hexateron (bittix)1260140150602t{3,3,3}---t{3,3,3}
6(0,0,0,1,2,3) or (0,1,2,3,3,3)Cantitruncated 5-simplexgreat rhombated hexateron (garx)27135290300120tr{3,3,3}--{ }×{3,3}t{3,3,3}
7(0,0,1,1,1,2) or (0,1,1,1,2,2)Runcinated 5-simplexsmall prismated hexateron (spix)4725542027060t0,3{3,3,3}-{3}×{3}{ }×r{3,3}r{3,3,3}
8(0,0,1,1,2,3) or (0,1,2,2,3,3)Runcitruncated 5-simplexprismatotruncated hexateron (pattix)47315720630180t0,1,3{3,3,3}-{6}×{3}{ }×r{3,3}rr{3,3,3}
9(0,0,1,2,2,3) or (0,1,1,2,3,3)Runcicantellated 5-simplexprismatorhombated hexateron (pirx)47255570540180t0,1,3{3,3,3}-{3}×{3}{ }×t{3,3}2t{3,3,3}
10(0,0,1,2,3,4) or (0,1,2,3,4,4)Runcicantitruncated 5-simplexgreat prismated hexateron (gippix)47315810900360Irr.5-cellt0,1,2,3{3,3,3}-{3}×{6}{ }×t{3,3}tr{3,3,3}
11(0,1,1,1,2,3) or (0,1,2,2,2,3)Steritruncated 5-simplexcelliprismated hexateron (cappix)62330570420120t{3,3,3}{ }×t{3,3}{3}×{6}{ }×{3,3}t0,3{3,3,3}
12(0,1,1,2,3,4) or (0,1,2,3,3,4)Stericantitruncated 5-simplexcelligreatorhombated hexateron (cograx)6248011401080360tr{3,3,3}{ }×tr{3,3}{3}×{6}{ }×rr{3,3}t0,1,3{3,3,3}
13(0,0,0,1,1,1)Birectified 5-simplexdodecateron (dot)12601209020{3}×{3}r{3,3,3}---r{3,3,3}
14(0,0,1,1,2,2)Bicantellated 5-simplexsmall birhombated dodecateron (sibrid)3218042036090rr{3,3,3}-{3}×{3}-rr{3,3,3}
15(0,0,1,2,3,3)Bicantitruncated 5-simplexgreat birhombated dodecateron (gibrid)32180420450180tr{3,3,3}-{3}×{3}-tr{3,3,3}
16(0,1,1,1,1,2)Stericated 5-simplexsmall cellated dodecateron (scad)6218021012030Irr.16-cell{3,3,3}{ }×{3,3}{3}×{3}{ }×{3,3}{3,3,3}
17(0,1,1,2,2,3)Stericantellated 5-simplexsmall cellirhombated dodecateron (card)62420900720180rr{3,3,3}{ }×rr{3,3}{3}×{3}{ }×rr{3,3}rr{3,3,3}
18(0,1,2,2,3,4)Steriruncitruncated 5-simplexcelliprismatotruncated dodecateron (captid)6245011101080360t0,1,3{3,3,3}{ }×t{3,3}{6}×{6}{ }×t{3,3}t0,1,3{3,3,3}
19(0,1,2,3,4,5)Omnitruncated 5-simplexgreat cellated dodecateron (gocad)6254015601800720Irr. {3,3,3}t0,1,2,3{3,3,3}{ }×tr{3,3}{6}×{6}{ }×tr{3,3}t0,1,2,3{3,3,3}
NonuniformOmnisnub 5-simplexsnub dodecateron (snod)snub hexateron (snix)422234040802520360ht0,1,2,3{3,3,3}ht0,1,2,3{3,3,2}ht0,1,2,3{3,2,3}ht0,1,2,3{3,3,2}ht0,1,2,3{3,3,3}(360)Irr. {3,3,3}

The B5 family

Further information: B5 polytope

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.

