Uniform 5-polytope

{{Short description|Five-dimensional geometric shape}} {{-}} {| class="wikitable" width="300" align="right" style="margin-left:1em;" |+Graphs of regular and uniform 5-polytopes. |- valign="top" align="center" | colspan="4" |100px|class=skin-invert<BR>5-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Rectified 5-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Truncated 5-simplex<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}} |- valign="top" align="center" | colspan="4" |100px|class=skin-invert<BR>Cantellated 5-simplex<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Runcinated 5-simplex<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node}} | colspan="4" |100px|class=skin-invert<BR>Stericated 5-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}} |- valign="top" align="center" | colspan="4" |100px|class=skin-invert<BR>5-orthoplex<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}} | colspan="4" |100px|class=skin-invert<BR>Truncated 5-orthoplex<BR>{{CDD|node_1|3|node_1|3|node|3|node|4|node}} | colspan="4" |100px|class=skin-invert<BR>Rectified 5-orthoplex<BR>{{CDD|node|3|node_1|3|node|3|node|4|node}} |- valign="top" align="center" | colspan="6" |150px|class=skin-invert<BR>Cantellated 5-orthoplex<BR>{{CDD|node_1|3|node|3|node_1|3|node|4|node}} | colspan="6" |150px|class=skin-invert<BR>Runcinated 5-orthoplex<BR>{{CDD|node_1|3|node|3|node|3|node_1|4|node}} |- valign="top" align="center" | colspan="4" |100px|class=skin-invert<BR>Cantellated 5-cube<BR>{{CDD|node_1|4|node|3|node_1|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Runcinated 5-cube<BR>{{CDD|node_1|4|node|3|node|3|node_1|3|node}} | colspan="4" |100px|class=skin-invert<BR>Stericated 5-cube<BR>{{CDD|node_1|4|node|3|node|3|node|3|node_1}} |- valign="top" align="center" | colspan="4" |100px|class=skin-invert<BR>5-cube<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Truncated 5-cube<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}} | colspan="4" |100px|class=skin-invert<BR>Rectified 5-cube<BR>{{CDD|node|4|node_1|3|node|3|node|3|node}} |- valign="top" align="center" | colspan="6" |150px|class=skin-invert<BR>5-demicube<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node}} | colspan="6" |150px|class=skin-invert<BR>Truncated 5-demicube<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node}} |- valign="top" align="center" | colspan="6" |150px|class=skin-invert<BR>Cantellated 5-demicube<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node}} | colspan="6" |150px|class=skin-invert<BR>Runcinated 5-demicube<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node_1}} |}

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''': (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''.<ref>T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900</ref> *'''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.<ref>[https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary], George Olshevsky</ref> **'''2000-2024''': Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,<ref>{{cite conference |url=https://archive.bridgesmathart.org/2000/bridges2000-239.pdf |title=Uniform Polychora |last1=Bowers |first1=Jonathan |author-link1= |last2= |first2= |author-link2= |date=2000 |publisher= |editor=Reza Sarhagi |book-title=Bridges 2000 |pages=239–246 |location= |conference=Bridges Conference |id=}}</ref> with a current count of 1333<!--was 1348, the references haven't yet been updated--> 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.<ref>[http://www.polytope.net/hedrondude/polytera.htm Uniform Polytera], Jonathan Bowers</ref><ref>[https://polytope.miraheze.org/wiki/Uniform_polytope Uniform polytope]</ref>

== Regular 5-polytopes == {{Main|List of regular polytopes#Dimension 5 and higher}} 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:

*{3,3,3,3} - 5-simplex *{4,3,3,3} - 5-cube *{3,3,3,4} - 5-orthoplex

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

== Convex uniform 5-polytopes == {{unsolved|mathematics|What is the complete set of convex uniform 5-polytopes?<ref>{{citation|url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes|work=Open Problem Garden|title=Convex uniform 5-polytopes|access-date=2016-10-04|date=May 24, 2012|author=ACW|archive-url=https://web.archive.org/web/20161005164840/http://www.openproblemgarden.org/op/convex_uniform_5_polytopes|archive-date=October 5, 2016|url-status=live}}</ref>}} 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.{{fact|date=February 2015|reason=all these need sourcing}}

=== 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, <nowiki></nowiki>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.320px|thumb|Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

;Fundamental families<ref>Regular and semi-regular polytopes III, p. 315 Three finite groups of 5-dimensions</ref>

{| class="wikitable sortable" !Group<BR>symbol || data-sort-type="number" |Order|| colspan="2" |Coxeter<BR>graph||Bracket<BR>notation||Commutator<BR>subgroup|| data-sort-type="number" |Coxeter<BR>number<BR>(h)|| colspan="2" data-sort-type="number" |Reflections<BR>''m''=5/2 ''h''<ref>Coxeter, ''Regular polytopes'', §12.6 The number of reflections, equation 12.61</ref> |- align="center" !A₅ || 720||{{CDD|node|3|node|3|node|3|node|3|node}}||{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1|3|node_c1}}|| [3,3,3,3]||[3,3,3,3]⁺||6 || || 15 {{CDD|node_c1}} |- align="center" !D₅ || 1920||{{CDD|nodes|split2|node|3|node|3|node}}||{{CDD|nodeab_c1|split2|node_c1|3|node_c1|3|node_c1}}|| [3,3,3^1,1]|| rowspan="2" |[3,3,3^1,1]⁺||8 || || 20 {{CDD|node_c1}} |- align="center" !B₅ || 3840||{{CDD|node|4|node|3|node|3|node|3|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|3|node_c1|3|node_c1}}|| [4,3,3,3] || 10 || 5 {{CDD|node_c2}}||20 {{CDD|node_c1}} |}

;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}×{&nbsp;}. {| class=wikitable |- valign=top !Coxeter<BR>group !Order !colspan=2|Coxeter<BR>diagram !Coxeter<BR>notation !Commutator<BR>subgroup !colspan=5|Reflections |- align=center !A₄A₁ || 120|| {{CDD|node|3|node|3|node|3|node|2|node}}|| {{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1|2|node_c5}}|| [3,3,3,2] = [3,3,3]×[ ] || [3,3,3]⁺ || || || 10 {{CDD|node_c1}}|| || 1 {{CDD|node_c5}} |- align=center !D₄A₁ ||384|| {{CDD|nodes|split2|node|3|node|2|node}}||{{CDD|nodeab_c1|split2|node_c1|3|node_c1|2|node_c5}}|| [3^1,1,1,2] = [3^1,1,1]×[ ] ||rowspan=2| [3^1,1,1]⁺ || || ||12 {{CDD|node_c1}} |||| 1 {{CDD|node_c5}} |- align=center !B₄A₁ || 768 || {{CDD|node|4|node|3|node|3|node|2|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|3|node_c1|2|node_c5}}|| [4,3,3,2] = [4,3,3]×[ ] || ||4 {{CDD|node_c2}} ||12 {{CDD|node_c1}} |||| 1 {{CDD|node_c5}} |- align=center !F₄A₁ || 2304|| {{CDD|node|3|node|4|node|3|node|2|node}}||{{CDD|node_c2|3|node_c2|4|node_c1|3|node_c1|2|node_c5}}|| [3,4,3,2] = [3,4,3]×[ ] ||[3⁺,4,3⁺] || ||12 {{CDD|node_c2}} ||12 {{CDD|node_c1}} |||| 1 {{CDD|node_c5}} |- align=center !H₄A₁ ||28800|| {{CDD|node|5|node|3|node|3|node|2|node}}||{{CDD|node_c1|5|node_c1|3|node_c1|3|node_c1|2|node_c5}}|| [5,3,3,2] = [3,4,3]×[ ] || [5,3,3]⁺|| || ||60 {{CDD|node_c1}} |||| 1 {{CDD|node_c5}} |- !colspan=12|Duoprismatic prisms (use 2p and 2q for evens) |- align=center !I₂(''p'')I₂(''q'')A₁ ||8''pq''|| {{CDD|node|p|node|2|node|q|node|2|node}}||{{CDD|node_c2|p|node_c2|2|node_c1|q|node_c1|2|node_c5}}|| [p,2,q,2] = [p]×[q]×[ ] ||rowspan=3|[p⁺,2,q⁺] || || ''p'' {{CDD|node_c2}} ||''q'' {{CDD|node_c1}} |||| 1 {{CDD|node_c5}}

|- align=center !I₂(2''p'')I₂(''q'')A₁ ||16''pq''|| {{CDD|node|2x|p|node|2|node|q|node|2|node}}||{{CDD|node_c3|2x|p|node_c2|2|node_c1|q|node_c1|2|node_c5}}|| [2p,2,q,2] = [2p]×[q]×[ ] ||p {{CDD|node_c3}}||''p'' {{CDD|node_c2}} || ''q'' {{CDD|node_c1}} |||| 1 {{CDD|node_c5}}

|- align=center !I₂(2''p'')I₂(2''q'')A₁ ||32''pq''|| {{CDD|node|2x|p|node|2|node|2x|q|node|2|node}}||{{CDD|node_c3|2x|p|node_c2|2|node_c1|2x|q|node_c4|2|node_c5}}|| [2p,2,2q,2] = [2p]×[2q]×[ ] ||''p'' {{CDD|node_c3}}||''p'' {{CDD|node_c2}}|| ''q'' {{CDD|node_c1}}|| ''q'' {{CDD|node_c4}}|| 1 {{CDD|node_c5}} |}

;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''}.

