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| {| align=right class=wikitable width=300
| | Nowomodna technologia spowodowała, że zaawansowany nakład, jakim istnieje niepodejrzanie druk 3d wykuło wiele ewentualności. Do służby, w jakich użytkuje się ów rodzaj wydruku przystaje formowanie a [http://pictwit.mobi/druk3d678016 skanowanie 3d]. Profilowanie 3d, owe służba, jaka liczy na złożeniu jednostce, jaka ją oznacza, rysunku technicznego.<br><br>Fachmani pracujący nad zarysem, na jego podstawie zbudują kostkę w trójwymiarze. Bryła ta będzie zdołała być wyeksploatowana do właściwego wydruku w modle 3d,ale również wizualizacji wskazywanej zwykle renderingiem a do przygotowania biegu obróbki skrawania na specjalistycznych machinach. A usługa, którą jest skanowanie 3d zawierzy na odtworzeniu wzorca w jego prostej tężyźnie.<br><br>Ma ono wiele przeznaczeń np. w stomatologii, można za pomocą owej technologii urzeczywistniać implanty czy w architekturze, zdumiewająco kameruje się w praktykowaniu makiet. Do innych przeznaczeń skanowania 3d wolno zaliczyć odbudowę designu w kryminalistyce czy w świecie zabawy zamieszkiwanie jakości, które do wyobrażenia przypominają drobiazgowe. W ostanim czasie procedura owa zoczyła również porządek wdrożeń w archiwistyce, owe imponujący sposób na zarchiwizowanie np.plonów muzeów. |
| |+ Graphs of [[List of regular polytopes#Five Dimensions|regular]] and [[uniform polytope]]s.
| |
| | || || || || || || || || || ||
| |
| |- align=center valign=top
| |
| |colspan=4|[[Image:5-simplex t0.svg|100px]]<BR>[[5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
| |
| |colspan=4|[[Image:5-simplex t1.svg|100px]]<BR>[[Rectified 5-simplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node}}
| |
| |colspan=4|[[Image:5-simplex t01.svg|100px]]<BR>[[Truncated 5-simplex]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}}
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:5-simplex t02.svg|100px]]<BR>[[Cantellated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node}}
| |
| |colspan=4|[[File:5-simplex t03.svg|100px]]<BR>[[Runcinated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node}}
| |
| |colspan=4|[[File:5-simplex t04.svg|100px]]<BR>[[Stericated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:5-cube t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}}
| |
| |colspan=4|[[File:5-cube t34.svg|100px]]<BR>[[Truncated 5-orthoplex]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|4|node}}
| |
| |colspan=4|[[File:5-cube t3.svg|100px]]<BR>[[Rectified 5-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|4|node}}
| |
| |- align=center valign=top
| |
| |colspan=6|[[File:5-cube t24.svg|150px]]<BR>[[Cantellated 5-orthoplex]]<BR>{{CDD|node_1|3|node|3|node_1|3|node|4|node}}
| |
| |colspan=6|[[File:5-cube t14.svg|150px]]<BR>[[Runcinated 5-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node_1|4|node}}
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:5-cube t02.svg|100px]]<BR>[[Cantellated 5-cube]]<BR>{{CDD|node_1|4|node|3|node_1|3|node|3|node}}
| |
| |colspan=4|[[File:5-cube t03.svg|100px]]<BR>[[Runcinated 5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node_1|3|node}}
| |
| |colspan=4|[[File:5-cube t04.svg|100px]]<BR>[[Stericated 5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node_1}}
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:5-cube t0.svg|100px]]<BR>[[5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}
| |
| |colspan=4|[[File:5-cube t01.svg|100px]]<BR>[[Truncated 5-cube]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}}
| |
| |colspan=4|[[File:5-cube t1.svg|100px]]<BR>[[Rectified 5-cube]]<BR>{{CDD|node|4|node_1|3|node|3|node|3|node}}
| |
| |- align=center valign=top
| |
| |colspan=6|[[File:5-demicube t0 D5.svg|150px]]<BR>[[5-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node}}
| |
| |colspan=6|[[File:5-demicube t01 D5.svg|150px]]<BR>[[Truncated 5-demicube]]<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node}}
| |
| |- align=center valign=top
| |
| |colspan=6|[[File:5-demicube t02 D5.svg|150px]]<BR>[[Cantellated 5-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node}}
| |
| |colspan=6|[[File:5-demicube t03 D5.svg|150px]]<BR>[[Runcinated 5-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node_1}}
| |
| |}
| |
| In [[geometry]], a '''uniform polyteron'''<ref>A [[5-polytope#A note on generality of terms for n-polytopes and elements|proposed name]] '''polyteron''' (plural: '''polytera''') has been advocated, from the [[Greek language|Greek]] root ''poly-'' meaning "many", a shortened ''[[Numerical prefix|tetra]]-'' meaning "four", and suffix ''-on''. "Four" refers to the dimension of the 5-polytope [[Facet (mathematics)|facets]].</ref><ref>http://www.steelpillow.com/polyhedra/ditela.html</ref> (or '''uniform [[5-polytope]]''') is a five-dimensional [[uniform polytope]]. By definition, a uniform polyteron is [[vertex-transitive]] and constructed from [[uniform polychoron]] [[Facet (geometry)|facets]].
| |
| | |
| The complete set of '''convex uniform polytera''' has not been determined, but most can be made as [[Wythoff construction]]s from a small set of [[Coxeter groups|symmetry groups]]. These construction operations are represented by the permutations of rings of the [[Coxeter-Dynkin diagram]]s.
| |
| | |
| == Regular 5-polytopes ==
| |
| {{Main|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} polychoral [[Facet (mathematics)|facets]] around each [[Face (geometry)|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 or more dimensions.
| |
| | |
| == Convex uniform 5-polytopes ==
| |
| | |
| There are 105 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 construction]]s, reflection symmetry generated with [[Coxeter group]]s.
| |
| | |
| === Reflection families ===
| |
| The [[5-simplex]] is the regular form in the A<sub>5</sub> family. The [[5-cube]] and [[5-orthoplex]] are the regular forms in the B<sub>5</sub> family. The bifurcating graph of the D<sub>6</sub> family contains the pentacross, as well as a [[5-demicube]] which is an [[alternation (geometry)|alternated]] 5-cube.
| |
| | |
| '''Fundamental families'''
| |
| | |
| {| class=wikitable
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| |-
| |
| |1||A<sub>5</sub>|| [3<sup>4</sup>]||{{CDD|node|3|node|3|node|3|node|3|node}}
| |
| |-
| |
| |2||B<sub>5</sub>||[4,3<sup>3</sup>]||{{CDD|node|4|node|3|node|3|node|3|node}}
| |
| |-
| |
| |3||D<sub>5</sub>||[3<sup>2,1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node}}
| |
| |}
| |
| | |
| {| class=wikitable width=480
| |
| |[[File:Coxeter diagram finite rank5 correspondence.png|480px]]<BR>Coxeter-Dynkin 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.
| |
| |}
| |
| | |
| '''Uniform prisms'''
| |
| There are 5 finite categorical [[Uniform polytope|uniform]] [[Prismatic polytope|prism]]atic families of polytopes based on the nonprismatic uniform [[4-polytope]]s:
| |
| | |
| {| class=wikitable
| |
| !#
| |
| !colspan=2|[[Coxeter group]]s
| |
| ![[Coxeter-Dynkin diagram|Coxeter graph]]
| |
| |- style="height:25px;"
| |
| | 1
| |
| | A<sub>4</sub> × A<sub>1</sub>
| |
| | [3,3,3,2]
| |
| | {{CDD|node|3|node|3|node|3|node|2|node}}
| |
| |- style="height:25px;"
| |
| | 2
| |
| | B<sub>4</sub> × A<sub>1</sub>
| |
| | [4,3,3,2]
| |
| | {{CDD|node|4|node|3|node|3|node|2|node}}
| |
| |- style="height:25px;"
| |
| | 3
| |
| | F<sub>4</sub> × A<sub>1</sub>
| |
| | [3,4,3,2]
| |
| | {{CDD|node|3|node|4|node|3|node|2|node}}
| |
| |- style="height:25px;"
| |
| | 4
| |
| | H<sub>4</sub> × A<sub>1</sub>
| |
| | [5,3,3,2]
| |
| | {{CDD|node|5|node|3|node|3|node|2|node}}
| |
| |- style="height:25px;"
| |
| | 5
| |
| | D<sub>4</sub> × A<sub>1</sub>
| |
| | [3<sup>1,1,1</sup>,2]
| |
| | {{CDD|nodes|split2|node|3|node|2|node}}
| |
| |}
| |
| | |
| There is one infinite family of 5-polytopes based on prisms of the uniform [[duoprism]]s {p}×{q}×{ }:
| |
| {| class=wikitable
| |
| !colspan=2|[[Coxeter group]]s
| |
| ![[Coxeter-Dynkin diagram|Coxeter graph]]
| |
| |- style="height:25px;"
| |
| | I<sub>2</sub>(p) × I<sub>2</sub>(q) × A<sub>1</sub>
| |
| | [p,2,q,2]
| |
| | {{CDD|node|p|node|2|node|q|node|2|node}}
| |
| |}
| |
| | |
| '''Uniform duoprisms'''
| |
| | |
| There are 3 categorical [[Uniform polytope|uniform]] [[duoprism]]atic families of polytopes based on [[Cartesian product]]s of the [[uniform polyhedron|uniform polyhedra]] and [[regular polygon]]s: {q,r}×{p}:
| |
| {| class=wikitable
| |
| !#
| |
| !colspan=2|[[Coxeter group]]s
| |
| ![[Coxeter-Dynkin diagram|Coxeter graph]]
| |
| |- style="height:25px;"
| |
| | 1
| |
| | A<sub>3</sub> × I<sub>2</sub>(p)
| |
| | [3,3,2,p]
| |
| | {{CDD|node|3|node|3|node|2|node|p|node}}
| |
| |- style="height:25px;"
| |
| | 2
| |
| | B<sub>3</sub> × I<sub>2</sub>(p)
| |
| | [4,3,2,p]
| |
| | {{CDD|node|4|node|3|node|2|node|p|node}}
| |
| |- style="height:25px;"
| |
| | 3.