#Base pointNameCoxeter diagramElement countsVertexfigureFacet counts by location: [4,3,3,3]
43210[4,3,3](10)[4,3,2](40)[4,2,3](80)[2,3,3](80)[3,3,3](32)Alt
20(0,0,0,0,1)√25-orthoplextriacontaditeron (tac)3280804010{3,3,4}----{3,3,3}
21(0,0,0,1,1)√2Rectified 5-orthoplexrectified triacontaditeron (rat)4224040024040{ }×{3,4}{3,3,4}---r{3,3,3}
22(0,0,0,1,2)√2Truncated 5-orthoplextruncated triacontaditeron (tot)4224040028080(Octah.pyr){3,3,4}---t{3,3,3}
23(0,0,1,1,1)√2Birectified 5-cubepenteractitriacontaditeron (nit)(Birectified 5-orthoplex)4228064048080{4}×{3}r{3,3,4}---r{3,3,3}
24(0,0,1,1,2)√2Cantellated 5-orthoplexsmall rhombated triacontaditeron (sart)8264015201200240Prism-wedger{3,3,4}{ }×{3,4}--rr{3,3,3}
25(0,0,1,2,2)√2Bitruncated 5-orthoplexbitruncated triacontaditeron (bittit)42280720720240t{3,3,4}---2t{3,3,3}
26(0,0,1,2,3)√2Cantitruncated 5-orthoplexgreat rhombated triacontaditeron (gart)8264015201440480t{3,3,4}{ }×{3,4}--t0,1,3{3,3,3}
27(0,1,1,1,1)√2Rectified 5-cuberectified penteract (rin)4220040032080{3,3}×{ }r{4,3,3}---{3,3,3}
28(0,1,1,1,2)√2Runcinated 5-orthoplexsmall prismated triacontaditeron (spat)162120021601440320r{4,3,3}{ }×r{3,4}{3}×{4}t0,3{3,3,3}
29(0,1,1,2,2)√2Bicantellated 5-cubesmall birhombated penteractitriacontaditeron (sibrant)(Bicantellated 5-orthoplex)12284021601920480rr{3,3,4}-{4}×{3}-rr{3,3,3}
30(0,1,1,2,3)√2Runcitruncated 5-orthoplexprismatotruncated triacontaditeron (pattit)162144036803360960rr{3,3,4}{ }×r{3,4}{6}×{4}-t0,1,3{3,3,3}
31(0,1,2,2,2)√2Bitruncated 5-cubebitruncated penteract (bittin)422807208003202t{4,3,3}---t{3,3,3}
32(0,1,2,2,3)√2Runcicantellated 5-orthoplexprismatorhombated triacontaditeron (pirt)1621200296028809602t{4,3,3}{ }×t{3,4}{3}×{4}-t0,1,3{3,3,3}
33(0,1,2,3,3)√2Bicantitruncated 5-cubegreat birhombated triacontaditeron (gibrant)(Bicantitruncated 5-orthoplex)12284021602400960tr{3,3,4}-{4}×{3}-rr{3,3,3}
34(0,1,2,3,4)√2Runcicantitruncated 5-orthoplexgreat prismated triacontaditeron (gippit)1621440416048001920tr{3,3,4}{ }×t{3,4}{6}×{4}-t0,1,2,3{3,3,3}
35(1,1,1,1,1)5-cubepenteract (pent)1040808032{3,3,3}{4,3,3}----
36(1,1,1,1,1)+ (0,0,0,0,1)√2Stericated 5-cubesmall cellated penteractitriacontaditeron (scant)(Stericated 5-orthoplex)2428001040640160Tetr.antiprm{4,3,3}{4,3}×{ }{4}×{3}{ }×{3,3}{3,3,3}
37(1,1,1,1,1)+ (0,0,0,1,1)√2Runcinated 5-cubesmall prismated penteract (span)202124021601440320t0,3{4,3,3}-{4}×{3}{ }×r{3,3}r{3,3,3}
38(1,1,1,1,1)+ (0,0,0,1,2)√2Steritruncated 5-orthoplexcelliprismated triacontaditeron (cappin)242152028802240640t0,3{4,3,3}{4,3}×{ }{6}×{4}{ }×t{3,3}t{3,3,3}
39(1,1,1,1,1)+ (0,0,1,1,1)√2Cantellated 5-cubesmall rhombated penteract (sirn)12268015201280320Prism-wedgerr{4,3,3}--{ }×{3,3}r{3,3,3}
40(1,1,1,1,1)+ (0,0,1,1,2)√2Stericantellated 5-cubecellirhombated penteractitriacontaditeron (carnit)(Stericantellated 5-orthoplex)242208047203840960rr{4,3,3}rr{4,3}×{ }{4}×{3}{ }×rr{3,3}rr{3,3,3}