{| class=wikitable |- valign=top !Coxeter<BR>group !Order !colspan=2|Coxeter<BR>diagram !Coxeter<BR>notation !Commutator<BR>subgroup !colspan=4|Reflections |- !colspan=12|Prismatic groups (use 2p for even) |- align=center !A₃''I''₂(''p'') || 48''p''|| {{CDD|node|3|node|3|node|2|node|p|node}}||{{CDD|node_c1|3|node_c1|3|node_c1|2|node_c3|p|node_c3}}|| [3,3,2,''p''] = [3,3]×[''p''] ||rowspan=4|[(3,3)⁺,2,''p''⁺] ||||6 {{CDD|node_c1}}||''p'' {{CDD|node_c3}}|| |- align=center !A₃''I''₂(''2p'') || 96''p''|| {{CDD|node|3|node|3|node|2|node|2x|p|node}}||{{CDD|node_c1|3|node_c1|3|node_c1|2|node_c3|2x|p|node_c4}}|| [3,3,2,2''p''] = [3,3]×[2''p''] ||||6 {{CDD|node_c1}}||''p'' {{CDD|node_c3}}||''p'' {{CDD|node_c4}} |- align=center !B₃''I''₂(''p'') ||96''p''|| {{CDD|node|4|node|3|node|2|node|p|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|2|node_c3|p|node_c3}}|| [4,3,2,''p''] = [4,3]×[''p''] ||3 {{CDD|node_c2}}||6{{CDD|node_c1}}||''p'' {{CDD|node_c3}} |- align=center !B₃''I''₂(''2p'') ||192''p''|| {{CDD|node|4|node|3|node|2|node|2x|p|node}}||{{CDD|node_c2|4|node_c1|3|node_c1|2|node_c3|2x|p|node_c4}}|| [4,3,2,2''p''] = [4,3]×[2''p''] ||3 {{CDD|node_c2}}||6 {{CDD|node_c1}}||''p'' {{CDD|node_c3}}||''p'' {{CDD|node_c4}} |- align=center !H₃''I''₂(''p'') ||240''p''|| {{CDD|node|5|node|3|node|2|node|p|node}}|| {{CDD|node_c1|5|node_c1|3|node_c1|2|node_c3|p|node_c3}}|| [5,3,2,''p''] = [5,3]×[''p''] ||rowspan=2|[(5,3)⁺,2,''p''⁺] || ||15 {{CDD|node_c1}}||''p'' {{CDD|node_c3}} |- align=center !H₃''I''₂(''2p'') ||480''p''|| {{CDD|node|5|node|3|node|2|node|2x|p|node}}|| {{CDD|node_c1|5|node_c1|3|node_c1|2|node_c3|2x|p|node_c4}}|| [5,3,2,2''p''] = [5,3]×[2''p''] || ||15 {{CDD|node_c1}}||''p'' {{CDD|node_c3}}||''p'' {{CDD|node_c4}} |}

=== 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: [3^2,1,1] ** 23 uniform 5-polytopes (8 unique) * Polychoral prisms: ** 56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[&nbsp;], [4,3,3]×[&nbsp;], [5,3,3]×[&nbsp;], [3^1,1,1]×[&nbsp;]. ** 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'']×[&nbsp;]. * Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[''p''], [4,3]×[''p''], [5,3]×[''p''].

=== The A₅ family === {{See|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).

{| class="wikitable" !rowspan=2|# !rowspan=2|Base point !rowspan=2|Johnson naming system<BR>Bowers name and (acronym)<BR>Coxeter diagram !colspan=5|k-face element counts !rowspan=2|Vertex<BR>figure !colspan=6 |Facet counts by location: [3,3,3,3] |- ! 4 ! 3 ! 2 ! 1 ! 0 ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(6) ! {{CDD|node|3|node|3|node|2|node}}<BR>[3,3,2]<BR>(15) ! {{CDD|node|3|node|2|node|3|node}}<BR>[3,2,3]<BR>(20) ! {{CDD|node|2|node|3|node|3|node}}<BR>[2,3,3]<BR>(15) ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(6) ! Alt |- !1 |(0,0,0,0,0,1) or (0,1,1,1,1,1) |5-simplex<BR>hexateron (hix)<BR>{{CDD|node|3|node|3|node|3|node|3|node_1}} | 6 | 15 | 20 | 15 | 6 | 60px<BR>{3,3,3} |60px<BR>{3,3,3} | - | - | - | - | |- !2 |(0,0,0,0,1,1) or (0,0,1,1,1,1) |Rectified 5-simplex<BR>rectified hexateron (rix)<BR>{{CDD|node|3|node|3|node|3|node_1|3|node}} | 12 | 45 | 80 | 60 | 15 | 60px<BR>t{3,3}×{&nbsp;} |60px<BR>r{3,3,3} | - | - | - |60px<BR>{3,3,3} |- !3 |(0,0,0,0,1,2) or (0,1,2,2,2,2) |Truncated 5-simplex<BR>truncated hexateron (tix)<BR>{{CDD|node|3|node|3|node|3|node_1|3|node_1}} | 12 | 45 | 80 | 75 | 30 | 60px<BR>Tetrah.pyr |60px<BR>t{3,3,3} | - | - | - |60px<BR>{3,3,3} | |- !4 |(0,0,0,1,1,2) or (0,1,1,2,2,2) |Cantellated 5-simplex<BR>small rhombated hexateron (sarx)<BR>{{CDD|node|3|node|3|node_1|3|node|3|node_1}}

| 27 | 135 | 290 | 240 | 60 |60px<BR>prism-wedge |60px<BR>rr{3,3,3} | - | - |60px<BR>{&nbsp;}×{3,3} |60px<BR>r{3,3,3} | |- !5 |(0,0,0,1,2,2) or (0,0,1,2,2,2) |Bitruncated 5-simplex<BR> bitruncated hexateron (bittix)<BR>{{CDD|node|3|node|3|node_1|3|node_1|3|node}}

| 12 | 60 | 140 | 150 | 60 | 60px |60px<BR>2t{3,3,3} | - | - | - |60px<BR>t{3,3,3} | |- !6 |(0,0,0,1,2,3) or (0,1,2,3,3,3) |Cantitruncated 5-simplex<BR>great rhombated hexateron (garx)<BR>{{CDD|node|3|node|3|node_1|3|node_1|3|node_1}}

| 27 | 135 | 290 | 300 | 120 |60px | 60px<BR>tr{3,3,3} | - | - | 60px<BR>{&nbsp;}×{3,3} | 60px<BR>t{3,3,3} | |- !7 |(0,0,1,1,1,2) or (0,1,1,1,2,2) |Runcinated 5-simplex<BR>small prismated hexateron (spix)<BR>{{CDD|node|3|node_1|3|node|3|node|3|node_1}} | 47 | 255 | 420 | 270 | 60 | 60px |60px<BR>t_0,3{3,3,3} | - |60px<BR>{3}×{3} |60px<BR>{&nbsp;}×r{3,3} |60px<BR>r{3,3,3} | |- !8 |(0,0,1,1,2,3) or (0,1,2,2,3,3) |Runcitruncated 5-simplex<BR>prismatotruncated hexateron (pattix)<BR>{{CDD|node|3|node_1|3|node|3|node_1|3|node_1}} | 47 | 315 | 720 | 630 | 180 |60px | 60px<BR>t_0,1,3{3,3,3} | - | 60px<BR>{6}×{3} | 60px<BR>{&nbsp;}×r{3,3} | 60px<BR>rr{3,3,3} | |- !9 |(0,0,1,2,2,3) or (0,1,1,2,3,3) |Runcicantellated 5-simplex<BR>prismatorhombated hexateron (pirx)<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node_1}} | 47 | 255 | 570 | 540 | 180 |60px | 60px<BR>t_0,1,3{3,3,3} | - | 60px<BR>{3}×{3} | 60px<BR>{&nbsp;}×t{3,3} | 60px<BR>2t{3,3,3} | |- !10 |(0,0,1,2,3,4) or (0,1,2,3,4,4) |Runcicantitruncated 5-simplex<BR>great prismated hexateron (gippix)<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1}} | 47 | 315 | 810 | 900 | 360 |60px<BR>Irr.5-cell | 60px<BR>t_0,1,2,3{3,3,3} | - | 60px<BR>{3}×{6} | 60px<BR>{&nbsp;}×t{3,3} | 60px<BR>tr{3,3,3} | |- !11 |(0,1,1,1,2,3) or (0,1,2,2,2,3) |Steritruncated 5-simplex<BR>celliprismated hexateron (cappix)<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node_1}} | 62 | 330 | 570 | 420 | 120 |60px | 60px<BR>t{3,3,3} | 60px<BR>{&nbsp;}×t{3,3} | 60px<BR>{3}×{6} | 60px<BR>{&nbsp;}×{3,3} | 60px<BR>t_0,3{3,3,3} | |- !12 |(0,1,1,2,3,4) or (0,1,2,3,3,4) |Stericantitruncated 5-simplex<BR>celligreatorhombated hexateron (cograx)<BR>{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1}} | 62 | 480 | 1140 | 1080 | 360 |60px | 60px<BR>tr{3,3,3} | 60px<BR>{&nbsp;}×tr{3,3} | 60px<BR>{3}×{6} | 60px<BR>{&nbsp;}×rr{3,3} | 60px<BR>t_0,1,3{3,3,3} | |- BGCOLOR="#e0f0e0" !13 |(0,0,0,1,1,1) |Birectified 5-simplex<BR>dodecateron (dot)<BR>{{dark mode invert|{{CDD|node|3|node|3|node_1|3|node|3|node}}}} | 12 | 60 | 120 | 90 | 20 | 60px<BR>{3}×{3} |60px<BR>r{3,3,3} | - | - | - |60px<BR>r{3,3,3} | |- BGCOLOR="#e0f0e0" !14 |(0,0,1,1,2,2) |Bicantellated 5-simplex<BR>small birhombated dodecateron (sibrid)<BR>{{dark mode invert|{{CDD|node|3|node_1|3|node|3|node_1|3|node}}}} | 32 | 180 | 420 | 360 | 90 |60px |60px<BR>rr{3,3,3} | - |60px<BR>{3}×{3} | - |60px<BR>rr{3,3,3} | |- BGCOLOR="#e0f0e0" !15 |(0,0,1,2,3,3) |Bicantitruncated 5-simplex<BR>great birhombated dodecateron (gibrid)<BR>{{dark mode invert|{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}}} | 32 | 180 | 420 | 450 | 180 |60px |60px<BR>tr{3,3,3} | - |60px<BR>{3}×{3} | - |60px<BR>tr{3,3,3} | |- BGCOLOR="#e0f0e0" !16 |(0,1,1,1,1,2) |Stericated 5-simplex<BR>small cellated dodecateron (scad)<BR>{{dark mode invert|{{CDD|node_1|3|node|3|node|3|node|3|node_1}}}} | 62 | 180 | 210 | 120 | 30 | 60px<BR>Irr.16-cell |60px<BR>{3,3,3} |60px<BR>{&nbsp;}×{3,3} |60px<BR>{3}×{3} |60px<BR>{&nbsp;}×{3,3} |60px<BR>{3,3,3} | |- BGCOLOR="#e0f0e0" !17 |(0,1,1,2,2,3) |Stericantellated 5-simplex<BR>small cellirhombated dodecateron (card)<BR>{{dark mode invert|{{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}}} | 62 | 420 | 900 | 720 | 180 |60px | 60px<BR>rr{3,3,3} | 60px<BR>{&nbsp;}×rr{3,3} | 60px<BR>{3}×{3} | 60px<BR>{&nbsp;}×rr{3,3} | 60px<BR>rr{3,3,3} | |- BGCOLOR="#e0f0e0" !18 |(0,1,2,2,3,4) |Steriruncitruncated 5-simplex<BR>celliprismatotruncated dodecateron (captid)<BR>{{dark mode invert|{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1}}}} | 62 | 450 | 1110 | 1080 | 360 |60px | 60px<BR>t_0,1,3{3,3,3} | 60px<BR>{&nbsp;}×t{3,3} | 60px<BR>{6}×{6} | 60px<BR>{&nbsp;}×t{3,3} | 60px<BR>t_0,1,3{3,3,3} | |- BGCOLOR="#e0f0e0" !19 |(0,1,2,3,4,5) |Omnitruncated 5-simplex<BR>great cellated dodecateron (gocad)<BR>{{dark mode invert|{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}}} | 62 | 540 | 1560 | 1800 | 720 |60px<BR>Irr. {3,3,3} |60px<BR>t_0,1,2,3{3,3,3} |60px<BR>{&nbsp;}×tr{3,3} |60px<BR>{6}×{6} |60px<BR>{&nbsp;}×tr{3,3} |60px<BR>t_0,1,2,3{3,3,3} | |- BGCOLOR="#d0f0f0" !Nonuniform | |Omnisnub 5-simplex<br>snub dodecateron (snod)<br>snub hexateron (snix)<br>{{dark mode invert|{{CDD|node_h|3|node_h|3|node_h|3|node_h|3|node_h}}}} | 422 | 2340 | 4080 | 2520 | 360 | |ht_0,1,2,3{3,3,3} |ht_0,1,2,3{3,3,2} |ht_0,1,2,3{3,2,3} |ht_0,1,2,3{3,3,2} |ht_0,1,2,3{3,3,3} |(360)<br>60px<br>Irr. {3,3,3} |}