| |
| | H<sub>3</sub> × I<sub>2</sub>(p)
| |
| | [5,3,2,p]
| |
| | {{CDD|node|5|node|3|node|2|node|p|node}}
| |
| |}
| |
| | |
| === Enumerating the convex uniform 5-polytopes ===
| |
| * [[Simplex]] family: A<sub>5</sub> [3<sup>4</sup>]
| |
| ** 19 uniform 5-polytopes
| |
| * [[Hypercube]]/[[Orthoplex]] family: BC<sub>5</sub> [4,3<sup>3</sup>]
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| ** 31 uniform 5-polytopes
| |
| * [[Demihypercube]] D<sub>5</sub>/E<sub>5</sub> family: [3<sup>2,1,1</sup>]
| |
| ** 23 uniform 5-polytopes (8 unique)
| |
| * Prisms and duoprisms:
| |
| ** 56 uniform 5-polytope (46 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [3<sup>1,1,1</sup>]×[ ].
| |
| ** One [[non-Wythoffian]] - The [[grand antiprism prism]] is the only known non-Wythoffian convex uniform 5-polytope, constructed from two [[grand antiprism]]s connected by polyhedral prisms.
| |
| | |
| That brings the tally to: 19+31+8+46+1=105
| |
| | |
| 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 A<sub>5</sub> family ===
| |
| | |
| There are 19 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. (16+4-1 cases)
| |
| | |
| They are named by [[Norman Johnson (mathematician)|Norman Johnson]] from the Wythoff construction operations upon regular 5-simplex (hexateron).
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| | |
| The [[Coxeter group#Finite Coxeter groups|A<sub>5</sub> family]] has symmetry of order 720 (6 [[factorial]]).
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| | |
| 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).
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| | |
| See symmetry graphs: [[List of A5 polytopes]]
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| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Base point
| |
| !rowspan=2|[[Norman Johnson (mathematician)|Johnson]] naming system<BR>Bowers name and (acronym)<BR>[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
| |
| !colspan=5|k-face element counts
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !colspan=5 |Facet counts by location: [3,3,3,3]
| |
| |-
| |
| ! 4
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| ! 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]×[ ]<BR>(15)
| |
| ! {{CDD|node|3|node|2|node|3|node}}<BR>[3]×[3]<BR>(20)
| |
| ! {{CDD|node|2|node|3|node|3|node}}<BR>[ ]×[3,3]<BR>(15)
| |
| ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(6)
| |
| |-
| |
| |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
| |
| | [[File:5-simplex verf.png|60px]]<BR>[[5-cell|{3,3,3}]]
| |
| |(5)<BR>[[File:4-simplex t0.svg|50px]]<BR>[[5-cell|{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
| |
| | [[File:Rectified 5-simplex verf.png|60px]]<BR>[[Truncated tetrahedral prism|t{3,3}×{ }]]
| |
| |(4)<BR>[[File:4-simplex t1.svg|50px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| | -
| |
| | -
| |
| | -
| |
| |(2)<BR>[[File:4-simplex t0.svg|50px]]<BR>[[5-cell|{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
| |
| | [[File:Truncated 5-simplex verf.png|60px]]<BR>[[5-cell|Tetrah.pyr]]
| |
| |(4)<BR>[[File:4-simplex t01.svg|50px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| | -
| |
| | -
| |
| | -
| |
| |(1)<BR>[[File:4-simplex t0.svg|50px]]<BR>[[5-cell|{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |4
| |
| |(0,0,0,1,1,1)
| |
| |[[Birectified 5-simplex]]<BR>dodecateron (dot)<BR>{{CDD|node|3|node|3|node_1|3|node|3|node}}
| |
| | 12
| |
| | 60
| |
| | 120
| |
| | 90
| |
| | 20
| |
| | [[File:Birectified hexateron verf.png|60px]]<BR>[[duoprism|{3}×{3}]]
| |
| |(3)<BR>[[File:4-simplex t1.svg|50px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| | -
| |
| | -
| |
| | -
| |
| |(3)<BR>[[File:4-simplex t1.svg|50px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |-
| |
| |5
| |
| |(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
| |
| |[[File:Cantellated hexateron verf.png|60px]]<BR>prism-wedge
| |
| |(3)<BR>[[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| | -
| |
| | -
| |
| |(1)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t0.svg|30px]]<BR>[[tetrahedral prism|{ }×{3,3}]]
| |
| |(1)<BR>[[File:4-simplex t1.svg|50px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |-
| |
| |6
| |
| |(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
| |
| | [[File:Bitruncated 5-simplex verf.png|60px]]
| |
| |(3)<BR>[[File:4-simplex t12.svg|50px]]<BR>[[Bitruncated 5-cell|2t{3,3,3}]]
| |
| | -
| |
| | -
| |
| | -
| |
| |(2)<BR>[[File:4-simplex t01.svg|50px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| |-
| |
| |7
| |
| |(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
| |
| |[[File:Canitruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t012.svg|50px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| | -
| |
| | -
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t0.svg|30px]]<BR>[[tetrahedral prism|{ }×{3,3}]]
| |
| | [[File:4-simplex t01.svg|50px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| |-
| |
| |8
| |
| |(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
| |
| | [[File:Runcinated 5-simplex verf.png|60px]]
| |
| |(2)<BR>[[File:4-simplex t03.svg|50px]]<BR>[[Runcinated 5-cell|t<sub>0,3</sub>{3,3,3}]]
| |
| | -
| |
| |(3)<BR>[[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| |(3)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t1.svg|30px]]<BR>[[Octahedral prism|{ }×r{3,3}]]
| |
| |(1)<BR>[[File:4-simplex t1.svg|50px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |9
| |
| |(0,0,1,1,2,2)
| |
| |[[Bicantellated 5-simplex]]<BR>small birhombated dodecateron (sibrid)<BR>{{CDD|node|3|node_1|3|node|3|node_1|3|node}}
| |
| | 32
| |
| | 180
| |
| | 420
| |
| | 360
| |
| | 90
| |
| |[[File:Bicantellated 5-simplex verf.png|60px]]
| |
| |(2)<BR>[[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| | -
| |
| |(8)<BR>[[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| | -
| |
| |(2)<BR>[[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |-
| |
| |10
| |
| |(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
| |
| |[[File:Runcitruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t013.svg|50px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]
| |
| | -
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{6}×{3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t1.svg|30px]]<BR>[[Octahedral prism|{ }×r{3,3}]]
| |
| | [[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |-
| |
| |11
| |
| |(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
| |
| |[[File:Runcicantellated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t03.svg|50px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]
| |
| | -
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:4-simplex t01.svg|30px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]
| |
| | [[File:4-simplex t12.svg|50px]]<BR>[[Bitruncated 5-cell|2t{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |12
| |
| |(0,0,1,2,3,3)
| |
| |[[Bicantitruncated 5-simplex]]<BR>great birhombated dodecateron (gibrid)<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}
| |
| | 32
| |
| | 180
| |
| | 420
| |
| | 450
| |
| | 180
| |