41(1,1,1,1,1)+ (0,0,1,2,2)√2Runcicantellated 5-cubeprismatorhombated penteract (prin)202124029602880960t0,2,3{4,3,3}-{4}×{3}{ }×t{3,3}2t{3,3,3}
42(1,1,1,1,1)+ (0,0,1,2,3)√2Stericantitruncated 5-orthoplexcelligreatorhombated triacontaditeron (cogart)2422320592057601920t0,2,3{4,3,3}rr{4,3}×{ }{6}×{4}{ }×tr{3,3}tr{3,3,3}
43(1,1,1,1,1)+ (0,1,1,1,1)√2Truncated 5-cubetruncated penteract (tan)42200400400160Tetrah.pyrt{4,3,3}---{3,3,3}
44(1,1,1,1,1)+ (0,1,1,1,2)√2Steritruncated 5-cubecelliprismated triacontaditeron (capt)242160029602240640t{4,3,3}t{4,3}×{ }{8}×{3}{ }×{3,3}t0,3{3,3,3}
45(1,1,1,1,1)+ (0,1,1,2,2)√2Runcitruncated 5-cubeprismatotruncated penteract (pattin)202156037603360960t0,1,3{4,3,3}-{8}×{3}{ }×r{3,3}rr{3,3,3}
46(1,1,1,1,1)+ (0,1,1,2,3)√2Steriruncitruncated 5-cubecelliprismatotruncated penteractitriacontaditeron (captint)(Steriruncitruncated 5-orthoplex)2422160576057601920t0,1,3{4,3,3}t{4,3}×{ }{8}×{6}{ }×t{3,3}t0,1,3{3,3,3}
47(1,1,1,1,1)+ (0,1,2,2,2)√2Cantitruncated 5-cubegreat rhombated penteract (girn)12268015201600640tr{4,3,3}--{ }×{3,3}t{3,3,3}
48(1,1,1,1,1)+ (0,1,2,2,3)√2Stericantitruncated 5-cubecelligreatorhombated penteract (cogrin)2422400600057601920 tr{4,3,3}tr{4,3}×{ }{8}×{3}{ }×rr{3,3}t0,1,3{3,3,3}
49(1,1,1,1,1)+ (0,1,2,3,3)√2Runcicantitruncated 5-cubegreat prismated penteract (gippin)2021560424048001920t0,1,2,3{4,3,3}-{8}×{3}{ }×t{3,3}tr{3,3,3}
50(1,1,1,1,1)+ (0,1,2,3,4)√2Omnitruncated 5-cubegreat cellated penteractitriacontaditeron (gacnet)(omnitruncated 5-orthoplex)2422640816096003840Irr. {3,3,3}tr{4,3}×{ }tr{4,3}×{ }{8}×{6}{ }×tr{3,3}t0,1,2,3{3,3,3}
515-demicubehemipenteract (hin) = 261201608016r{3,3,3}h{4,3,3}----(16){3,3,3}
52Cantic 5-cubeTruncated hemipenteract (thin) = 42280640560160h2{4,3,3}---(16)r{3,3,3}(16)t{3,3,3}
53Runcic 5-cubeSmall rhombated hemipenteract (sirhin) = 42360880720160h3{4,3,3}---(16)r{3,3,3}(16)rr{3,3,3}
54Steric 5-cubeSmall prismated hemipenteract (siphin) = 8248072040080h{4,3,3}h{4,3}×{}--(16){3,3,3}(16)t0,3{3,3,3}
55Runcicantic 5-cubeGreat rhombated hemipenteract (girhin) = 4236010401200480h2,3{4,3,3}---(16)2t{3,3,3}(16)tr{3,3,3}
56Stericantic 5-cubePrismatotruncated hemipenteract (pithin) = 8272018401680480h2{4,3,3}h2{4,3}×{}--(16)rr{3,3,3}(16)t0,1,3{3,3,3}
57Steriruncic 5-cubePrismatorhombated hemipenteract (pirhin) = 8256012801120320h3{4,3,3}h{4,3}×{}--(16)t{3,3,3}(16)t0,1,3{3,3,3}
58Steriruncicantic 5-cubeGreat prismated hemipenteract (giphin) = 8272020802400960h2,3{4,3,3}h2{4,3}×{}--(16)tr{3,3,3}(16)t0,1,2,3{3,3,3}
NonuniformAlternated runcicantitruncated 5-orthoplexSnub prismatotriacontaditeron (snippit)Snub hemipenteract (snahin) = 11226240108806720960sr{3,3,4}sr{2,3,4}sr{3,2,4}-ht0,1,2,3{3,3,3}(960)Irr. {3,3,3}
NonuniformEdge-snub 5-orthoplexPyritosnub penteract (pysnan)1202792015360105601920sr3{3,3,4}sr3{2,3,4}sr3{3,2,4}s{3,3}×{ }ht0,1,2,3{3,3,3}(960)Irr. {3,3}×{ }
NonuniformSnub 5-cubeSnub penteract (snan)2162122402160013440960ht0,1,2,3{3,3,4}ht0,1,2,3{2,3,4}ht0,1,2,3{3,2,4}ht0,1,2,3{3,3,2}ht0,1,2,3{3,3,3}(1920)Irr. {3,3,3}