=== The B₅ family === {{See|B5 polytope}} The B₅ family has symmetry of order 3840 (5!&times;2⁵).

This family has 2⁵&minus;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 {{CDD|node_h1|4|node|3}}... = {{CDD|nodes_10ru|split2}}..... (There are more alternations that are not listed because they produce only repetitions, as {{CDD|node_h0|4|node_1|3}}... = {{CDD|nodes_11|split2}}.... and {{CDD|node_h0|4|node|3}}... = {{CDD|nodes|split2}}.... 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.

{|class="wikitable" !rowspan=2|# !rowspan=2|Base point !rowspan=2|Name<BR>Coxeter diagram !colspan=5|Element counts !rowspan=2|Vertex<BR>figure !colspan=6 |Facet counts by location: [4,3,3,3] |- BGCOLOR="#e0e0f0" !4||3||2||1||0 ! {{CDD|node|4|node|3|node|3|node}}<BR>[4,3,3]<BR>(10) ! {{CDD|node|4|node|3|node|2||node}}<BR>[4,3,2]<BR>(40) ! {{CDD|node|4|node|2|node|3|node}}<BR>[4,2,3]<BR>(80) ! {{CDD|node|2|node|3|node|3|node}}<BR>[2,3,3]<BR>(80) ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(32) ! Alt |- BGCOLOR="#f0e0e0" !20 ||(0,0,0,0,1)√2||5-orthoplex<br>triacontaditeron (tac)<BR>{{dark mode invert|{{CDD||node|4|node|3|node|3|node|3|node_1}}}}||32||80||80||40||10 ||60px<BR>{3,3,4}|| - || - || - || - ||60px<BR>{3,3,3}|| |- BGCOLOR="#f0e0e0" !21 ||(0,0,0,1,1)√2||Rectified 5-orthoplex<br>rectified triacontaditeron (rat)<BR>{{dark mode invert|{{CDD||node|4|node|3|node|3|node_1|3|node}}}}||42||240||400||240||40 ||60px<BR>{&nbsp;}×{3,4}|| 60px<BR>{3,3,4} || - || - || - ||60px<BR>r{3,3,3} || |- BGCOLOR="#f0e0e0" !22 ||(0,0,0,1,2)√2||Truncated 5-orthoplex<br>truncated triacontaditeron (tot)<BR>{{dark mode invert|{{CDD||node|4|node|3|node|3|node_1|3|node_1}}}}||42||240||400||280||80 ||60px<BR>(Octah.pyr)||60px<BR>{3,3,4} || - || - || - ||60px<BR>t{3,3,3}|| |- BGCOLOR="#e0f0e0" !23 ||(0,0,1,1,1)√2||Birectified 5-cube<br>penteractitriacontaditeron (nit)<BR>(Birectified 5-orthoplex)<BR>{{dark mode invert|{{CDD||node|4|node|3|node_1|3|node|3|node}}}}||42||280||640||480||80 ||60px<BR>{4}×{3}|| 60px<BR>r{3,3,4} || - || - || - || 60px<BR>r{3,3,3} || |-BGCOLOR="#f0e0e0" !24 ||(0,0,1,1,2)√2||Cantellated 5-orthoplex<br>small rhombated triacontaditeron (sart)<BR>{{dark mode invert|{{CDD||node|4|node|3|node_1|3|node|3|node_1}}}}||82||640||1520||1200||240 ||60px<BR>Prism-wedge|| 60px<BR>r{3,3,4}|| 60px<br>{&nbsp;}×{3,4} || - || - || 60px<BR>rr{3,3,3} || |- BGCOLOR="#f0e0e0" !25 ||(0,0,1,2,2)√2||Bitruncated 5-orthoplex<br>bitruncated triacontaditeron (bittit)<BR>{{dark mode invert|{{CDD||node|4|node|3|node_1|3|node_1|3|node}}}}||42||280||720||720||240 ||60px|| 60px<BR>t{3,3,4} || - || - || - || 60px<BR>2t{3,3,3} || |- BGCOLOR="#f0e0e0" !26 ||(0,0,1,2,3)√2||Cantitruncated 5-orthoplex<br>great rhombated triacontaditeron (gart)<BR>{{dark mode invert|{{CDD||node|4|node|3|node_1|3|node_1|3|node_1}}}}||82||640||1520||1440||480 ||60px||60px<BR>t{3,3,4}|| 60px<br>{&nbsp;}×{3,4} || -|| - || 60px<BR>t_0,1,3{3,3,3} || |- BGCOLOR="#e0e0f0" !27 ||(0,1,1,1,1)√2||Rectified 5-cube<br>rectified penteract (rin)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node|3|node|3|node}}}}||42||200||400||320||80 || 60px<BR>{3,3}×{&nbsp;}|| 60px<BR>r{4,3,3}|| - || - || - || 60px<BR>{3,3,3} || |-BGCOLOR="#f0e0e0" !28 ||(0,1,1,1,2)√2||Runcinated 5-orthoplex<br>small prismated triacontaditeron (spat)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node|3|node|3|node_1}}}}||162||1200||2160||1440||320 || 60px||60px<BR>r{4,3,3} || 60px<br>{&nbsp;}×r{3,4} || 60px<BR>{3}×{4}|| || 60px<BR>t_0,3{3,3,3} || |- BGCOLOR="#e0f0e0" !29 ||(0,1,1,2,2)√2||Bicantellated 5-cube<br>small birhombated penteractitriacontaditeron (sibrant)<BR>(Bicantellated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node|3|node_1|3|node}}}}||122||840||2160||1920||480 || 60px|| 60px<BR>rr{3,3,4}|| - || 60px<BR>{4}×{3}|| - || 60px<BR>rr{3,3,3} || |- BGCOLOR="#f0e0e0" !30 ||(0,1,1,2,3)√2||Runcitruncated 5-orthoplex<br>prismatotruncated triacontaditeron (pattit)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node|3|node_1|3|node_1}}}}||162||1440||3680||3360||960 ||60px|| 60px<BR>rr{3,3,4} || 60px<br>{&nbsp;}×r{3,4} || 60px<BR>{6}×{4}|| - || 60px<BR>t_0,1,3{3,3,3} || |- BGCOLOR="#e0e0f0" !31 ||(0,1,2,2,2)√2||Bitruncated 5-cube<br>bitruncated penteract (bittin)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node_1|3|node|3|node}}}}||42||280||720||800||320 || 60px|| 60px<BR>2t{4,3,3}|| - || - || - || 60px<BR>t{3,3,3} || |- BGCOLOR="#f0e0e0" !32 ||(0,1,2,2,3)√2||Runcicantellated 5-orthoplex<br>prismatorhombated triacontaditeron (pirt)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node_1|3|node|3|node_1}}}}||162||1200||2960||2880||960 ||60px|| 60px<BR>2t{4,3,3}||60px<br>{&nbsp;}×t{3,4}|| 60px<BR>{3}×{4} || - || 60px<BR>t_0,1,3{3,3,3} || |- BGCOLOR="#e0f0e0" !33 ||(0,1,2,3,3)√2||Bicantitruncated 5-cube<br>great birhombated triacontaditeron (gibrant)<BR>(Bicantitruncated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node_1|3|node_1|3|node}}}}||122||840||2160||2400||960 || 60px|| 60px<BR>tr{3,3,4}|| - || 60px<BR>{4}×{3}|| - || 60px<BR>rr{3,3,3} || |- BGCOLOR="#f0e0e0" !34 ||(0,1,2,3,4)√2||Runcicantitruncated 5-orthoplex<br>great prismated triacontaditeron (gippit)<BR>{{dark mode invert|{{CDD||node|4|node_1|3|node_1|3|node_1|3|node_1}}}}||162||1440||4160||4800||1920 ||60px|| 60px<BR>tr{3,3,4} || 60px<br>{&nbsp;}×t{3,4} || 60px<BR>{6}×{4}|| - || 60px<BR>t_0,1,2,3{3,3,3} || |- BGCOLOR="#e0e0f0" !35 ||(1,1,1,1,1)||5-cube<br>penteract (pent)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node|3|node|3|node}}}}||10||40||80||80||32 ||60px<BR>{3,3,3}|| 60px<BR>{4,3,3}|| - || - || - || - || |- BGCOLOR="#e0f0e0" !36 ||(1,1,1,1,1)<BR>+ (0,0,0,0,1)√2||Stericated 5-cube<br>small cellated penteractitriacontaditeron (scant)<BR>(Stericated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node|3|node|3|node_1}}}}||242||800||1040||640||160 || 60px<BR>Tetr.antiprm|| 60px<BR>{4,3,3}|| 60px<BR>{4,3}×{&nbsp;}|| 60px<BR>{4}×{3}|| 60px<BR>{&nbsp;}×{3,3}|| 60px<BR>{3,3,3} || |- BGCOLOR="#e0e0f0" !37 ||(1,1,1,1,1)<BR>+ (0,0,0,1,1)√2||Runcinated 5-cube<br>small prismated penteract (span)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node|3|node_1|3|node}}}}||202||1240||2160||1440||320 || 60px|| 60px<BR>t_0,3{4,3,3}|| - || 60px<BR>{4}×{3}|| 60px<BR>{&nbsp;}×r{3,3}|| 60px<BR>r{3,3,3} || |- BGCOLOR="#f0e0e0" !38 ||(1,1,1,1,1)<BR>+ (0,0,0,1,2)√2||Steritruncated 5-orthoplex<br>celliprismated triacontaditeron (cappin)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node|3|node_1|3|node_1}}}}||242||1520||2880||2240||640 ||60px|| 60px<BR>t_0,3{4,3,3} || 60px<BR>{4,3}×{&nbsp;} || 60px<BR>{6}×{4} || 60px<br>{&nbsp;}×t{3,3} || 60px<BR>t{3,3,3} || |- BGCOLOR="#e0e0f0" !39 ||(1,1,1,1,1)<BR>+ (0,0,1,1,1)√2||Cantellated 5-cube<br>small rhombated penteract (sirn)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node_1|3|node|3|node}}}}||122||680||1520||1280||320 || 60px<BR>Prism-wedge|| 60px<BR>rr{4,3,3}|| - || - || 60px<BR>{&nbsp;}×{3,3}|| 60px<BR>r{3,3,3} || |- BGCOLOR="#e0f0e0" !40 ||(1,1,1,1,1)<BR>+ (0,0,1,1,2)√2||Stericantellated 5-cube<br>cellirhombated penteractitriacontaditeron (carnit)<BR>(Stericantellated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node_1|3|node|3|node_1}}}}||242||2080||4720||3840||960 ||60px|| 60px<BR>rr{4,3,3}|| 60px<BR>rr{4,3}×{&nbsp;}|| 60px<BR>{4}×{3}|| 