| |[[File:Bicanitruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t012.svg|50px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| | -
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| | -
| |
| | [[File:4-simplex t012.svg|50px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| |-
| |
| |13
| |
| |(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
| |
| |[[File:Runcicantitruncated 5-simplex verf.png|60px]]<BR>Irr.[[5-cell]]
| |
| | [[File:4-simplex t0123.svg|50px]]<BR>[[Omnitruncated 5-cell|t<sub>0,1,2,3</sub>{3,3,3}]]
| |
| | -
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{3}×{6}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:4-simplex t01.svg|30px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]
| |
| | [[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |14
| |
| |(0,1,1,1,1,2)
| |
| |[[Stericated 5-simplex]]<BR>small cellated dodecateron (scad)<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
| |
| | 62
| |
| | 180
| |
| | 210
| |
| | 120
| |
| | 30
| |
| | [[File:Stericated hexateron verf.png|60px]]<BR>Irr.[[16-cell]]
| |
| |(1)<BR>[[File:4-simplex t0.svg|50px]]<BR>[[5-cell|{3,3,3}]]
| |
| |(4)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t0.svg|30px]]<BR>[[tetrahedral prism|{ }×{3,3}]]
| |
| |(6)<BR>[[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| |(4)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t0.svg|30px]]<BR>[[tetrahedral prism|{ }×{3,3}]]
| |
| |(1)<BR>[[File:4-simplex t0.svg|50px]]<BR>[[5-cell|{3,3,3}]]
| |
| |-
| |
| |15
| |
| |(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
| |
| |[[File:Steritruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t01.svg|50px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:4-simplex t01.svg|30px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{3}×{6}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t0.svg|30px]]<BR>[[Tetrahedral prism|{ }×{3,3}]]
| |
| | [[File:4-simplex t03.svg|50px]]<BR>[[Runcinated 5-cell|t<sub>0,3</sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |16
| |
| |(0,1,1,2,2,3)
| |
| |[[Stericantellated 5-simplex]]<BR>small cellirhombated dodecateron (card)<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}
| |
| | 62
| |
| | 420
| |
| | 900
| |
| | 720
| |
| | 180
| |
| |[[File:Stericantellated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t02.svg|30px]]<BR>[[Cuboctahedral prism|{ }×rr{3,3}]]
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t0.svg|30px]]<BR>[[Duoprism|{3}×{3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t02.svg|30px]]<BR>[[Cuboctahedral prism|{ }×rr{3,3}]]
| |
| | [[File:4-simplex t02.svg|50px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |-
| |
| |17
| |
| |(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
| |
| |[[File:Stericanitruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t012.svg|50px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t012.svg|30px]]<BR>[[Truncated octahedral prism|{ }×tr{3,3}]]
| |
| | [[File:2-simplex t0.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{3}×{6}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:3-simplex t02.svg|30px]]<BR>[[Cuboctahedral prism|{ }×rr{3,3}]]
| |
| | [[File:4-simplex t013.svg|50px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |18
| |
| |(0,1,2,2,3,4)
| |
| |[[Steriruncitruncated 5-simplex]]<BR>celliprismatotruncated dodecateron (captid)<BR>{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1}}
| |
| | 62
| |
| | 450
| |
| | 1110
| |
| | 1080
| |
| | 360
| |
| |[[File:Steriruncitruncated 5-simplex verf.png|60px]]
| |
| | [[File:4-simplex t013.svg|50px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:4-simplex t01.svg|30px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]
| |
| | [[File:2-simplex t01.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{6}×{6}]]
| |
| | [[File:1-simplex t0.svg|30px]]×[[File:4-simplex t01.svg|30px]]<BR>[[Truncated tetrahedral prism|{ }×t<sub>0,1,3</sub>{3,3}]]
| |
| | [[File:4-simplex t013.svg|50px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |19
| |
| |(0,1,2,3,4,5)
| |
| |[[Omnitruncated 5-simplex]]<BR>great cellated dodecateron (gocad)<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| | 62
| |
| | 540
| |
| | 1560
| |
| | 1800
| |
| | 720
| |
| |[[File:Omnitruncated 5-simplex verf.png|60px]]<BR>[[5-cell|Irr. {3,3,3}]]
| |
| |(1)<BR>[[File:4-simplex t0123.svg|50px]]<BR>[[Omnitruncated 5-cell|t<sub>0,1,2,3</sub>{3,3,3}]]
| |
| |(1)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t012.svg|30px]]<BR>[[Truncated octahedral prism|{ }×tr{3,3}]]
| |
| |(1)<BR>[[File:2-simplex t01.svg|30px]]×[[File:2-simplex t01.svg|30px]]<BR>[[Duoprism|{6}×{6}]]
| |
| |(1)<BR>[[File:1-simplex t0.svg|30px]]×[[File:3-simplex t012.svg|30px]]<BR>[[Truncated octahedral prism|{ }×tr{3,3}]]
| |
| |(1)<BR>[[File:4-simplex t0123.svg|50px]]<BR>[[Omnitruncated 5-cell|t<sub>0,1,2,3</sub>{3,3,3}]]
| |
| |}
| |
| | |
| === The B<sub>5</sub> family ===
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|B<sub>5</sub> family]] has symmetry of order 3840 (5!×2<sup>5</sup>).
| |
| | |
| This family has 2<sup>5</sub>−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the [[Coxeter-Dynkin diagram]].
| |
| | |
| For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.
| |
| | |
| The 5-cube family of polytera 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 polyteron. All coordinates correspond with uniform polytera of edge length 2.
| |
| | |
| See symmetry graph: [[List of B5 polytopes]]
| |
| | |
| {|class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Base point
| |
| !rowspan=2|Name<BR>[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
| |
| !colspan=5|Element counts
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !colspan=5 |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]×[ ]<BR>(40)
| |
| ! {{CDD|node|4|node|2|node|3|node}}<BR>[4]×[3]<BR>(80)
| |
| ! {{CDD|node|2|node|3|node|3|node}}<BR>[ ]×[3,3]<BR>(80)
| |
| ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(32)
| |
| |- BGCOLOR="#f0e0e0"
| |
| |1||(0,0,0,0,1)√2||[[5-orthoplex]]<BR>(Quadrirectified 5-cube)<BR>{{CDD||node|4|node|3|node|3|node|3|node_1}}||32||80||80||40||10
| |
| ||[[File:pentacross verf.png|60px]]<BR>[[16-cell|{3,3,4}]]||[[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]]|| - || - || - || -
| |
| |- BGCOLOR="#f0e0e0"
| |
| |2||(0,0,0,1,1)√2||[[Rectified 5-orthoplex]]<BR>(Trirectified 5-cube)<BR>{{CDD||node|4|node|3|node|3|node_1|3|node}}||42||240||400||240||40
| |
| ||[[File:Rectified pentacross verf.png|60px]]<BR>[[Octahedral prism|{ }×{3,4}]]|| [[File:Schlegel wireframe 16-cell.png|60px]]<BR><BR>[[16-cell|{3,3,4}]] || - || - || - ||[[File:Schlegel half-solid rectified 5-cell.png|60px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |3||(0,0,0,1,2)√2||[[Truncated 5-orthoplex]]<BR>(Quadritruncated 5-cube)<BR>{{CDD||node|4|node|3|node|3|node_1|3|node_1}}||42||240||400||280||80
| |
| ||[[File:Truncated pentacross.png|60px]]<BR>(Octah.pyr)||[[File:Schlegel half-solid truncated pentachoron.png|60px]]<BR>[[Truncated 5-cell|t{3,3,3}]]||[[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]] || - || - || -
| |
| |- BGCOLOR="#e0f0e0"
| |