The D5 family

Further information: D5 polytope

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.

#Coxeter diagramSchläfli symbol symbolsJohnson and Bowers namesElement countsVertexfigureFacets by location: [31,2,1]
43210[3,3,3](16)[31,1,1](10)[3,3]×[ ](40)[ ]×[3]×[ ](80)[3,3,3](16)Alt
[51] = h{4,3,3,3}, 5-demicubeHemipenteract (hin)261201608016r{3,3,3}{3,3,3}h{4,3,3}---
[52] = h2{4,3,3,3}, cantic 5-cubeTruncated hemipenteract (thin)42280640560160t{3,3,3}h2{4,3,3}--r{3,3,3}
[53] = h3{4,3,3,3}, runcic 5-cubeSmall rhombated hemipenteract (sirhin)42360880720160rr{3,3,3}h3{4,3,3}--r{3,3,3}
[54] = h4{4,3,3,3}, steric 5-cubeSmall prismated hemipenteract (siphin)8248072040080t0,3{3,3,3}h{4,3,3}h{4,3}×{}-{3,3,3}
[55] = h2,3{4,3,3,3}, runcicantic 5-cubeGreat rhombated hemipenteract (girhin)42360104012004802t{3,3,3}h2,3{4,3,3}--tr{3,3,3}
[56] = h2,4{4,3,3,3}, stericantic 5-cubePrismatotruncated hemipenteract (pithin)8272018401680480t0,1,3{3,3,3}h2{4,3,3}h2{4,3}×{}-rr{3,3,3}
[57] = h3,4{4,3,3,3}, steriruncic 5-cubePrismatorhombated hemipenteract (pirhin)8256012801120320t0,1,3{3,3,3}h3{4,3,3}h{4,3}×{}-t{3,3,3}
[58] = h2,3,4{4,3,3,3}, steriruncicantic 5-cubeGreat prismated hemipenteract (giphin)8272020802400960t0,1,2,3{3,3,3}h2,3{4,3,3}h2{4,3}×{}-tr{3,3,3}
Nonuniform = ht0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplexSnub hemipenteract (snahin)11226240108806720960ht0,1,2,3{3,3,3}sr{3,3,4}sr{2,3,4}sr{3,2,4}ht0,1,2,3{3,3,3}(960)Irr. {3,3,3}

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!).

#Coxeter diagram and SchläflisymbolsNameElement counts
FacetsCellsFacesEdgesVertices
59 = {3,3,3}×{ }5-cell prism (penp)720302510
60 = r{3,3,3}×{ }Rectified 5-cell prism (rappip)1250907020
61 = t{3,3,3}×{ }Truncated 5-cell prism (tippip)125010010040
62 = rr{3,3,3}×{ }Cantellated 5-cell prism (srippip)2212025021060
63 = t0,3{3,3,3}×{ }Runcinated 5-cell prism (spiddip)3213020014040
64 = 2t{3,3,3}×{ }Bitruncated 5-cell prism (decap)126014015060
65 = tr{3,3,3}×{ }Cantitruncated 5-cell prism (grippip)22120280300120
66 = t0,1,3{3,3,3}×{ }Runcitruncated 5-cell prism (prippip)32180390360120
67 = t0,1,2,3{3,3,3}×{ }Omnitruncated 5-cell prism (gippiddip)32210540600240

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.