60px<BR>{&nbsp;}×rr{3,3}|| 60px<BR>rr{3,3,3} || |- BGCOLOR="#e0e0f0" !41 ||(1,1,1,1,1)<BR>+ (0,0,1,2,2)√2||Runcicantellated 5-cube<br>prismatorhombated penteract (prin)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node_1|3|node_1|3|node}}}}||202||1240||2960||2880||960 ||60px|| 60px<BR>t_0,2,3{4,3,3}|| - || 60px<BR>{4}×{3}|| 60px<BR>{&nbsp;}×t{3,3}|| 60px<BR>2t{3,3,3} || |- BGCOLOR="#f0e0e0" !42 ||(1,1,1,1,1)<BR>+ (0,0,1,2,3)√2||Stericantitruncated 5-orthoplex<br>celligreatorhombated triacontaditeron (cogart)<BR>{{dark mode invert|{{CDD||node_1|4|node|3|node_1|3|node_1|3|node_1}}}}||242||2320||5920||5760||1920 ||60px|| 60px<BR>t_0,2,3{4,3,3}|| 60px<BR>rr{4,3}×{&nbsp;}|| 60px<BR>{6}×{4}|| 60px<BR>{&nbsp;}×tr{3,3}|| 60px<BR>tr{3,3,3} || |- BGCOLOR="#e0e0f0" !43 ||(1,1,1,1,1)<BR>+ (0,1,1,1,1)√2||Truncated 5-cube<br>truncated penteract (tan)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node|3|node|3|node}}}}||42||200||400||400||160 || 60px<BR>Tetrah.pyr|| 60px<BR>t{4,3,3}|| - || - || - || 60px<BR>{3,3,3} || |- BGCOLOR="#e0e0f0" !44 ||(1,1,1,1,1)<BR>+ (0,1,1,1,2)√2||Steritruncated 5-cube<br>celliprismated triacontaditeron (capt)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node|3|node|3|node_1}}}}||242||1600||2960||2240||640 ||60px|| 60px<BR>t{4,3,3}|| 60px<BR>t{4,3}×{&nbsp;}|| 60px<BR>{8}×{3}|| 60px<BR>{&nbsp;}×{3,3}|| 60px<BR>t_0,3{3,3,3} || |- BGCOLOR="#e0e0f0" !45 ||(1,1,1,1,1)<BR>+ (0,1,1,2,2)√2||Runcitruncated 5-cube<br>prismatotruncated penteract (pattin)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node|3|node_1|3|node}}}}||202||1560||3760||3360||960 ||60px||60px<BR>t_0,1,3{4,3,3} || - || 60px<BR>{8}×{3}|| 60px<br>{&nbsp;}×r{3,3} || 60px<BR>rr{3,3,3} || |- BGCOLOR="#e0f0e0" !46 ||(1,1,1,1,1)<BR>+ (0,1,1,2,3)√2||Steriruncitruncated 5-cube<br>celliprismatotruncated penteractitriacontaditeron (captint)<BR>(Steriruncitruncated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node|3|node_1|3|node_1}}}}||242||2160||5760||5760||1920 ||60px|| 60px<BR>t_0,1,3{4,3,3}|| 60px<BR>t{4,3}×{&nbsp;}|| 60px<BR>{8}×{6}|| 60px<BR>{&nbsp;}×t{3,3}|| 60px<BR>t_0,1,3{3,3,3} || |- BGCOLOR="#e0e0f0" !47 ||(1,1,1,1,1)<BR>+ (0,1,2,2,2)√2||Cantitruncated 5-cube<br>great rhombated penteract (girn)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node_1|3|node|3|node}}}}||122||680||1520||1600||640 ||60px|| 60px<BR>tr{4,3,3}|| - || - || 60px<BR>{&nbsp;}×{3,3}|| 60px<BR>t{3,3,3} || |- BGCOLOR="#e0e0f0" !48 ||(1,1,1,1,1)<BR>+ (0,1,2,2,3)√2||Stericantitruncated 5-cube<br>celligreatorhombated penteract (cogrin)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node_1|3|node|3|node_1}}}}||242||2400||6000||5760||1920 ||60px|| 60px<BR> tr{4,3,3}|| 60px<BR>tr{4,3}×{&nbsp;}|| 60px<BR>{8}×{3}|| 60px<BR>{&nbsp;}×rr{3,3}|| 60px<BR>t_0,1,3{3,3,3} || |- BGCOLOR="#e0e0f0" !49 ||(1,1,1,1,1)<BR>+ (0,1,2,3,3)√2||Runcicantitruncated 5-cube<br>great prismated penteract (gippin)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node_1|3|node_1|3|node}}}}||202||1560||4240||4800||1920 ||60px|| 60px<BR>t_0,1,2,3{4,3,3}|| - || 60px<BR>{8}×{3}|| 60px<BR>{&nbsp;}×t{3,3}|| 60px<BR>tr{3,3,3} || |- BGCOLOR="#e0f0e0" !50 ||(1,1,1,1,1)<BR>+ (0,1,2,3,4)√2||Omnitruncated 5-cube<br>great cellated penteractitriacontaditeron (gacnet)<BR>(omnitruncated 5-orthoplex)<BR>{{dark mode invert|{{CDD||node_1|4|node_1|3|node_1|3|node_1|3|node_1}}}}||242||2640||8160||9600||3840 || 60px<BR>Irr. {3,3,3}|| 60px<BR>tr{4,3}×{&nbsp;}|| 60px<BR>tr{4,3}×{&nbsp;}|| 60px<BR>{8}×{6}|| 60px<BR>{&nbsp;}×tr{3,3}|| 60px<BR>t_0,1,2,3{3,3,3} || |- BGCOLOR="#d0f0f0" !51 | |5-demicube<br>hemipenteract (hin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node|3|node|3|node}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node|3|node|3|node}}}} |26 |120 |160 |80 |16 |60px<br>r{3,3,3} |60px<br>h{4,3,3} | - | - | - | - |(16)<br>60px<br>{3,3,3} |- BGCOLOR="#d0f0f0" !52 | |Cantic 5-cube<br>Truncated hemipenteract (thin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node_1|3|node|3|node}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node_1|3|node|3|node}}}} |42 |280 |640 |560 |160 |60px |60px<br>h₂{4,3,3} | - | - | - |(16)<br>60px<br>r{3,3,3} |(16)<br>60px<br>t{3,3,3} |- BGCOLOR="#d0f0f0" !53 | | Runcic 5-cube<br>Small rhombated hemipenteract (sirhin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node|3|node_1|3|node}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node|3|node_1|3|node}}}} |42 |360 |880 |720 |160 | |60px<BR>h₃{4,3,3} | - | - | - |(16)<br>60px<BR>r{3,3,3} |(16)<br>60px<BR>rr{3,3,3} |- BGCOLOR="#d0f0f0" !54 | | Steric 5-cube<br>Small prismated hemipenteract (siphin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node|3|node|3|node_1}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node|3|node|3|node_1}}}} |82 |480 |720 |400 |80 | |60px<br>h{4,3,3} |60px<br>h{4,3}×{} | - | - |(16)<br>60px<BR>{3,3,3} |(16)<br>60px<BR>t_0,3{3,3,3} |- BGCOLOR="#d0f0f0" !55 | | Runcicantic 5-cube<br>Great rhombated hemipenteract (girhin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}}}} |42 |360 |1040 |1200 |480 | |60px<BR>h_2,3{4,3,3} | - | - | - |(16)<br>60px<BR>2t{3,3,3} |(16)<br>60px<br>tr{3,3,3} |- BGCOLOR="#d0f0f0" !56 | | Stericantic 5-cube<br>Prismatotruncated hemipenteract (pithin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node_1|3|node|3|node_1}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1}}}} |82 |720 |1840 |1680 |480 | |60px<br>h₂{4,3,3} |60px<br>h₂{4,3}×{} | - | - |(16)<br>60px<br>rr{3,3,3} |(16)<br>60px<br>t_0,1,3{3,3,3} |- BGCOLOR="#d0f0f0" !57 | |Steriruncic 5-cube<br>Prismatorhombated hemipenteract (pirhin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node|3|node_1|3|node_1}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1}}}} |82 |560 |1280 |1120 |320 | |60px<BR>h₃{4,3,3} |60px<br>h{4,3}×{} | - | - |(16)<br>60px<br>t{3,3,3} |(16)<br>60px<br>t_0,1,3{3,3,3} |- BGCOLOR="#d0f0f0" !58 | |Steriruncicantic 5-cube<br>Great prismated hemipenteract (giphin)<br>{{dark mode invert|{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1}}}} = {{dark mode invert|{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1}}}} |82 |720 |2080 |2400 |960 | |60px<BR>h_2,3{4,3,3} |60px<br>h₂{4,3}×{} | - | - |(16)<br>60px<br>tr{3,3,3} |(16)<br>60px<br>t_0,1,2,3{3,3,3} |- BGCOLOR="#d0f0f0" !Nonuniform | |Alternated runcicantitruncated 5-orthoplex<br>Snub prismatotriacontaditeron (snippit)<BR>Snub hemipenteract (snahin)<BR>{{dark mode invert|{{CDD|node|4|node_h|3|node_h|3|node_h|3|node_h}}}} = {{dark mode invert|{{CDD|nodes_hh|split2|node_h|3|node_h|3|node_h}}}} |1122 |6240 |10880 |6720 |960 | |60px<br>sr{3,3,4} |sr{2,3,4} |sr{3,2,4} | - |ht_0,1,2,3{3,3,3} |(960)<br>60px<br>Irr. {3,3,3} |- BGCOLOR="#d0f0f0" !Nonuniform | |Edge-snub 5-orthoplex<br>Pyritosnub penteract (pysnan)<BR>{{dark mode invert|{{CDD|node_1|4|node_h|3|node_h|3|node_h|3|node_h}}}} |1202 |7920 |15360 |10560 |1920 | |sr₃{3,3,4} |sr₃{2,3,4} |sr₃{3,2,4} |60px<br>s{3,3}×{&nbsp;} |ht_0,1,2,3{3,3,3} |(960)<br>60px<br>Irr. {3,3}×{&nbsp;} |- BGCOLOR="#d0f0f0" !Nonuniform | |Snub 5-cube<br>Snub penteract (snan)<BR>{{dark mode invert|{{CDD|node_h|4|node_h|3|node_h|3|node_h|3|node_h}}}} |2162 |12240 |21600 |13440 |960 | |ht_0,1,2,3{3,3,4} |ht_0,1,2,3{2,3,4} |ht_0,1,2,3{3,2,4} |ht_0,1,2,3{3,3,2} |ht_0,1,2,3{3,3,3} |(1920)<br>60px<br>Irr. {3,3,3} |}