| |4||(0,0,1,1,1)√2||[[Birectified 5-cube]]<BR>(Birectified 5-orthoplex)<BR>{{CDD||node|4|node|3|node_1|3|node|3|node}}||42||280||640||480||80
| |
| ||[[File:Birectified penteract verf.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| [[File:Schlegel half-solid rectified 16-cell.png|60px]]<BR>[[Rectified 16-cell|r{3,3,4}]] || - || - || - || [[File:Schlegel half-solid rectified 5-cell.png|60px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |-BGCOLOR="#f0e0e0"
| |
| |5||(0,0,1,1,2)√2||[[Cantellated 5-orthoplex]]<BR>(Tricantellated 5-cube)<BR>{{CDD||node|4|node|3|node_1|3|node|3|node_1}}||82||640||1520||1200||240
| |
| ||[[File:Cantellated pentacross verf.png|60px]]<BR>Prism-wedge|| r{3,3,4}|| { }×{3,4} || - || - || [[File:Schlegel half-solid cantellated 5-cell.png|60px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |6||(0,0,1,2,2)√2||[[Bitruncated 5-orthoplex]]<BR>(tritruncated 5-cube)<BR>{{CDD||node|4|node|3|node_1|3|node_1|3|node}}||42||280||720||720||240
| |
| ||[[File:Bitruncated pentacross verf.png|60px]]|| t{3,3,4} || - || - || - || [[File:Schlegel half-solid bitruncated 5-cell.png|60px]]<BR>[[Bitruncated 5-cell|2t{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |7||(0,0,1,2,3)√2||[[Cantitruncated 5-orthoplex]]<BR>(tricantitruncated 5-orthoplex)<BR>{{CDD||node|4|node|3|node_1|3|node_1|3|node_1}}||82||640||1520||1440||480
| |
| ||[[File:Canitruncated 5-orthoplex verf.png|60px]]||rr{3,3,4} || { }×r{3,4} || [[File:6-4 duoprism.png|60px]]<BR>[[Duoprism|{6}×{4}]]|| - || [[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |8||(0,1,1,1,1)√2||[[Rectified 5-cube]]<BR>{{CDD||node|4|node_1|3|node|3|node|3|node}}||42||200||400||320||80
| |
| || [[File:Rectified 5-cube verf.png|60px]]<BR>[[Tetrahedral prism|{3,3}×{ }]]|| [[File:Schlegel half-solid rectified 8-cell.png|60px]]<BR>[[Rectified tesseract|r{4,3,3}]]|| - || - || - || [[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]]
| |
| |-BGCOLOR="#f0e0e0"
| |
| |9||(0,1,1,1,2)√2||[[Runcinated 5-orthoplex]]<BR>{{CDD||node|4|node_1|3|node|3|node|3|node_1}}||162||1200||2160||1440||320
| |
| ||[[File:Runcinated pentacross verf.png|60px]]|| r{4,3,3} || - || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{3}×{4}]]|| || [[File:Schlegel half-solid runcinated 5-cell.png|60px]]<BR>[[Runcinated 5-cell|t<sub><small>0,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |10||(0,1,1,2,2)√2||[[Bicantellated 5-cube]]<BR>(Bicantellated 5-orthoplex)<BR>{{CDD||node|4|node_1|3|node|3|node_1|3|node}}||122||840||2160||1920||480
| |
| || [[File:Bicantellated penteract verf.png|60px]]|| [[File:Schlegel half-solid cantellated 8-cell.png|60px]]<BR>[[Cantellated tesseract|rr{4,3,3}]]|| - || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| - || [[File:Schlegel half-solid cantellated 5-cell.png|60px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |11||(0,1,1,2,3)√2||[[Runcitruncated 5-orthoplex]]<BR>{{CDD||node|4|node_1|3|node|3|node_1|3|node_1}}||162||1440||3680||3360||960
| |
| ||[[File:Runcitruncated 5-orthoplex verf.png|60px]]|| rr{3,3,4} || { }×r{3,4} || [[File:6-4 duoprism.png|60px]]<BR>[[Duoprism|{6}×{4}]]|| - || [[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |12||(0,1,2,2,2)√2||[[Bitruncated 5-cube]]<BR>{{CDD||node|4|node_1|3|node_1|3|node|3|node}}||42||280||720||800||320
| |
| || [[File:Bitruncated penteract verf.png|60px]]|| [[File:Schlegel half-solid bitruncated 8-cell.png|60px]]<BR>[[Bitruncated tesseract|2t{4,3,3}]]|| - || - || - || [[File:Schlegel half-solid truncated pentachoron.png|60px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |13||(0,1,2,2,3)√2||[[Runcicantellated 5-orthoplex]]<BR>{{CDD||node|4|node_1|3|node_1|3|node|3|node_1}}||162||1200||2960||2880||960
| |
| ||[[File:Runcicantellated 5-orthoplex verf.png|60px]]|| { }×t{3,4}|| 2t{3,3,4} || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{3}×{4}]] || - || [[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |14||(0,1,2,3,3)√2||[[Bicantitruncated 5-cube]]<BR>(Bicantitruncated 5-orthoplex)<BR>{{CDD||node|4|node_1|3|node_1|3|node_1|3|node}}||122||840||2160||2400||960
| |
| || [[File:Bicantellated penteract verf.png|60px]]|| [[File:Schlegel half-solid cantellated 8-cell.png|60px]]<BR>[[Cantellated tesseract|rr{4,3,3}]]|| - || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| - || [[File:Schlegel half-solid cantellated 5-cell.png|60px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |15||(0,1,2,3,4)√2||[[Runcicantitruncated 5-orthoplex]]<BR>{{CDD||node|4|node_1|3|node_1|3|node_1|3|node_1}}||162||1440||4160||4800||1920
| |
| ||[[File:Runcicantitruncated 5-orthoplex verf.png|60px]]|| tr{3,3,4} || { }×t{3,4} || [[File:6-4 duoprism.png|60px]]<BR>[[Duoprism|{6}×{4}]]|| - || [[File:Schlegel half-solid omnitruncated 5-cell.png|60px]]<BR>[[Omnitruncated 5-cell|t<sub><small>0,1,2,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |16||(1,1,1,1,1)||[[5-cube]]<BR>{{CDD||node_1|4|node|3|node|3|node|3|node}}||10||40||80||80||32
| |
| ||[[File:5-cube verf.png|60px]]<BR>[[5-cell|{3,3,3}]]|| [[File:Schlegel wireframe 8-cell.png|60px]]<BR>[[Tesseract|{4,3,3}]]|| - || - || - || -
| |
| |- BGCOLOR="#e0f0e0"
| |
| |17||(1,1,1,1,1)<BR>+ (0,0,0,0,1)√2||[[Stericated 5-cube]]<BR>(Stericated 5-orthoplex)<BR>{{CDD||node_1|4|node|3|node|3|node|3|node_1}}||242||800||1040||640||160
| |
| || [[File:Stericated penteract verf.png|60px]]<BR>Tetr.antiprm|| [[File:Schlegel wireframe 8-cell.png|60px]]<BR>[[tesseract|{4,3,3}]]|| [[File:Schlegel wireframe 8-cell.png|60px]]<BR>[[Tesseract|{4,3}×{ }]]|| [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| [[File:Tetrahedral prism.png|60px]]<BR>[[Tetrahedral prism|{ }×{3,3}]]|| [[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |18||(1,1,1,1,1)<BR>+ (0,0,0,1,1)√2||[[Runcinated 5-cube]]<BR>{{CDD||node_1|4|node|3|node|3|node_1|3|node}}||202||1240||2160||1440||320
| |
| || [[File:Runcinated penteract verf.png|60px]]|| [[File:Schlegel half-solid runcinated 8-cell.png|60px]]<BR>[[Runcinated tesseract|t<sub><small>0,3</small></sub>{4,3,3}]]|| - || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| [[File:Octahedral prism.png|60px]]<BR>[[Rectified tetrahedral prism|{ }×r{3,3}]]|| [[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |19||(1,1,1,1,1)<BR>+ (0,0,0,1,2)√2||[[Steritruncated 5-orthoplex]]<BR>{{CDD||node_1|4|node|3|node|3|node_1|3|node_1}}||242||1520||2880||2240||640
| |
| ||[[File:Steritruncated 5-orthoplex verf.png|60px]]|| t<sub>0,3</sub>{3,3,4} || { }×{4,3} || - || - || [[File:Schlegel half-solid truncated pentachoron.png|60px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |20||(1,1,1,1,1)<BR>+ (0,0,1,1,1)√2||[[Cantellated 5-cube]]<BR>{{CDD||node_1|4|node|3|node_1|3|node|3|node}}||122||680||1520||1280||320
| |
| || [[File:Cantellated 5-cube vertf.png|60px]]<BR>Prism-wedge|| [[File:Schlegel half-solid cantellated 8-cell.png|60px]]<BR>[[Cantellated tesseract|rr{4,3,3}]]|| - || - || [[File:Tetrahedral prism.png|60px]]<BR>[[Tetrahedral prism|{ }×{3,3}]]|| [[File:Schlegel half-solid rectified 5-cell.png|60px]]<BR>[[Rectified 5-cell|r{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |21||(1,1,1,1,1)<BR>+ (0,0,1,1,2)√2||[[Stericantellated 5-cube]]<BR>(Stericantellated 5-orthoplex)<BR>{{CDD||node_1|4|node|3|node_1|3|node|3|node_1}}||242||2080||4720||3840||960