#Coxeter diagram and SchläflisymbolsNameElement counts
FacetsCellsFacesEdgesVertices
[16] = {4,3,3}×{ }Tesseractic prism (pent)(Same as 5-cube)1040808032
68 = r{4,3,3}×{ }Rectified tesseractic prism (rittip)2613627222464
69 = t{4,3,3}×{ }Truncated tesseractic prism (tattip)26136304320128
70 = rr{4,3,3}×{ }Cantellated tesseractic prism (srittip)58360784672192
71 = t0,3{4,3,3}×{ }Runcinated tesseractic prism (sidpithip)82368608448128
72 = 2t{4,3,3}×{ }Bitruncated tesseractic prism (tahp)26168432480192
73 = tr{4,3,3}×{ }Cantitruncated tesseractic prism (grittip)58360880960384
74 = t0,1,3{4,3,3}×{ }Runcitruncated tesseractic prism (prohp)8252812161152384
75 = t0,1,2,3{4,3,3}×{ }Omnitruncated tesseractic prism (gidpithip)8262416961920768
76 = {3,3,4}×{ }16-cell prism (hexip)1864885616
77 = r{3,3,4}×{ }Rectified 16-cell prism (icope)(Same as 24-cell prism)2614428821648
78 = t{3,3,4}×{ }Truncated 16-cell prism (thexip)2614431228896
79 = rr{3,3,4}×{ }Cantellated 16-cell prism (ricope)(Same as rectified 24-cell prism)50336768672192
80 = tr{3,3,4}×{ }Cantitruncated 16-cell prism (ticope)(Same as truncated 24-cell prism)50336864960384
81 = t0,1,3{3,3,4}×{ }Runcitruncated 16-cell prism (prittip)8252812161152384
82 = sr{3,3,4}×{ }snub 24-cell prism (sadip)1467681392960192
Nonuniformrectified tesseractic alterprism (rita)5028846428864
Nonuniformtruncated 16-cell alterprism (thexa)2616838433696
Nonuniformbitruncated tesseractic alterprism (taha)50288624576192

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.

#Coxeter diagram and SchläflisymbolsNameElement counts
FacetsCellsFacesEdgesVertices
[77] = {3,4,3}×{ }24-cell prism (icope)2614428821648
[79] = r{3,4,3}×{ }rectified 24-cell prism (ricope)50336768672192
[80] = t{3,4,3}×{ }truncated 24-cell prism (ticope)50336864960384
83 = rr{3,4,3}×{ }cantellated 24-cell prism (sricope)146100823042016576
84 = t0,3{3,4,3}×{ }runcinated 24-cell prism (spiccup)242115219201296288
85 = 2t{3,4,3}×{ } bitruncated 24-cell prism (contip)5043212481440576
86 = tr{3,4,3}×{ }cantitruncated 24-cell prism (gricope)1461008259228801152
87 = t0,1,3{3,4,3}×{ }runcitruncated 24-cell prism (pricope)2421584364834561152
88 = t0,1,2,3{3,4,3}×{ } omnitruncated 24-cell prism (gippiccup)2421872508857602304
[82] = s{3,4,3}×{ }snub 24-cell prism (sadip)1467681392960192

H4 × A1

This prismatic family has 15 forms:

The A1 x H4 family has symmetry of order 28800 (2*14400).