=== The D₅ family === {{See|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 {{CDD|node|3|node|3|node|3|node}} element are identical and the symmetry doubles: the relations are {{CDD|node_h0|4|node_1|3}}... = {{CDD|nodes_11|split2}}.... and {{CDD|node_h0|4|node|3}}... = {{CDD|nodes|split2}}..., 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 {{CDD|node_h1|4|node|3}}... = {{CDD|nodes_10ru|split2}}... duplicating uniform 5-polytopes 51 through 58 above.

{| class="wikitable" !rowspan=2|# !rowspan=2|Coxeter diagram<BR>Schläfli symbol symbols<BR>Johnson and Bowers names !colspan=5|Element counts !rowspan=2|Vertex<BR>figure !colspan=6 |Facets by location: File:CD B5 nodes.png [3^1,2,1] |- !4 !3 !2 !1 !0 ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(16) ! {{CDD|nodes|split2|node|3|node}}<BR>[3^1,1,1]<BR>(10) ! {{CDD|nodes|split2|node|2|node}}<BR>[3,3]×[&nbsp;]<BR>(40) ! {{CDD|node|2|node|3|node|2|node}}<BR>[&nbsp;]×[3]×[&nbsp;]<BR>(80) ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(16) ! Alt |- ![51] | {{CDD|nodes_10ru|split2|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node}}<BR>h{4,3,3,3}, 5-demicube<BR>Hemipenteract (hin) | 26 | 120 | 160 | 80 | 16 | 50px<BR>r{3,3,3} | 60px<br>{3,3,3} | 60px<br>h{4,3,3} | - | - | - | |- ![52] | {{CDD|nodes_10ru|split2|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node}}<BR>h₂{4,3,3,3}, cantic 5-cube<BR>Truncated hemipenteract (thin) | 42 | 280 | 640 | 560 | 160 |60px |60px<br>t{3,3,3} |60px<br>h₂{4,3,3} | - | - |60px<br>r{3,3,3} | |- ![53] | {{CDD|nodes_10ru|split2|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node}}<BR>h₃{4,3,3,3}, runcic 5-cube<BR>Small rhombated hemipenteract (sirhin) | 42 | 360 | 880 | 720 | 160 | |60px<BR>rr{3,3,3} |60px<BR>h₃{4,3,3} | - | - |60px<br>r{3,3,3} | |- ![54] | {{CDD|nodes_10ru|split2|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1}}<BR>h₄{4,3,3,3}, steric 5-cube<BR>Small prismated hemipenteract (siphin) | 82 | 480 | 720 | 400 | 80 | |60px<BR>t_0,3{3,3,3} |60px<br>h{4,3,3} |60px<br>h{4,3}×{} | - |60px<BR>{3,3,3} | |- ![55] | {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node}}<BR>h_2,3{4,3,3,3}, runcicantic 5-cube<BR>Great rhombated hemipenteract (girhin) | 42 | 360 | 1040 | 1200 | 480 | |60px<BR>2t{3,3,3} |60px<BR>h_2,3{4,3,3} | - | - |60px<br>tr{3,3,3} | |- ![56] | {{CDD|nodes_10ru|split2|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1}}<BR>h_2,4{4,3,3,3}, stericantic 5-cube<BR>Prismatotruncated hemipenteract (pithin) | 82 | 720 | 1840 | 1680 | 480 | |60px<br>t_0,1,3{3,3,3} |60px<br>h₂{4,3,3} |60px<br>h₂{4,3}×{} | - |60px<br>rr{3,3,3} | |- ![57] | {{CDD|nodes_10ru|split2|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1}}<BR>h_3,4{4,3,3,3}, steriruncic 5-cube<BR>Prismatorhombated hemipenteract (pirhin) | 82 | 560 | 1280 | 1120 | 320 | |60px<br>t_0,1,3{3,3,3} |60px<BR>h₃{4,3,3} |60px<br>h{4,3}×{} | - |60px<br>t{3,3,3} | |- ![58] | {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1}}<BR>h_2,3,4{4,3,3,3}, steriruncicantic 5-cube<BR>Great prismated hemipenteract (giphin) | 82 | 720 | 2080 | 2400 | 960 | | 60px<br>t_0,1,2,3{3,3,3} | 60px<BR>h_2,3{4,3,3} | 60px<br>h₂{4,3}×{} | - | 60px<br>tr{3,3,3} | |- bgcolor="#D0F0F0" ! Nonuniform | {{dark mode invert|{{CDD|nodes_hh|split2|node_h|3|node_h|3|node_h}}}} = {{dark mode invert|{{CDD|node|4|node_h|3|node_h|3|node_h|3|node_h}}}}<BR>ht_0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex<br>Snub hemipenteract (snahin) |1122 |6240 |10880 |6720 |960 | | ht_0,1,2,3{3,3,3} | 60px<br>sr{3,3,4} | sr{2,3,4} | sr{3,2,4} | ht_0,1,2,3{3,3,3} | (960)<br>60px<br>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!).

{| class="wikitable" !rowspan=2|# !rowspan=2|Coxeter diagram<BR> and Schläfli<BR>symbols<BR>Name !colspan=5 rowspan=1|Element counts |- ! Facets|| Cells|| Faces|| Edges|| Vertices |- |59 |{{CDD|node_1|3|node|3|node|3|node|2|node_1}} = {3,3,3}×{&nbsp;}<BR>5-cell prism (penp) |7||20||30||25||10 |- |60 |{{CDD|node|3|node_1|3|node|3|node|2|node_1}} = r{3,3,3}×{&nbsp;}<BR>Rectified 5-cell prism (rappip) |12||50||90||70||20 |- |61 |{{CDD|node_1|3|node_1|3|node|3|node|2|node_1}} = t{3,3,3}×{&nbsp;}<BR>Truncated 5-cell prism (tippip) |12||50||100||100||40 |- |62 |{{CDD|node_1|3|node|3|node_1|3|node|2|node_1}} = rr{3,3,3}×{&nbsp;}<BR>Cantellated 5-cell prism (srippip) |22||120||250||210||60 |- BGCOLOR="#e0f0e0" |63 |{{dark mode invert|{{CDD|node_1|3|node|3|node|3|node_1|2|node_1}}}} = t_0,3{3,3,3}×{&nbsp;}<BR>Runcinated 5-cell prism (spiddip) |32||130||200||140||40 |- BGCOLOR="#e0f0e0" |64 |{{dark mode invert|{{CDD|node|3|node_1|3|node_1|3|node|2|node_1}}}} = 2t{3,3,3}×{&nbsp;}<BR>Bitruncated 5-cell prism (decap) |12||60||140||150||60 |- |65 |{{CDD|node_1|3|node_1|3|node_1|3|node|2|node_1}} = tr{3,3,3}×{&nbsp;}<BR>Cantitruncated 5-cell prism (grippip) |22||120||280||300||120 |- |66 |{{CDD|node_1|3|node_1|3|node|3|node_1|2|node_1}} = t_0,1,3{3,3,3}×{&nbsp;}<BR>Runcitruncated 5-cell prism (prippip) |32||180||390||360||120 |- BGCOLOR="#e0f0e0" |67 |{{dark mode invert|{{CDD|node_1|3|node_1|3|node_1|3|node_1|2|node_1}}}} = t_0,1,2,3{3,3,3}×{&nbsp;}<BR>Omnitruncated 5-cell prism (gippiddip) |32||210||540||600||240 |}

==== B₄ × A₁ ==== This prismatic family has 16 forms. (Three are shared with [3,4,3]×[&nbsp;] 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.