| |
| ||[[File:Stericantellated 5-orthoplex verf.png|60px]]|| [[File:Schlegel half-solid cantellated 8-cell.png|60px]]<BR>[[Cantellated tesseract|rr{4,3,3}]]|| [[File:Rhombicuboctahedral prism.png|60px]]<BR>[[Cantellated tesseract|rr{4,3}×{ }]]|| [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| [[File:Cuboctahedral prism.png|60px]]<BR>[[Cantellated tetrahedral prism|{ }×rr{3,3}]]|| [[File:Schlegel half-solid cantellated 5-cell.png|60px]]<BR>[[Cantellated 5-cell|rr{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |22||(1,1,1,1,1)<BR>+ (0,0,1,2,2)√2||[[Runcicantellated 5-cube]]<BR>{{CDD||node_1|4|node|3|node_1|3|node_1|3|node}}||202||1240||2960||2880||960
| |
| ||[[File:Runcicantellated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid runcitruncated 8-cell.png|60px]]<BR>[[Runcitruncated tesseract|t<sub><small>0,1,3</small></sub>{4,3,3}]]|| - || [[File:3-4 duoprism.png|60px]]<BR>[[Duoprism|{4}×{3}]]|| [[File:Truncated tetrahedral prism.png|60px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]|| [[File:Schlegel half-solid bitruncated 5-cell.png|60px]]<BR>[[Bitruncated 5-cell|2t{3,3,3}]]
| |
| |- BGCOLOR="#f0e0e0"
| |
| |23||(1,1,1,1,1)<BR>+ (0,0,1,2,3)√2||[[Stericantitruncated 5-orthoplex]]<BR>{{CDD||node_1|4|node|3|node_1|3|node_1|3|node_1}}||242||2320||5920||5760||1920
| |
| ||[[File:Stericanitruncated 5-orthoplex verf.png|60px]]|| [[File:Truncated tetrahedral prism.png|60px]]<BR>[[Rhombicuboctahedral prism|{ }×rr{3,4}]]|| [[File:Runcitruncated 16-cell.png|60px]]<BR>[[Runcitruncated 16-cell|t<sub>0,1,3</sub>{3,3,4}]]|| [[File:6-4 duoprism.png|60px]]<BR>[[Duoprism|{6}×{4}]]|| [[File:Truncated tetrahedral prism.png|60px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]|| [[File:Schlegel half-solid cantitruncated 5-cell.png|60px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |24||(1,1,1,1,1)<BR>+ (0,1,1,1,1)√2||[[Truncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node|3|node|3|node}}||42||200||400||400||160
| |
| || [[File:Truncated 5-cube verf.png|60px]]<BR>[[5-cell|Tetrah.pyr]]|| [[File:Schlegel half-solid truncated tesseract.png|60px]]<BR>[[Truncated tesseract|t{4,3,3}]]|| - || - || - || [[File:Schlegel wireframe 5-cell.png|60px]]<BR>[[5-cell|{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |25||(1,1,1,1,1)<BR>+ (0,1,1,1,2)√2||[[Steritruncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node|3|node|3|node_1}}||242||1600||2960||2240||640
| |
| ||[[File:Steritruncated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid truncated tesseract.png|60px]]<BR>[[Truncated tesseract|t{4,3,3}]]|| [[File:Truncated cubic prism.png|60px]]<BR>[[Truncated cubic prism|t{4,3}×{ }]]|| [[File:8-3 duoprism.png|60px]]<BR>[[Duoprism|{8}×{3}]]|| [[File:Tetrahedral prism.png|60px]]<BR>[[Tetrahedral prism|{ }×{3,3}]]|| [[File:Schlegel half-solid runcinated 5-cell.png|60px]]<BR>[[Runcinated 5-cell|t<sub><small>0,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |26||(1,1,1,1,1)<BR>+ (0,1,1,2,2)√2||[[Runcitruncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node|3|node_1|3|node}}||202||1560||3760||3360||960
| |
| ||[[File:Runcitruncated 5-cube verf.png|60px]]||[[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub>0,1,3</sub>{4,3,3}]] || { }×t{4,3} || [[File:6-8 duoprism.png|60px]]<BR>[[Duoprism|{6}×{8}]]|| { }×t{3,3} || t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |27||(1,1,1,1,1)<BR>+ (0,1,1,2,3)√2||[[Steriruncitruncated 5-cube]]<BR>(Steriruncitruncated 5-orthoplex)<BR>{{CDD||node_1|4|node_1|3|node|3|node_1|3|node_1}}||242||2160||5760||5760||1920
| |
| ||[[File:Steriruncitruncated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid runcitruncated 8-cell.png|60px]]<BR>[[Runcitruncated tesseract|t<sub><small>0,1,3</small></sub>{4,3,3}]]|| [[File:Truncated cubic prism.png|60px]]<BR>[[Truncated cubic prism|t{4,3}×{ }]]|| [[File:8-6 duoprism.png|60px]]<BR>[[Duoprism|{8}×{6}]]|| [[File:Truncated tetrahedral prism.png|60px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]|| [[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |28||(1,1,1,1,1)<BR>+ (0,1,2,2,2)√2||[[Cantitruncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node_1|3|node|3|node}}||122||680||1520||1600||640
| |
| ||[[File:Canitruncated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid cantitruncated 8-cell.png|60px]]<BR>[[Cantitruncated tesseract|tr{4,3,3}]]|| - || - || [[File:Tetrahedral prism.png|60px]]<BR>[[Tetrahedral prism|{ }×{3,3}]]|| [[File:Schlegel half-solid truncated pentachoron.png|60px]]<BR>[[Truncated 5-cell|t{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |29||(1,1,1,1,1)<BR>+ (0,1,2,2,3)√2||[[Stericantitruncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node_1|3|node|3|node_1}}||242||2400||6000||5760||1920
| |
| ||[[File:Stericanitruncated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid cantitruncated 8-cell.png|60px]]<BR> [[Cantitruncated tesseract|tr{4,3,3}]]|| [[File:Truncated cuboctahedral prism.png|60px]]<BR>[[truncated cuboctahedral prism|tr{4,3}×{ }]]|| [[File:8-3 duoprism.png|60px]]<BR>[[Duoprism|{8}×{3}]]|| [[File:Cuboctahedral prism.png|60px]]<BR>[[Cantellated tetrahedral prism|{ }×t<sub><small>0,2</small></sub>{3,3}]]|| [[File:Schlegel half-solid runcitruncated 5-cell.png|60px]]<BR>[[Runcitruncated 5-cell|t<sub><small>0,1,3</small></sub>{3,3,3}]]
| |
| |- BGCOLOR="#e0e0f0"
| |
| |30||(1,1,1,1,1)<BR>+ (0,1,2,3,3)√2||[[Runcicantitruncated 5-cube]]<BR>{{CDD||node_1|4|node_1|3|node_1|3|node_1|3|node}}||202||1560||4240||4800||1920
| |
| ||[[File:Runcicantitruncated 5-cube verf.png|60px]]|| [[File:Schlegel half-solid omnitruncated 8-cell.png|60px]]<BR>[[Omnitruncated tesseract|t<sub><small>0,1,2,3</small></sub>{4,3,3}]]|| - || [[File:8-3 duoprism.png|60px]]<BR>[[Duoprism|{8}×{3}]]|| [[File:Truncated tetrahedral prism.png|60px]]<BR>[[Truncated tetrahedral prism|{ }×t{3,3}]]|| [[File:Schlegel half-solid cantitruncated 5-cell.png|60px]]<BR>[[Cantitruncated 5-cell|tr{3,3,3}]]
| |
| |- BGCOLOR="#e0f0e0"
| |
| |31||(1,1,1,1,1)<BR>+ (0,1,2,3,4)√2||[[Omnitruncated 5-cube]]<BR>(omnitruncated 5-orthoplex)<BR>{{CDD||node_1|4|node_1|3|node_1|3|node_1|3|node_1}}||242||2640||8160||9600||3840
| |
| || [[File:Omnitruncated 5-cube verf.png|60px]]<BR>[[5-cell|Irr. {3,3,3}]]|| [[File:Schlegel half-solid omnitruncated 8-cell.png|60px]]<BR>[[truncated cuboctahedral prism|tr{4,3}×{ }]]|| [[File:Truncated cuboctahedral prism.png|60px]]<BR>[[truncated cuboctahedral prism|tr{4,3}×{ }]]|| [[File:8-6 duoprism.png|60px]]<BR>[[Duoprism|{8}×{6}]]|| [[File:Truncated octahedral prism.png|60px]]<BR>[[Omnitruncated tetrahedral prism|{ }×tr{3,3}]]|| [[File:Schlegel half-solid omnitruncated 5-cell.png|60px]]<BR>[[Omnitruncated 5-cell|t<sub><small>0,1,2,3</small></sub>{3,3,3}]]
| |
| |}
| |
| | |
| === The D<sub>5</sub> family ===
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|D<sub>5</sub> family]] has symmetry of order 1920 (5! x 2<sup>4</sup>).
| |
| | |
| This family has 23 Wythoffian uniform polyhedra, from ''3x8-1'' permutations of the D<sub>5</sub> [[Coxeter-Dynkin diagram]] with one or more rings. 15 (2x8-1) are repeated from the B<sub>5</sub> family and 8 are unique to this family.