#Coxeter diagram and SchläflisymbolsNameElement counts
FacetsCellsFacesEdgesVertices
89 = {5,3,3}×{ }120-cell prism (hipe)122960264030001200
90 = r{5,3,3}×{ }Rectified 120-cell prism (rahipe)7224560984084002400
91 = t{5,3,3}×{ }Truncated 120-cell prism (thipe)722456011040120004800
92 = rr{5,3,3}×{ }Cantellated 120-cell prism (srahip)19221296029040252007200
93 = t0,3{5,3,3}×{ }Runcinated 120-cell prism (sidpixhip)26421272022080168004800
94 = 2t{5,3,3}×{ }Bitruncated 120-cell prism (xhip)722576015840180007200
95 = tr{5,3,3}×{ }Cantitruncated 120-cell prism (grahip)192212960326403600014400
96 = t0,1,3{5,3,3}×{ }Runcitruncated 120-cell prism (prixip)264218720448804320014400
97 = t0,1,2,3{5,3,3}×{ }Omnitruncated 120-cell prism (gidpixhip)264222320628807200028800
98 = {3,3,5}×{ }600-cell prism (exip)602240031201560240
99 = r{3,3,5}×{ }Rectified 600-cell prism (roxip)72250401080079201440
100 = t{3,3,5}×{ }Truncated 600-cell prism (texip)722504011520100802880
101 = rr{3,3,5}×{ }Cantellated 600-cell prism (srixip)14421152028080252007200
102 = tr{3,3,5}×{ }Cantitruncated 600-cell prism (grixip)144211520316803600014400
103 = t0,1,3{3,3,5}×{ }Runcitruncated 600-cell prism (prahip)264218720448804320014400

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) = (2pq,5pq,4pq+2p+2q,3pq+3p+3q,p+q+2,1).

Coxeter diagramNamesElement counts
4-facesCellsFacesEdgesVertices
{p}×{q}×{ }9p+q+23pq+3p+3q4pq+2p+2q5pq2pq
{p}2×{ }2(p+1)3p(p+1)4p(p+1)5p22p2
{3}2×{ }836484518
{4}2×{ } = 5-cube1040808032

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 ).

#NameElement counts
FacetsCellsFacesEdgesVertices
104grand antiprism prism (gappip)10322136019401100200

Notes on the Wythoff construction for the uniform 5-polytopes

Construction of the reflective 5-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 5-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.

Here are the primary operators available for constructing and naming the uniform 5-polytopes.

The last operation, the snub, and more generally the alternation, are the operations that can create nonreflective forms. These are drawn with "hollow rings" at the nodes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

OperationExtendedSchläfli symbolCoxeter diagramDescription
Parentt0{p,q,r,s}{p,q,r,s}Any regular 5-polytope
Rectifiedt1{p,q,r,s}r{p,q,r,s}The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.
Birectifiedt2{p,q,r,s}2r{p,q,r,s}Birectification reduces faces to points, cells to their duals.
Trirectifiedt3{p,q,r,s}3r{p,q,r,s}Trirectification reduces cells to points. (Dual rectification)
Quadrirectifiedt4{p,q,r,s}4r{p,q,r,s}Quadrirectification reduces 4-faces to points. (Dual)
Truncatedt0,1{p,q,r,s}t{p,q,r,s}Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.
Cantellatedt0,2{p,q,r,s}rr{p,q,r,s}In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place.
Runcinatedt0,3{p,q,r,s}Runcination reduces cells and creates new cells at the vertices and edges.
Stericatedt0,4{p,q,r,s}2r2r{p,q,r,s}Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.)
Omnitruncatedt0,1,2,3,4{p,q,r,s}All four operators, truncation, cantellation, runcination, and sterication are applied.
Halfh{2p,3,q,r}Alternation, same as
Cantich2{2p,3,q,r}Same as
Runcich3{2p,3,q,r}Same as
Runcicantich2,3{2p,3,q,r}Same as
Sterich4{2p,3,q,r}Same as
Steriruncich3,4{2p,3,q,r}Same as
Stericantich2,4{2p,3,q,r}Same as
Steriruncicantich2,3,4{2p,3,q,r}Same as
Snubs{p,2q,r,s}Alternated truncation
Snub rectifiedsr{p,q,2r,s}Alternated truncated rectification
ht0,1,2,3{p,q,r,s}Alternated runcicantitruncation
Full snubht0,1,2,3,4{p,q,r,s}Alternated omnitruncation