{| class="wikitable" !rowspan=2|# !rowspan=2|Coxeter diagram<BR> and Schläfli<BR>symbols<BR>Name !colspan=5 rowspan=1|Element counts |- ! Facets|| Cells|| Faces|| Edges|| Vertices |- BGCOLOR="#f0e0e0" |'''[16]'''||{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|2|node_1}}}} = {4,3,3}×{&nbsp;}<BR>Tesseractic prism (pent)<BR>(Same as 5-cube) |10||40||80||80||32 |- BGCOLOR="#f0e0e0" |'''68'''||{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|2|node_1}}}} = r{4,3,3}×{&nbsp;}<BR>Rectified tesseractic prism (rittip) |26||136||272||224||64 |- BGCOLOR="#f0e0e0" |'''69'''||{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|2|node_1}}}} = t{4,3,3}×{&nbsp;}<BR>Truncated tesseractic prism (tattip) |26||136||304||320||128 |- BGCOLOR="#f0e0e0" |'''70'''||{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|2|node_1}}}} = rr{4,3,3}×{&nbsp;}<BR>Cantellated tesseractic prism (srittip) |58||360||784||672||192 |- BGCOLOR="#e0f0e0" |'''71'''||{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|2|node_1}}}} = t_0,3{4,3,3}×{&nbsp;}<BR>Runcinated tesseractic prism (sidpithip) |82||368||608||448||128 |- BGCOLOR="#e0f0e0" |'''72'''||{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|2|node_1}}}} = 2t{4,3,3}×{&nbsp;}<BR>Bitruncated tesseractic prism (tahp) |26||168||432||480||192 |- BGCOLOR="#f0e0e0" |'''73'''||{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|2|node_1}}}} = tr{4,3,3}×{&nbsp;}<BR>Cantitruncated tesseractic prism (grittip) |58||360||880||960||384 |- BGCOLOR="#f0e0e0" |'''74'''||{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|2|node_1}}}} = t_0,1,3{4,3,3}×{&nbsp;}<BR>Runcitruncated tesseractic prism (prohp) |82||528||1216||1152||384 |- BGCOLOR="#e0f0e0" |'''75'''||{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|2|node_1}}}} = t_0,1,2,3{4,3,3}×{&nbsp;}<BR>Omnitruncated tesseractic prism (gidpithip) |82||624||1696||1920||768 |- BGCOLOR="#e0e0f0" |'''76'''||{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|2|node_1}}}} = {3,3,4}×{&nbsp;}<BR>16-cell prism (hexip) |18||64||88||56||16 |- BGCOLOR="#e0e0f0" |'''77'''||{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|2|node_1}}}} = r{3,3,4}×{&nbsp;}<BR>Rectified 16-cell prism (icope)<BR>(Same as '''24-cell prism''') |26||144||288||216||48 |- BGCOLOR="#e0e0f0" |'''78'''||{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|2|node_1}}}} = t{3,3,4}×{&nbsp;}<BR>Truncated 16-cell prism (thexip) |26||144||312||288||96 |- BGCOLOR="#e0e0f0" |'''79'''||{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|2|node_1}}}} = rr{3,3,4}×{&nbsp;}<BR>Cantellated 16-cell prism (ricope)<BR>(Same as '''rectified 24-cell prism''') |50||336||768||672||192 |- BGCOLOR="#e0e0f0" |'''80'''||{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|2|node_1}}}} = tr{3,3,4}×{&nbsp;}<BR>Cantitruncated 16-cell prism (ticope)<BR>(Same as '''truncated 24-cell prism''') |50||336||864||960||384 |- BGCOLOR="#e0e0f0" |'''81'''||{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|2|node_1}}}} = t_0,1,3{3,3,4}×{&nbsp;}<BR>Runcitruncated 16-cell prism (prittip) |82||528||1216||1152||384 |- BGCOLOR="#a0e0f0" |'''82'''||{{dark mode invert|{{CDD|node_h|3|node_h|3|node_h|4|node|2|node_1}}}} = sr{3,3,4}×{&nbsp;}<BR>snub 24-cell prism (sadip) |146||768||1392||960||192 |- BGCOLOR="#a0e0f0" |Nonuniform||{{dark mode invert|{{CDD|node_h|2x|node_1|3|node|3|node|4|node_h}}}}<br>rectified tesseractic alterprism (rita) |50||288||464||288||64 |- BGCOLOR="#a0e0f0" |Nonuniform||{{dark mode invert|{{CDD|node_h|2x|node|3|node_1|3|node|4|node_h}}}}<br>truncated 16-cell alterprism (thexa) |26||168||384||336||96 |- BGCOLOR="#a0e0f0" |Nonuniform||{{dark mode invert|{{CDD|node_h|2x|node_1|3|node_1|3|node|4|node_h}}}}<br>bitruncated tesseractic alterprism (taha) |50||288||624||576||192 |}

==== 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.

{| class="wikitable" !rowspan=2|# !rowspan=2|Coxeter diagram<BR> and Schläfli<BR>symbols<BR>Name !colspan=5 rowspan=1|Element counts |- ! Facets|| Cells|| Faces|| Edges|| Vertices |- |[77]||{{CDD|node_1|3|node|4|node|3|node|2|node_1}} = {3,4,3}×{&nbsp;}<BR>24-cell prism (icope) |26||144||288||216||48 |- |[79]||{{CDD|node|3|node_1|4|node|3|node|2|node_1}} = r{3,4,3}×{&nbsp;}<BR>rectified 24-cell prism (ricope) |50||336||768||672||192 |- |[80]||{{CDD|node_1|3|node_1|4|node|3|node|2|node_1}} = t{3,4,3}×{&nbsp;}<BR>truncated 24-cell prism (ticope) |50||336||864||960||384 |- |'''83'''||{{CDD|node_1|3|node|4|node_1|3|node|2|node_1}} = rr{3,4,3}×{&nbsp;}<BR>cantellated 24-cell prism (sricope) |146||1008||2304||2016||576 |- BGCOLOR="#b0f0b0" |'''84'''||{{dark mode invert|{{CDD|node_1|3|node|4|node|3|node_1|2|node_1}}}} = t_0,3{3,4,3}×{&nbsp;}<BR>runcinated 24-cell prism (spiccup) |242||1152||1920||1296||288 |- BGCOLOR="#b0f0b0" |'''85'''||{{dark mode invert|{{CDD|node|3|node_1|4|node_1|3|node|2|node_1}}}} = 2t{3,4,3}×{&nbsp;}<BR> bitruncated 24-cell prism (contip) |50||432||1248||1440||576 |- |'''86'''||{{CDD|node_1|3|node_1|4|node_1|3|node|2|node_1}} = tr{3,4,3}×{&nbsp;}<BR>cantitruncated 24-cell prism (gricope) |146||1008||2592||2880||1152 |- |'''87'''||{{CDD|node_1|3|node_1|4|node|3|node_1|2|node_1}} = t_0,1,3{3,4,3}×{&nbsp;}<BR>runcitruncated 24-cell prism (pricope) |242||1584||3648||3456||1152 |- BGCOLOR="#b0f0b0" |'''88'''||{{dark mode invert|{{CDD|node_1|3|node_1|4|node_1|3|node_1|2|node_1}}}} = t_0,1,2,3{3,4,3}×{&nbsp;}<BR> omnitruncated 24-cell prism (gippiccup) |242||1872||5088||5760||2304 |- BGCOLOR="#b0e0f0" |[82]||{{dark mode invert|{{CDD|node_h|3|node_h|4|node|3|node|2|node_1}}}} = s{3,4,3}×{&nbsp;}<BR>snub 24-cell prism (sadip) |146||768||1392||960||192 |}

==== H₄ × A₁ ==== This prismatic family has 15 forms:

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

{| class="wikitable" !rowspan=2|# !rowspan=2|Coxeter diagram<BR> and Schläfli<BR>symbols<BR>Name !colspan=5 rowspan=1|Element counts |- ! Facets|| Cells|| Faces|| Edges|| Vertices |- BGCOLOR="#f0e0e0" |'''89'''||{{dark mode invert|{{CDD|node_1|5|node|3|node|3|node|2|node_1}}}} = {5,3,3}×{&nbsp;}<BR>120-cell prism (hipe) |122||960||2640||3000||1200 |- BGCOLOR="#f0e0e0" |'''90'''||{{dark mode invert|{{CDD|node|5|node_1|3|node|3|node|2|node_1}}}} = r{5,3,3}×{&nbsp;}<BR>Rectified 120-cell prism (rahipe) |722||4560||9840||8400||2400 |- BGCOLOR="#f0e0e0" |'''91'''||{{dark mode invert|{{CDD|node_1|5|node_1|3|node|3|node|2|node_1}}}} = t{5,3,3}×{&nbsp;}<BR>Truncated 120-cell prism (thipe) |722||4560||11040||12000||4800 |- BGCOLOR="#f0e0e0" |'''92'''||{{dark mode invert|{{CDD|node_1|5|node|3|node_1|3|node|2|node_1}}}} = rr{5,3,3}×{&nbsp;}<BR>Cantellated 120-cell prism (srahip) |1922||12960||29040||25200||7200 |- BGCOLOR="#e0f0e0" |'''93'''||{{dark mode invert|{{CDD|node_1|5|node|3|node|3|node_1|2|node_1}}}} = t_0,3{5,3,3}×{&nbsp;}<BR>Runcinated 120-cell prism (sidpixhip) |2642||12720||22080||16800||4800 |- BGCOLOR="#e0f0e0" |'''94'''||{{dark mode invert|{{CDD|node|5|node_1|3|node_1|3|node|2|node_1}}}} = 2t{5,3,3}×{&nbsp;}<BR>Bitruncated 120-cell prism (xhip) |722||5760||15840||18000||7200 |- BGCOLOR="#f0e0e0" |'''95'''||{{dark mode invert|{{CDD|node_1|5|node_1|3|node_1|3|node|2|node_1}}}} = tr{5,3,3}×{&nbsp;}<BR>Cantitruncated 120-cell prism (grahip) |1922||12960||32640||36000||14400 |- BGCOLOR="#f0e0e0" |'''96'''||{{dark mode invert|{{CDD|node_1|5|node_1|3|node|3|node_1|2|node_1}}}} = t_0,1,3{5,3,3}×{&nbsp;}<BR>Runcitruncated 120-cell prism (prixip) |2642||18720||44880||43200||14400 |- BGCOLOR="#e0f0e0" |'''97'''||{{dark mode invert|{{CDD|node_1|5|node_1|3|node_1|3|node_1|2|node_1}}}} = t_0,1,2,3{5,3,3}×{&nbsp;}<BR>Omnitruncated 120-cell prism (gidpixhip) |2642||22320||62880||72000||28800 |- BGCOLOR="#e0e0f0" |'''98'''||{{dark mode invert|{{CDD|node|5|node|3|node|3|node_1|2|node_1}}}} = {3,3,5}×{&nbsp;}<BR>600-cell prism (exip) |602||2400||3120||1560||240 |- BGCOLOR="#e0e0f0" |'''99'''||{{dark mode invert|{{CDD|node|5|node|3|node_1|3|node|2|node_1}}}} = r{3,3,5}×{&nbsp;}<BR>Rectified 600-cell prism (roxip) |722||5040||10800||7920||1440 |- BGCOLOR="#e0e0f0" |'''100'''||{{dark mode invert|{{CDD|node|5|node|3|node_1|3|node_1|2|node_1}}}} = t{3,3,5}×{&nbsp;}<BR>Truncated 600-cell prism (texip) |722||5040||11520||10080||2880 |- BGCOLOR="#e0e0f0" |'''101'''||{{dark mode invert|{{CDD|node|5|node_1|3|node|3|node_1|2|node_1}}}} = rr{3,3,5}×{&nbsp;}<BR>Cantellated 600-cell prism (srixip) |1442||11520||28080||25200||7200 |- BGCOLOR="#e0e0f0" |'''102'''||{{dark mode invert|{{CDD|node|5|node_1|3|node_1|3|node_1|2|node_1}}}} = tr{3,3,5}×{&nbsp;}<BR>Cantitruncated 600-cell prism (grixip) |1442||11520||31680||36000||14400 |- BGCOLOR="#e0e0f0" |'''103'''||{{dark mode invert|{{CDD|node_1|5|node|3|node_1|3|node_1|2|node_1}}}} = t_0,1,3{3,3,5}×{&nbsp;}<BR>Runcitruncated 600-cell prism (prahip) |2642||18720||44880||43200||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''}×{&nbsp;} 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'''). {| class="wikitable" |- !rowspan=2|Coxeter diagram !rowspan=2|Names !colspan=6|Element counts |- ! 4-faces ! Cells ! Faces ! Edges ! Vertices |- align=center |{{CDD|branch_10|labelp|2|branch_10|labelq|2|node_1}}||{''p''}×{''q''}×{&nbsp;}<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/n-m-dippip.htm | title=N,k-dippip }}</ref>||''p''+''q''+2||3''pq''+3''p''+3''q''||4''pq''+2''p''+2''q''||5''pq''||2''pq'' |- align=center |{{CDD|branch_10|labelp|2|branch_10|labelp|2|node_1}}||{''p''}²×{&nbsp;}||2(''p''+1)||3''p''(''p''+1)||4''p''(''p''+1)||5''p''²||2''p''² |- align=center |{{CDD|branch_10|2|branch_10|2|node_1}}||{3}²×{&nbsp;}||8||36||48||45||18 |- align=center |{{CDD|branch_10|label4|2|branch_10|label4|2|node_1}}||{4}²×{ } = 5-cube||10||40||80||80||32 |}

==== 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 50px, 20 pentagonal antiprism prisms 50px, and 300 tetrahedral prisms 50px).

{| class="wikitable" !rowspan=2|# !rowspan=2| Name !colspan=5|Element counts |- ! Facets|| Cells|| Faces|| Edges|| Vertices |- |'''104'''|| grand antiprism prism (gappip)<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/gappip.htm | title=Gappip }}</ref>|| 322|| 1360|| 1940|| 1100|| 200 |}

== 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.

{|class="wikitable" !Operation !width=200 colspan=2|Extended<BR>Schläfli symbol !width=80|Coxeter diagram !Description |- align=center ! Parent |t₀{p,q,r,s} |{p,q,r,s} |{{CDD|node_1|p|node|q|node|r|node|s|node}} | Any regular 5-polytope |- align=center ! Rectified | t₁{p,q,r,s}||r{p,q,r,s} |{{CDD|node|p|node_1|q|node|r|node|s|node}} |align=left|The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual. |- align=center ! Birectified | t₂{p,q,r,s}||2r{p,q,r,s} |{{CDD|node|p|node|q|node_1|r|node|s|node}} |align=left|Birectification reduces faces to points, cells to their duals. |- align=center ! Trirectified | t₃{p,q,r,s}||3r{p,q,r,s} |{{CDD|node|p|node|q|node|r|node_1|s|node}} |align=left|Trirectification reduces cells to points. (Dual rectification) |- align=center ! Quadrirectified | t₄{p,q,r,s}||4r{p,q,r,s} |{{CDD|node|p|node|q|node|r|node|s|node_1}} |align=left|Quadrirectification reduces 4-faces to points. (Dual) |- align=center !Truncated | t_0,1{p,q,r,s}||t{p,q,r,s} |{{CDD|node_1|p|node_1|q|node|r|node|s|node}} |align=left|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.<BR>400px|class=skin-invert |- align=center ! Cantellated | t_0,2{p,q,r,s}||rr{p,q,r,s} |{{CDD|node_1|p|node|q|node_1|r|node|s|node}} |align=left|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. <BR>400px |- align=center ! Runcinated |colspan=2| t_0,3{p,q,r,s} |{{CDD|node_1|p|node|q|node|r|node_1|s|node}} |align=left|Runcination reduces cells and creates new cells at the vertices and edges. |- align=center ! Stericated |t_0,4{p,q,r,s}|| 2r2r{p,q,r,s} |{{CDD|node_1|p|node|q|node|r|node|s|node_1}} |align=left|Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.) |- align=center !Omnitruncated |colspan=2| t_0,1,2,3,4{p,q,r,s} |{{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1}} |align=left|All four operators, truncation, cantellation, runcination, and sterication are applied.

|- align=center !Half |colspan=2|h{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node|q|node|r|node}} |align=left|Alternation, same as {{CDD|labelp|branch_10ru|split2|node|q|node|r|node}} |- align=center !Cantic |colspan=2|h₂{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node_1|q|node|r|node}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node_1|q|node|r|node}} |- align=center !Runcic |colspan=2|h₃{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node|q|node_1|r|node}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node|q|node_1|r|node}} |- align=center !Runcicantic |colspan=2|h_2,3{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node_1|q|node_1|r|node}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node_1|q|node_1|r|node}} |- align=center !Steric |colspan=2|h₄{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node|q|node|r|node_1}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node|q|node|r|node_1}} |- align=center !Steriruncic |colspan=2|h_3,4{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node|q|node_1|r|node_1}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node|q|node_1|r|node_1}} |- align=center !Stericantic |colspan=2|h_2,4{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node_1|q|node|r|node_1}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node_1|q|node|r|node_1}} |- align=center !Steriruncicantic |colspan=2|h_2,3,4{2p,3,q,r} |{{CDD|node_h1|2x|p|node|3|node_1|q|node_1|r|node_1}} |align=left|Same as {{CDD|labelp|branch_10ru|split2|node_1|q|node_1|r|node_1}} |- align=center !Snub |colspan=2|s{p,2q,r,s} |{{CDD|node_h|p|node_h|2x|q|node|r|node|s|node}} |align=left|Alternated truncation |- align=center !Snub rectified |colspan=2|sr{p,q,2r,s} |{{CDD|node_h|p|node_h|q|node_h|2x|r|node|s|node}} |align=left|Alternated truncated rectification |- align=center ! |colspan=2|ht_0,1,2,3{p,q,r,s} |{{CDD|node_h|p|node_h|q|node_h|r|node_h|2x|s|node}} |align=left|Alternated runcicantitruncation |- align=center !Full snub |colspan=2|ht_0,1,2,3,4{p,q,r,s} |{{CDD|node_h|p|node_h|q|node_h|r|node_h|s|node_h}} |align=left|Alternated omnitruncation |}

== Regular and uniform honeycombs == 436px|thumb|Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence. There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.<ref>Regular polytopes, p. 297. Table IV, Fundamental regions for irreducible groups generated by reflections.</ref><ref>Regular and Semiregular polytopes, II, pp. 298–302 Four-dimensional honeycombs</ref>

{| class=wikitable |+ Fundamental groups |- !# !colspan=3|Coxeter group !Coxeter diagram !Forms |- align=center |1||<math>{\tilde{A}}_4</math>||[3^[5]]||[(3,3,3,3,3)]||{{CDD|branch|3ab|nodes|split2|node}}||7 |- align=center |2||<math>{\tilde{C}}_4</math>||[4,3,3,4]|| ||{{CDD|node|4|node|3|node|3|node|4|node}}||19 |- align=center |3||<math>{\tilde{B}}_4</math>||[4,3,3^1,1]||[4,3,3,4,1⁺]||{{CDD|nodes|split2|node|3|node|4|node}} = {{CDD|node_h0|4|node|3|node|3|node|4|node}}||23 (8 new) |- align=center |4||<math>{\tilde{D}}_4</math>||[3^1,1,1,1]||[1⁺,4,3,3,4,1⁺]||{{CDD|nodes|split2|node|split1|nodes}} = {{CDD|node_h0|4|node|3|node|3|node|4|node_h0}}||9 (0 new) |- align=center |5||<math>{\tilde{F}}_4</math>||[3,4,3,3]|| ||{{CDD|node|3|node|4|node|3|node|3|node}}||31 (21 new) |} There are three regular honeycombs of Euclidean 4-space: *tesseractic honeycomb, with symbols {4,3,3,4}, {{CDD|node_1|4|node|3|node|3|node|4|node}} = {{CDD|node_1|4|node|3|node|split1|nodes}}. There are 19 uniform honeycombs in this family. * 24-cell honeycomb, with symbols {3,4,3,3}, {{CDD|node_1|3|node|4|node|3|node|3|node}}. There are 31 reflective uniform honeycombs in this family, and one alternated form. ** Truncated 24-cell honeycomb with symbols t{3,4,3,3}, {{CDD|node_1|3|node_1|4|node|3|node|3|node}} ** Snub 24-cell honeycomb, with symbols s{3,4,3,3}, {{CDD|node_h|3|node_h|4|node|3|node|3|node}} and {{CDD|node_h|3|node_h|3|node_h|4|node|3|node}} constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex. * 16-cell honeycomb, with symbols {3,3,4,3}, {{CDD|node_1|3|node|3|node|4|node|3|node}}

Other families that generate uniform honeycombs: * There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h{4,3²,4} it is geometrically identical to the 16-cell honeycomb, {{CDD|node|4|node|3|node|3|node|4|node_h1}} = {{CDD|node|4|node|3|node|split1|nodes_10lu}} * There are 7 uniquely ringed forms from the <math>{\tilde{A}}_4</math>, {{CDD|branch|3ab|nodes|split2|node}} family, all new, including: ** 4-simplex honeycomb {{CDD|branch|3ab|nodes|split2|node_1}} ** Truncated 4-simplex honeycomb {{CDD|branch_11|3ab|nodes|split2|node}} ** Omnitruncated 4-simplex honeycomb {{CDD|branch_11|3ab|nodes_11|split2|node_1}} * There are 9 uniquely ringed forms in the <math>{\tilde{D}}_4</math>: [3^1,1,1,1] {{CDD|nodes|split2|node|split1|nodes}} family, two new ones, including the quarter tesseractic honeycomb, {{CDD|nodes_10ru|split2|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node|3|node|4|node_h1}}, and the bitruncated tesseractic honeycomb, {{CDD|nodes_10ru|split2|node_1|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node|4|node_h1}}.