| |
| | |
| See symmetry graphs: [[List of D5 polytopes]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]<BR>[[Schläfli symbol]] symbols<BR>Johnson and Bowers names
| |
| !colspan=5|Element counts
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !colspan=5 |Facets by location: [[File:CD B5 nodes.png]] [3<sup>1,2,1</sup>]
| |
| |-
| |
| !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<sup>1,1,1</sup>]<BR>(10)
| |
| ! {{CDD|nodes|split2|node|2|node}}<BR>[3,3]×[ ]<BR>(40)
| |
| ! {{CDD|node|2|node|3|node|2|node}}<BR>[ ]×[3]×[ ]<BR>(80)
| |
| ! {{CDD|node|3|node|3|node|3|node}}<BR>[3,3,3]<BR>(16)
| |
| |-
| |
| !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
| |
| | [[File:Demipenteract verf.png|50px]]<BR>[[rectified 5-cell|t<sub>1</sub>{3,3,3}]]
| |
| | {3,3,3}
| |
| | t<sub>0</sub>(1<sub>11</sub>)
| |
| | -
| |
| | -
| |
| | -
| |
| |-
| |
| !52
| |
| | {{CDD|nodes_10ru|split2|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node}}<BR>h<sub>2</sub>{4,3,3,3}, [[cantic 5-cube]]<BR>Truncated hemipenteract (thin)
| |
| | 42
| |
| | 280
| |
| | 640
| |
| | 560
| |
| | 160
| |
| |[[File:Truncated 5-demicube verf.png|60px]]
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !53
| |
| | {{CDD|nodes_10ru|split2|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node}}<BR>h<sub>3</sub>{4,3,3,3}, [[runcic 5-cube]]<BR>Small rhombated hemipenteract (sirhin)
| |
| | 42
| |
| | 360
| |
| | 880
| |
| | 720
| |
| | 160
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !54
| |
| | {{CDD|nodes_10ru|split2|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1}}<BR>h<sub>4</sub>{4,3,3,3}, [[steric 5-cube]]<BR>Small prismated hemipenteract (siphin)
| |
| | 82
| |
| | 480
| |
| | 720
| |
| | 400
| |
| | 80
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !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<sub>2,3</sub>{4,3,3,3}, [[runcicantic 5-cube]]<BR>Great rhombated hemipenteract (girhin)
| |
| | 42
| |
| | 360
| |
| | 1040
| |
| | 1200
| |
| | 480
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !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<sub>2,4</sub>{4,3,3,3}, [[stericantic 5-cube]]<BR>Prismatotruncated hemipenteract (pithin)
| |
| | 82
| |
| | 720
| |
| | 1840
| |
| | 1680
| |
| | 480
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !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<sub>3,4</sub>{4,3,3,3}, [[steriruncic 5-cube]]<BR>Prismatorhombated hemipenteract (pirhin)
| |
| | 82
| |
| | 560
| |
| | 1280
| |
| | 1120
| |
| | 320
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| !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<sub>2,3,4</sub>{4,3,3,3}, [[steriruncicantic 5-cube]]<BR>Great prismated hemipenteract (giphin)
| |
| | 82
| |
| | 720
| |
| | 2080
| |
| | 2400
| |
| | 960
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |}
| |
| | |
| === Uniform prismatic forms ===
| |
| | |
| There are 5 finite categorical [[Uniform polytope|uniform]] [[Prismatic polytope|prism]]atic families of polytopes based on the nonprismatic uniform [[4-polytope]]s:
| |
| | |
| ==== A<sub>4</sub> × A<sub>1</sub> ====
| |
| | |
| This prismatic family has [[Uniform polychoron#The A4 .5B3.2C3.2C3.5D family - .285-cell.29|9 forms]]:
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|A<sub>1</sub> x A<sub>4</sub> family]] has symmetry of order 240 (2*5!).
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR> and [[Schläfli symbol|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}}<BR>{3,3,3}×{ }<BR>[[5-cell prism]]
| |
| |7||20||30||25||10
| |
| |-
| |
| |60
| |
| |{{CDD|node|3|node_1|3|node|3|node|2|node_1}}<BR>r{3,3,3}×{ }<BR>[[Rectified 5-cell prism]]
| |
| |12||50||90||70||20
| |
| |-
| |
| |61
| |
| |{{CDD|node_1|3|node_1|3|node|3|node|2|node_1}}<BR>t{3,3,3}×{ }<BR>[[Truncated 5-cell prism]]
| |
| |12||50||100||100||40
| |
| |-
| |
| |62
| |
| |{{CDD|node_1|3|node|3|node_1|3|node|2|node_1}}<BR>rr{3,3,3}×{ }<BR>[[Cantellated 5-cell prism]]
| |
| |22||120||250||210||60
| |
| |- BGCOLOR="#e0f0e0"
| |
| |63
| |
| |{{CDD|node_1|3|node|3|node|3|node_1|2|node_1}}<BR>t<sub>0,3</sub>{3,3,3}×{ }<BR>[[Runcinated 5-cell prism]]
| |
| |32||130||200||140||40
| |
| |- BGCOLOR="#e0f0e0"
| |
| |64
| |
| |{{CDD|node|3|node_1|3|node_1|3|node|2|node_1}}<BR>2t{3,3,3}×{ }<BR>[[Bitruncated 5-cell prism]]
| |
| |12||60||140||150||60
| |
| |-
| |
| |65
| |
| |{{CDD|node_1|3|node_1|3|node_1|3|node|2|node_1}}<BR>tr{3,3,3}×{ }<BR>[[Cantitruncated 5-cell prism]]
| |
| |22||120||280||300||120
| |
| |-
| |
| |66
| |
| |{{CDD|node_1|3|node_1|3|node|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{3,3,3}×{ }<BR>[[Runcitruncated 5-cell prism]]
| |
| |32||180||390||360||120
| |
| |- BGCOLOR="#e0f0e0"
| |
| |67
| |
| |{{CDD|node_1|3|node_1|3|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,2,3</sub>{3,3,3}×{ }<BR>[[Omnitruncated 5-cell prism]]
| |
| |32||210||540||600||240
| |
| |}
| |
| | |
| ==== B<sub>4</sub> × A<sub>1</sub> ====
| |
| | |
| This prismatic family has [[Uniform polychoron#The B.2FC4 .5B4.2C3.2C3.5D family - .28tesseract.2F16-cell.29|16 forms]]. (Three are shared with [3,4,3]×[ ] family)
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|A<sub>1</sub> x B<sub>4</sub> family]] has symmetry of order 768 (2*2^4*4!).
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR> and [[Schläfli symbol|Schläfli]]<BR>symbols<BR>Name
| |
| !colspan=5 rowspan=1|Element counts
| |
| |-
| |
| ! Facets|| Cells|| Faces|| Edges|| Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''[16]'''||{{CDD|node_1|4|node|3|node|3|node|2|node_1}}<BR>{4,3,3}×{ }<BR>Tesseractic prism<BR>(Same as [[5-cube]])
| |
| |10||40||80||80||32
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''68'''||{{CDD|node|4|node_1|3|node|3|node|2|node_1}}<BR>r{4,3,3}×{ }<BR>[[Rectified tesseractic prism]]
| |
| |26||136||272||224||64
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''69'''||{{CDD|node_1|4|node_1|3|node|3|node|2|node_1}}<BR>t{4,3,3}×{ }<BR>[[Truncated tesseractic prism]]
| |
| |26||136||304||320||128
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''70'''||{{CDD|node_1|4|node|3|node_1|3|node|2|node_1}}<BR>rr{4,3,3}×{ }<BR>[[Cantellated tesseractic prism]]
| |
| |58||360||784||672||192
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''71'''||{{CDD|node_1|4|node|3|node|3|node_1|2|node_1}}<BR>t<sub>0,3</sub>{4,3,3}×{ }<BR>[[Runcinated tesseractic prism]]
| |
| |82||368||608||448||128
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''72'''||{{CDD|node|4|node_1|3|node_1|3|node|2|node_1}}<BR>2t{4,3,3}×{ }<BR>[[Bitruncated tesseractic prism]]
| |
| |26||168||432||480||192
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''73'''||{{CDD|node_1|4|node_1|3|node_1|3|node|2|node_1}}<BR>tr{4,3,3}×{ }<BR>[[Cantitruncated tesseractic prism]]
| |
| |58||360||880||960||384
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''74'''||{{CDD|node_1|4|node_1|3|node|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{4,3,3}×{ }<BR>[[Runcitruncated tesseractic prism]]
| |
| |82||528||1216||1152||384
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''75'''||{{CDD|node_1|4|node_1|3|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,2,3</sub>{4,3,3}×{ }<BR>[[Omnitruncated tesseractic prism]]
| |
| |82||624||1696||1920||768
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''76'''||{{CDD|node|4|node|3|node|3|node_1|2|node_1}}<BR>{3,3,4}×{ }<BR>[[16-cell prism]]
| |
| |18||64||88||56||16
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''77'''||{{CDD|node|4|node|3|node_1|3|node|2|node_1}}<BR>r{3,3,4}×{ }<BR>[[Rectified 16-cell prism]]<BR>(Same as '''24-cell prism''')
| |
| |26||144||288||216||48
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''78'''||{{CDD|node|4|node|3|node_1|3|node_1|2|node_1}}<BR>t{3,3,4}×{ }<BR>[[Truncated 16-cell prism]]
| |
| |26||144||312||288||96
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''79'''||{{CDD|node|4|node_1|3|node|3|node_1|2|node_1}}<BR>rr{3,3,4}×{ }<BR>[[Cantellated 16-cell prism]]<BR>(Same as '''rectified 24-cell prism''')
| |
| |50||336||768||672||192
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''80'''||{{CDD|node|4|node_1|3|node_1|3|node_1|2|node_1}}<BR>tr{3,3,4}×{ }<BR>[[Cantitruncated 16-cell prism]]<BR>(Same as '''truncated 24-cell prism''')
| |
| |50||336||864||960||384
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''81'''||{{CDD|node_1|4|node|3|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{3,3,4}×{ }<BR>[[Runcitruncated 16-cell prism]]
| |
| |82||528||1216||1152||384
| |
| |- BGCOLOR="#a0e0f0"
| |
| |'''82'''||{{CDD|node_h|3|node_h|3|node_h|4|node|2|node_1}}<BR>sr{3,3,4}×{ }<BR>[[snub 24-cell prism]]
| |
| |146||768||1392||960||192
| |
| |}
| |
| | |
| ==== F<sub>4</sub> × A<sub>1</sub> ====
| |
| | |
| This prismatic family has [[Uniform polychoron#The F4 .5B3.2C4.2C3.5D family - .2824-cell.29|10 forms]].