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.1112

Fundamental groups
#Coxeter groupCoxeter diagramForms
1 A ~ 4 {\displaystyle {\tilde {A}}_{4}} [3[5]][(3,3,3,3,3)]7
2 C ~ 4 {\displaystyle {\tilde {C}}_{4}} [4,3,3,4]19
3 B ~ 4 {\displaystyle {\tilde {B}}_{4}} [4,3,31,1][4,3,3,4,1+] = 23 (8 new)
4 D ~ 4 {\displaystyle {\tilde {D}}_{4}} [31,1,1,1][1+,4,3,3,4,1+] = 9 (0 new)
5 F ~ 4 {\displaystyle {\tilde {F}}_{4}} [3,4,3,3]31 (21 new)

There are three regular honeycombs of Euclidean 4-space:

Other families that generate uniform honeycombs:

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

Prismatic groups
#Coxeter groupCoxeter diagram
1 C ~ 3 {\displaystyle {\tilde {C}}_{3}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,3,4,2,∞]
2 B ~ 3 {\displaystyle {\tilde {B}}_{3}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,31,1,2,∞]
3 A ~ 3 {\displaystyle {\tilde {A}}_{3}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[4],2,∞]
4 C ~ 2 {\displaystyle {\tilde {C}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,4,2,∞,2,∞]
5 H ~ 2 {\displaystyle {\tilde {H}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [6,3,2,∞,2,∞]
6 A ~ 2 {\displaystyle {\tilde {A}}_{2}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[3],2,∞,2,∞]
7 I ~ 1 {\displaystyle {\tilde {I}}_{1}} × I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [∞,2,∞,2,∞,2,∞]
8 A ~ 2 {\displaystyle {\tilde {A}}_{2}} x A ~ 2 {\displaystyle {\tilde {A}}_{2}} [3[3],2,3[3]]
9 A ~ 2 {\displaystyle {\tilde {A}}_{2}} × B ~ 2 {\displaystyle {\tilde {B}}_{2}} [3[3],2,4,4]
10 A ~ 2 {\displaystyle {\tilde {A}}_{2}} × G ~ 2 {\displaystyle {\tilde {G}}_{2}} [3[3],2,6,3]
11 B ~ 2 {\displaystyle {\tilde {B}}_{2}} × B ~ 2 {\displaystyle {\tilde {B}}_{2}} [4,4,2,4,4]
12 B ~ 2 {\displaystyle {\tilde {B}}_{2}} × G ~ 2 {\displaystyle {\tilde {G}}_{2}} [4,4,2,6,3]
13 G ~ 2 {\displaystyle {\tilde {G}}_{2}} × G ~ 2 {\displaystyle {\tilde {G}}_{2}} [6,3,2,6,3]

Regular and uniform hyperbolic honeycombs

Hyperbolic compact groups

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.

A F ^ 4 {\displaystyle {\widehat {AF}}_{4}} = [(3,3,3,3,4)]:

D H ¯ 4 {\displaystyle {\bar {DH}}_{4}} = [5,3,31,1]:

H ¯ 4 {\displaystyle {\bar {H}}_{4}} = [3,3,3,5]:

B H ¯ 4 {\displaystyle {\bar {BH}}_{4}} = [4,3,3,5]: K ¯ 4 {\displaystyle {\bar {K}}_{4}} = [5,3,3,5]:

There are 5 regular compact convex hyperbolic honeycombs in H4 space:13

Compact regular convex hyperbolic honeycombs
Honeycomb nameSchläfliSymbol{p,q,r,s}Coxeter diagramFacettype{p,q,r}Celltype{p,q}Facetype{p}Facefigure{s}Edgefigure{r,s}Vertexfigure{q,r,s}Dual
Order-5 5-cell (pente){3,3,3,5}{3,3,3}{3,3}{3}{5}{3,5}{3,3,5}{5,3,3,3}
Order-3 120-cell (hitte){5,3,3,3}{5,3,3}{5,3}{5}{3}{3,3}{3,3,3}{3,3,3,5}
Order-5 tesseractic (pitest){4,3,3,5}{4,3,3}{4,3}{4}{5}{3,5}{3,3,5}{5,3,3,4}
Order-4 120-cell (shitte){5,3,3,4}{5,3,3}{5,3}{5}{4}{3,4}{3,3,4}{4,3,3,5}
Order-5 120-cell (phitte){5,3,3,5}{5,3,3}{5,3}{5}{5}{3,5}{3,3,5}Self-dual