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

{| class=wikitable |+ Prismatic groups |- !# !colspan=2|Coxeter group !Coxeter diagram |- |1||<math>{\tilde{C}}_3</math>×<math>{\tilde{I}}_1</math>||[4,3,4,2,∞]||{{CDD|node|4|node|3|node|4|node|2|node|infin|node}} |- |2||<math>{\tilde{B}}_3</math>×<math>{\tilde{I}}_1</math>||[4,3^1,1,2,∞]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|infin|node}} |- |3||<math>{\tilde{A}}_3</math>×<math>{\tilde{I}}_1</math>||[3^[4],2,∞]||{{CDD|branch|3ab|branch|2|node|infin|node}} |- |4||<math>{\tilde{C}}_2</math>×<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,4,2,∞,2,∞]||{{CDD|node|4|node|4|node|2|node|infin|node|2|node|infin|node}} |- |5||<math>{\tilde{H}}_2</math>×<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[6,3,2,∞,2,∞]||{{CDD|node|6|node|3|node|2|node|infin|node|2|node|infin|node}} |- |6||<math>{\tilde{A}}_2</math>×<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3^[3],2,∞,2,∞]||{{CDD|node|split1|branch|2|node|infin|node|2|node|infin|node}} |- |7||<math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[∞,2,∞,2,∞,2,∞]||{{CDD|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |8||<math>{\tilde{A}}_2</math>x<math>{\tilde{A}}_2</math>||[3^[3],2,3^[3]]||{{CDD|node|split1|branch|2|node|split1|branch}} |- |9||<math>{\tilde{A}}_2</math>×<math>{\tilde{B}}_2</math>||[3^[3],2,4,4]||{{CDD|node|split1|branch|2|node|4|node|4|node}} |- |10||<math>{\tilde{A}}_2</math>×<math>{\tilde{G}}_2</math>||[3^[3],2,6,3]||{{CDD|node|split1|branch|2|node|6|node|3|node}} |- |11||<math>{\tilde{B}}_2</math>×<math>{\tilde{B}}_2</math>||[4,4,2,4,4]||{{CDD|node|4|node|4|node|2|node|4|node|4|node}} |- |12||<math>{\tilde{B}}_2</math>×<math>{\tilde{G}}_2</math>||[4,4,2,6,3]||{{CDD|node|4|node|4|node|2|node|6|node|3|node}} |- |13||<math>{\tilde{G}}_2</math>×<math>{\tilde{G}}_2</math>||[6,3,2,6,3]||{{CDD|node|6|node|3|node|2|node|6|node|3|node}} |}

=== 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.

{| class="wikitable" | valign=top align=right| <math>{\widehat{AF}}_4</math> = [(3,3,3,3,4)]: {{CDD|label4|branch|3ab|nodes|split2|node}} | valign=top align=right| <math>{\bar{DH}}_4</math> = [5,3,3^1,1]: {{CDD|node|5|node|3|node|split1|nodes}} | valign=top align=right|<math>{\bar{H}}_4</math> = [3,3,3,5]: {{CDD|node|3|node|3|node|3|node|5|node}}<BR> <math>{\bar{BH}}_4</math> = [4,3,3,5]: {{CDD|node|4|node|3|node|3|node|5|node}}<BR> <math>{\bar{K}}_4</math> = [5,3,3,5]: {{CDD|node|5|node|3|node|3|node|5|node}} |}

There are 5 regular compact convex hyperbolic honeycombs in H⁴ space:<ref>Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p. 213</ref> {| class="wikitable" |+ Compact regular convex hyperbolic honeycombs |- !Honeycomb name !Schläfli<BR>Symbol<BR>{p,q,r,s} !Coxeter diagram !Facet<BR>type<BR>{p,q,r} !Cell<BR>type<BR>{p,q} !Face<BR>type<BR>{p} !Face<BR>figure<BR>{s} !Edge<BR>figure<BR>{r,s} !Vertex<BR>figure<BR>{q,r,s} !Dual |- BGCOLOR="#ffe0e0" align=center |Order-5 5-cell (pente)||{3,3,3,5}||{{dark mode invert|{{CDD|node|5|node|3|node|3|node|3|node_1}}}}||{3,3,3}||{3,3}||{3}||{5}||{3,5}||{3,3,5}||{5,3,3,3} |- BGCOLOR="#e0e0ff" align=center |Order-3 120-cell (hitte)||{5,3,3,3}||{{dark mode invert|{{CDD|node_1|5|node|3|node|3|node|3|node}}}}||{5,3,3}||{5,3}||{5}||{3}||{3,3}||{3,3,3}||{3,3,3,5} |- BGCOLOR="#ffe0e0" align=center |Order-5 tesseractic (pitest)||{4,3,3,5}||{{dark mode invert|{{CDD|node|5|node|3|node|3|node|4|node_1}}}}||{4,3,3}||{4,3}||{4}||{5}||{3,5}||{3,3,5}||{5,3,3,4} |- BGCOLOR="#e0e0ff" align=center |Order-4 120-cell (shitte)||{5,3,3,4}||{{dark mode invert|{{CDD|node_1|5|node|3|node|3|node|4|node}}}}||{5,3,3}||{5,3}||{5}||{4}||{3,4}||{3,3,4}||{4,3,3,5} |- BGCOLOR="#e0ffe0" align=center |Order-5 120-cell (phitte)||{5,3,3,5}||{{dark mode invert|{{CDD|node_1|5|node|3|node|3|node|5|node}}}}||{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: {| class="wikitable" |+ Compact regular hyperbolic star-honeycombs |- !Honeycomb name !Schläfli<BR>Symbol<BR>{p,q,r,s} !Coxeter diagram !Facet<BR>type<BR>{p,q,r} !Cell<BR>type<BR>{p,q} !Face<BR>type<BR>{p} !Face<BR>figure<BR>{s} !Edge<BR>figure<BR>{r,s} !Vertex<BR>figure<BR>{q,r,s} !Dual |- BGCOLOR="#ffe0e0" align=center |Order-3 small stellated 120-cell (sishitte)||{5/2,5,3,3}||{{dark mode invert|{{CDD|node_1|5|rat|d2|node|5|node|3|node|3|node}}}}||{5/2,5,3}||{5/2,5}||{5}||{5}||{3,3}||{5,3,3}||{3,3,5,5/2} |- BGCOLOR="#e0e0ff" align=center |Pentagrammic-order 600-cell (fipte)||{3,3,5,5/2}||{{dark mode invert|{{CDD|node|5|rat|d2|node|5|node|3|node|3|node_1}}}}||{3,3,5}||{3,3}||{3}||{5/2}||{5,5/2}||{3,5,5/2}||{5/2,5,3,3} |- BGCOLOR="#ffe0e0" align=center |Order-5 icosahedral 120-cell (tifipte)||{3,5,5/2,5}||{{dark mode invert|{{CDD|node_1|3|node|5|node|5|rat|d2|node|5|node}}}}||{3,5,5/2}||{3,5}||{3}||{5}||{5/2,5}||{5,5/2,5}||{5,5/2,5,3} |- BGCOLOR="#e0e0ff" align=center |Order-3 great 120-cell (gohitte)||{5,5/2,5,3}||{{dark mode invert|{{nowrap|{{CDD|node|3|node|5|node|5|rat|d2|node|5|node_1}}}}}}||{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.

{| class=wikitable |align=right| <math>{\bar{P}}_4</math> = [3,3^[4]]: {{CDD|node|split1|nodes|split2|node|3|node}}

<math>{\bar{BP}}_4</math> = [4,3^[4]]: {{CDD|node|split1|nodes|split2|node|4|node}}<BR> <math>{\bar{FR}}_4</math> = [(3,3,4,3,4)]: {{CDD|branch|4-4|nodes|split2|node}}<BR> <math>{\bar{DP}}_4</math> = [3^[3]×[]]: {{CDD|node|split1|branchbranch|split2|node}}

|align=right| <math>{\bar{N}}_4</math> = [4,/3\,3,4]: {{CDD|nodes|split2-43|node|3|node|4|node}}<BR> <math>{\bar{O}}_4</math> = [3,4,3^1,1]: {{CDD|nodes|split2|node|4|node|3|node}}<BR> <math>{\bar{S}}_4</math> = [4,3^2,1]: {{CDD|nodes|split2-43|node|3|node|3|node}}<BR> <math>{\bar{M}}_4</math> = [4,3^1,1,1]: {{CDD|nodes|split2-43|node|split1|nodes}}

|align=right| <math>{\bar{R}}_4</math> = [3,4,3,4]: {{CDD|node|4|node|3|node|4|node|3|node}}

|}

== Notes == {{reflist}}

== References == * 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) * {{cite journal|year=1910|author=A. Boole Stott|authorlink=Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|journal=Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam.|volume=XI|number=1|publisher=Johannes Müller|location=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-url=https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-date=29 April 2025}} * H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 (p.&nbsp;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 p. 213) * '''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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}} ** (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.&nbsp;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) [https://books.google.com/books?id=ODfjmOeNLMUC&dq=%22Reflection%20groups%20and%20Coxeter%20groups%22&pg=PA141]

== External links == * {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} – includes nonconvex forms as well as the duplicate constructions from the B₅ and D₅ families

{{Polytopes}} {{Honeycombs}}

Category:5-polytopes

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