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|A<sub>1</sub> x F<sub>4</sub> 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<sup>+</sup>,4,3,2] symmetry, order 1152.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR> and [[Schläfli symbol|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}}<BR>{3,4,3}×{ }<BR>[[24-cell prism]]
| |
| |26||144||288||216||48
| |
| |-
| |
| |[79]||{{CDD|node|3|node_1|4|node|3|node|2|node_1}}<BR>r{3,4,3}×{ }<BR>[[rectified 24-cell prism]]
| |
| |50||336||768||672||192
| |
| |-
| |
| |[80]||{{CDD|node_1|3|node_1|4|node|3|node|2|node_1}}<BR>t{3,4,3}×{ }<BR>[[truncated 24-cell prism]]
| |
| |50||336||864||960||384
| |
| |-
| |
| |'''83'''||{{CDD|node_1|3|node|4|node_1|3|node|2|node_1}}<BR>rr{3,4,3}×{ }<BR>[[cantellated 24-cell prism]]
| |
| |146||1008||2304||2016||576
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''84'''||{{CDD|node_1|3|node|4|node|3|node_1|2|node_1}}<BR>t<sub>0,3</sub>{3,4,3}×{ }<BR>[[runcinated 24-cell prism]]
| |
| |242||1152||1920||1296||288
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''85'''||{{CDD|node|3|node_1|4|node_1|3|node|2|node_1}}<BR>2t{3,4,3}×{ }<BR> [[bitruncated 24-cell prism]]
| |
| |50||432||1248||1440||576
| |
| |-
| |
| |'''86'''||{{CDD|node_1|3|node_1|4|node_1|3|node|2|node_1}}<BR>tr{3,4,3}×{ }<BR>[[cantitruncated 24-cell prism]]
| |
| |146||1008||2592||2880||1152
| |
| |-
| |
| |'''87'''||{{CDD|node_1|3|node_1|4|node|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{3,4,3}×{ }<BR>[[runcitruncated 24-cell prism]]
| |
| |242||1584||3648||3456||1152
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''88'''||{{CDD|node_1|3|node_1|4|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,2,3</sub>{3,4,3}×{ }<BR> [[omnitruncated 24-cell prism]]
| |
| |242||1872||5088||5760||2304
| |
| |- BGCOLOR="#a0e0f0"
| |
| |[82]||{{CDD|node_h|3|node_h|4|node|3|node|2|node_1}}<BR>s{3,4,3}×{ }<BR>[[snub 24-cell prism]]
| |
| |146||768||1392||960||192
| |
| |}
| |
| | |
| ==== H<sub>4</sub> × A<sub>1</sub> ====
| |
| | |
| This prismatic family has [[Uniform polychoron#The H4 .5B5.2C3.2C3.5D family .E2.80.94 .28120-cell.2F600-cell.29|15 forms]]:
| |
| | |
| The [[Coxeter group#Finite Coxeter groups|A<sub>1</sub> x H<sub>4</sub> family]] has symmetry of order 28800 (2*14400).
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR> and [[Schläfli symbol|Schläfli]]<BR>symbols<BR>Name
| |
| !colspan=5 rowspan=1|Element counts
| |
| |-
| |
| ! Facets|| Cells|| Faces|| Edges|| Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''89'''||{{CDD|node_1|5|node|3|node|3|node|2|node_1}}<BR>{5,3,3}×{ }<BR>[[120-cell prism]]
| |
| |122||960||2640||3000||1200
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''90'''||{{CDD|node|5|node_1|3|node|3|node|2|node_1}}<BR>r{5,3,3}×{ }<BR>[[Rectified 120-cell prism]]
| |
| |722||4560||9840||8400||2400
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''91'''||{{CDD|node_1|5|node_1|3|node|3|node|2|node_1}}<BR>t{5,3,3}×{ }<BR>[[Truncated 120-cell prism]]
| |
| |722||4560||11040||12000||4800
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''92'''||{{CDD|node_1|5|node|3|node_1|3|node|2|node_1}}<BR>rr{5,3,3}×{ }<BR>[[Cantellated 120-cell prism]]
| |
| |1922||12960||29040||25200||7200
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''93'''||{{CDD|node_1|5|node|3|node|3|node_1|2|node_1}}<BR>t<sub>0,3</sub>{5,3,3}×{ }<BR>[[Runcinated 120-cell prism]]
| |
| |2642||12720||22080||16800||4800
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''94'''||{{CDD|node|5|node_1|3|node_1|3|node|2|node_1}}<BR>2t{5,3,3}×{ }<BR>[[Bitruncated 120-cell prism]]
| |
| |722||5760||15840||18000||7200
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''95'''||{{CDD|node_1|5|node_1|3|node_1|3|node|2|node_1}}<BR>tr{5,3,3}×{ }<BR>[[Cantitruncated 120-cell prism]]
| |
| |1922||12960||32640||36000||14400
| |
| |- BGCOLOR="#f0e0e0"
| |
| |'''96'''||{{CDD|node_1|5|node_1|3|node|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{5,3,3}×{ }<BR>[[Runcitruncated 120-cell prism]]
| |
| |2642||18720||44880||43200||14400
| |
| |- BGCOLOR="#e0f0e0"
| |
| |'''97'''||{{CDD|node_1|5|node_1|3|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,2,3</sub>{5,3,3}×{ }<BR>[[Omnitruncated 120-cell prism]]
| |
| |2642||22320||62880||72000||28800
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''98'''||{{CDD|node|5|node|3|node|3|node_1|2|node_1}}<BR>{3,3,5}×{ }<BR>[[600-cell prism]]
| |
| |602||2400||3120||1560||240
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''99'''||{{CDD|node|5|node|3|node_1|3|node|2|node_1}}<BR>r{3,3,5}×{ }<BR>[[Rectified 600-cell prism]]
| |
| |722||5040||10800||7920||1440
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''100'''||{{CDD|node|5|node|3|node_1|3|node_1|2|node_1}}<BR>t{3,3,5}×{ }<BR>[[Truncated 600-cell prism]]
| |
| |722||5040||11520||10080||2880
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''101'''||{{CDD|node|5|node_1|3|node|3|node_1|2|node_1}}<BR>rr{3,3,5}×{ }<BR>[[Cantellated 600-cell prism]]
| |
| |1442||11520||28080||25200||7200
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''102'''||{{CDD|node|5|node_1|3|node_1|3|node_1|2|node_1}}<BR>tr{3,3,5}×{ }<BR>[[Cantitruncated 600-cell prism]]
| |
| |1442||11520||31680||36000||14400
| |
| |- BGCOLOR="#e0e0f0"
| |
| |'''103'''||{{CDD|node_1|5|node|3|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,3</sub>{3,3,5}×{ }<BR>[[Runcitruncated 600-cell prism]]
| |
| |2642||18720||44880||43200||14400
| |
| |}
| |
| | |
| ==== Grand antiprism prism ====
| |
| | |
| The '''grand antiprism prism''' is the only known convex non-Wythoffian uniform polyteron. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600 [[tetrahedron|tetrahedra]], 40 [[pentagonal antiprism]]s, 700 [[triangular prism]]s, 20 [[pentagonal prism]]s), 322 hypercells (2 [[grand antiprism]]s [[Image:Grand antiprism.png|50px]], 20 [[pentagonal antiprism]] prisms [[Image:Pentagonal antiprismatic prism.png|50px]], and 300 [[tetrahedral prism]]s [[Image:Tetrahedral prism.png|50px]]).