There are also 4 regular compact hyperbolic star-honeycombs in H4 space:

Compact regular hyperbolic star-honeycombs
Honeycomb nameSchläfliSymbol{p,q,r,s}Coxeter diagramFacettype{p,q,r}Celltype{p,q}Facetype{p}Facefigure{s}Edgefigure{r,s}Vertexfigure{q,r,s}Dual
Order-3 small stellated 120-cell{5/2,5,3,3}{5/2,5,3}{5/2,5}{5}{5}{3,3}{5,3,3}{3,3,5,5/2}
Order-5/2 600-cell{3,3,5,5/2}{3,3,5}{3,3}{3}{5/2}{5,5/2}{3,5,5/2}{5/2,5,3,3}
Order-5 icosahedral 120-cell{3,5,5/2,5}{3,5,5/2}{3,5}{3}{5}{5/2,5}{5,5/2,5}{5,5/2,5,3}
Order-3 great 120-cell{5,5/2,5,3}{5,5/2,5}{5,5/2}{5}{3}{5,3}{5/2,5,3}{3,5,5/2,5}
Hyperbolic paracompact groups

There are 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.

P ¯ 4 {\displaystyle {\bar {P}}_{4}} = [3,3[4]]:

B P ¯ 4 {\displaystyle {\bar {BP}}_{4}} = [4,3[4]]: F R ¯ 4 {\displaystyle {\bar {FR}}_{4}} = [(3,3,4,3,4)]: D P ¯ 4 {\displaystyle {\bar {DP}}_{4}} = [3[3]×[]]:

N ¯ 4 {\displaystyle {\bar {N}}_{4}} = [4,/3\,3,4]: O ¯ 4 {\displaystyle {\bar {O}}_{4}} = [3,4,31,1]: S ¯ 4 {\displaystyle {\bar {S}}_{4}} = [4,32,1]: M ¯ 4 {\displaystyle {\bar {M}}_{4}} = [4,31,1,1]:

R ¯ 4 {\displaystyle {\bar {R}}_{4}} = [3,4,3,4]:

Notes

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
  • 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, Regular Polytopes, 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)
    • H.S.M. Coxeter, The Beauty of Geometry: Twelve Essays (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
  • 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 [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] (p. 287 5D Euclidean groups, p. 298 Four-dimensionsal honeycombs)
    • (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
  • James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2) [2]
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Fundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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  • e
Fundamental convex regular and uniform honeycombs in dimensions 2–9
SpaceFamily A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E2Uniform tiling0[3]δ3hδ3qδ3Hexagonal
E3Uniform convex honeycomb0[4]δ4hδ4qδ4
E4Uniform 4-honeycomb0[5]δ5hδ5qδ524-cell honeycomb
E5Uniform 5-honeycomb0[6]δ6hδ6qδ6
E6Uniform 6-honeycomb0[7]δ7hδ7qδ7222
E7Uniform 7-honeycomb0[8]δ8hδ8qδ8133 • 331
E8Uniform 8-honeycomb0[9]δ9hδ9qδ9152 • 251 • 521
E9Uniform 9-honeycomb0[10]δ10hδ10qδ10
E10Uniform 10-honeycomb0[11]δ11hδ11qδ11
En−1Uniform (n−1)-honeycomb0[n]δnhδnqδn1k2 • 2k1 • k21

References

  1. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 /wiki/Thorold_Gosset ↩

  2. Multidimensional Glossary, George Olshevsky https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html ↩

  3. Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246. https://archive.bridgesmathart.org/2000/bridges2000-239.pdf ↩

  4. Uniform Polytera, Jonathan Bowers http://www.polytope.net/hedrondude/polytera.htm ↩

  5. Uniform polytope https://polytope.miraheze.org/wiki/Uniform_polytope ↩

  6. ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04 http://www.openproblemgarden.org/op/convex_uniform_5_polytopes ↩

  7. Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions ↩

  8. Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61 /wiki/Coxeter ↩

  9. "N,k-dippip". https://bendwavy.org/klitzing/incmats/n-m-dippip.htm ↩

  10. "Gappip". https://bendwavy.org/klitzing/incmats/gappip.htm ↩

  11. Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections. ↩

  12. Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs ↩

  13. Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213 ↩