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2| Name
| |
| !colspan=5|Element counts
| |
| |-
| |
| ! Facets|| Cells|| Faces|| Edges|| Vertices
| |
| |-
| |
| |'''104'''|| [[grand antiprism prism]]<BR>Gappip|| 322|| 1360|| 1940|| 1100|| 200
| |
| |}
| |
| | |
| == Notes on the Wythoff construction for the uniform 5-polytopes ==
| |
| | |
| Construction of the reflective 5-dimensional [[uniform polytope]]s are done through a [[Wythoff construction]] process, and represented through a [[Coxeter-Dynkin 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 polytope]]s 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 operation 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
| |
| |t<sub>0</sub>{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
| |
| ! [[Rectification (geometry)|Rectified]]
| |
| | t<sub>1</sub>{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
| |
| ! [[Rectification (geometry)|Birectified]]
| |
| | t<sub>2</sub>{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, [[Cell (geometry)|cells]] to their [[Dual polytope|duals]].
| |
| |- align=center
| |
| ! [[Rectification (geometry)|Trirectified]]
| |
| | t<sub>3</sub>{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
| |
| ! [[Rectification (geometry)|Quadrirectified]]
| |
| | t<sub>4</sub>{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
| |
| ![[Truncation (geometry)|Truncated]]
| |
| | t<sub>0,1</sub>{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>[[Image:Cube truncation sequence.svg|400px]]
| |
| |- align=center
| |
| ! [[Cantellation (geometry)|Cantellated]]
| |
| | t<sub>0,2</sub>{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. A uniform cantellation is half way between both the parent and dual forms.<BR>[[Image:Cube cantellation sequence.svg|400px]]
| |
| |- align=center
| |
| ! [[Runcination (geometry)|Runcinated]]
| |
| |colspan=2| t<sub>0,3</sub>{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
| |
| ! [[Sterication (geometry)|Stericated]]
| |
| |colspan=2| t<sub>0,4</sub>{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 (geometry)|expansion]] operation for polyterons.)
| |
| |- align=center
| |
| ![[Omnitruncation (geometry)|Omnitruncated]]
| |
| |colspan=2| t<sub>0,1,2,3,4</sub>{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 (geometry)|Alternation]], same as {{CDD|labelp|branch_10ru|split2|node|q|node|r|node}}
| |
| |- align=center
| |
| !Cantic
| |
| |colspan=2|h<sub>2</sub>{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|h<sub>3</sub>{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<sub>2,3</sub>{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|h<sub>4</sub>{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
| |
| !Runcisteric
| |
| |colspan=2|h<sub>3,4</sub>{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<sub>2,4</sub>{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<sub>2,3,4</sub>{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<sub>0,1,2,3</sub>{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<sub>0,1,2,3,4</sub>{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 ==
| |
| [[File:Coxeter diagram affine rank5 correspondence.png|436px|thumb|Coxeter-Dynkin 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>[[George Olshevsky]] (2006), ''Uniform Panoploid Tetracombs'', manuscript. Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs.</ref>
| |
| | |
| {| class=wikitable
| |
| |+ Fundamental groups
| |
| |-
| |
| !#
| |
| !colspan=3|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| !Forms
| |
| |- align=center
| |
| |1||<math>{\tilde{A}}_4</math>||[3<sup>[5]</sup>]||[(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<sup>1,1</sup>]||[4,3,3,4,1<sup>+</sup>]||{{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<sup>1,1,1,1</sup>]||[1<sup>+</sup>,4,3,3,4,1<sup>+</sup>]||{{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 [[List of regular polytopes#Higher dimensions 3|regular honeycomb]]s 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}} constructed by four [[snub 24-cell]], one [[16-cell]], and five [[5-cell]]s 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{4,3<sup>2</sup>,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<sup>1,1,1,1</sup>] {{CDD|nodes|split2|node|split1|nodes}} family, two new ones, including the [[quarter tesseractic honeycomb]], {{CDD|nodes_11|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|4|node_h1}}, and the [[bitruncated tesseractic honeycomb]], {{CDD|nodes_11|split2|node_1|split1|nodes}} = {{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-Dynkin 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<sup>1,1</sup>,2,∞]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|infin|node}}
| |
| |-
| |
| |3||<math>{\tilde{A}}_3</math>×<math>{\tilde{I}}_1</math>||[3<sup>[4]</sup>,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<sup>[3]</sup>,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<sup>[3]</sup>,2,3<sup>[3]</sup>]||{{CDD|node|split1|branch|2|node|split1|branch}}
| |
| |-
| |
| |9||<math>{\tilde{A}}_2</math>×<math>{\tilde{B}}_2</math>||[3<sup>[3]</sup>,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<sup>[3]</sup>,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 tessellations of hyperbolic 4-space===
| |
| | |
| There are five kinds of convex regular [[Honeycomb (geometry)|honeycombs]] and four kinds of star-honeycombs in H<sup>4</sup> space:<ref>Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213</ref>
| |
| {| class="wikitable"
| |
| |-
| |
| !Honeycomb name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>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 figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[dual polyhedron|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-5 5-cell honeycomb|Order-5 5-cell]]||{3,3,3,5}||{{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 honeycomb|Order-3 120-cell]]||{5,3,3,3}||{{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 honeycomb|Order-5 tesseractic]]||{4,3,3,5}||{{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 honeycomb|Order-4 120-cell]]||{5,3,3,4}||{{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 honeycomb|Order-5 120-cell]]||{5,3,3,5}||{{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 four regular star-honeycombs in H<sup>4</sup> space:
| |
| {| class="wikitable"
| |
| |-
| |
| !Honeycomb name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>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 figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[dual polyhedron|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-3 small stellated 120-cell honeycomb|Order-3 small stellated 120-cell]]||{5/2,5,3,3}||{{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 honeycomb|Order-5/2 600-cell]]||{3,3,5,5/2}||{{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 600-cell honeycomb|Order-5 icosahedral 120-cell]]||{3,5,5/2,5}||{{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 honeycomb|Order-3 great 120-cell]]||{5,5/2,5,3}||{{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}
| |
| |}
| |
| | |
| === Regular and uniform hyperbolic honeycombs ===
| |
| | |
| There are 5 [[Coxeter-Dynkin diagram#Compact|compact hyperbolic Coxeter groups]] of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams. There are also 9 [[Coxeter-Dynkin diagram#Rank 4 to 10|noncompact hyperbolic Coxeter groups of rank 5]], each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Noncompact groups generate honeycombs with infinite [[Facet (geometry)|facets]] or [[vertex figure]]s.
| |
| | |
| {| class="wikitable"
| |
| |+ Compact hyperbolic groups
| |
| | 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<sup>1,1</sup>]: {{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}}
| |
| |}
| |
| {| class=wikitable
| |
| |+ Noncompact hyperbolic groups
| |
| |align=right|
| |
| <math>{\bar{P}}_4</math> = [3,3<sup>[4]</sup>]: {{CDD|node|split1|nodes|split2|node|3|node}}
| |
| | |
| <math>{\bar{BP}}_4</math> = [4,3<sup>[4]</sup>]: {{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<sup>[3]×[ ]</sup>]: {{CDD|node|split1|branchbranch|split2|node}}
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| | |
| |align=right|
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| <math>{\bar{N}}_4</math> = [4,/3\,3,4]: {{CDD|nodes|split2-43|node|3|node|4|node}}<BR>
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| <math>{\bar{O}}_4</math> = [3,4,3<sup>1,1</sup>]: {{CDD|nodes|split2|node|4|node|3|node}}<BR>
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| <math>{\bar{S}}_4</math> = [4,3<sup>2,1</sup>]: {{CDD|nodes|split2-43|node|3|node|3|node}}<BR>
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| <math>{\bar{M}}_4</math> = [4,3<sup>1,1,1</sup>]: {{CDD|nodes|split2-43|node|split1|nodes}}
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| | |
| |align=right|
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| <math>{\bar{R}}_4</math> = [3,4,3,4]: {{CDD|node|4|node|3|node|4|node|3|node}}
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| | |
| |}
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| | |
| ==Notes==
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| {{reflist}}
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| | |
| == References ==
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| * [[Thorold Gosset|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)
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| * [[Alicia Boole Stott|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
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
| |
| ** [[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)
| |
| ** [[Coxeter|H.S.M. Coxeter]], ''The Beauty of Geometry: Twelve Essays'' (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
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| * '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
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| ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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| ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] (p. 287 5D Euclidean groups)
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| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
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| * {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}}
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| * 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) [http://books.google.com/books?id=ODfjmOeNLMUC&lpg=PP1&ots=AX5SYxPQ9S&dq=%22Reflection%20groups%20and%20Coxeter%20groups%22&pg=PA141]
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| == External links ==
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| * [http://www.steelpillow.com/polyhedra/ditela.html Polytope names], Guy Inchbald
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| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
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| * [http://members.aol.com/Polycell/glossary.html#Polytope Glossary for hyperspace: Polytope], [[George Olshevsky]]
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| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary], Garrett Jones
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| * [http://www.os2fan2.com/gloss/pglossp.html#PGPOLYTOPE polytope names], Wendy Krieger
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| {{Polytopes}}
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| {{Honeycombs}}
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| [[Category:5-polytopes]]
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| [[eo:5-hiperpluredro]]
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