uniform, polytope, uniform, dimensional, polytope, graphs, three, regular, related, uniform, polytopes, simplex, truncated, simplex, rectified, simplex, cantellated, simplex, runcinated, simplex, stericated, simplex, pentellated, simplex, orthoplex, truncated,. Uniform 6 dimensional polytope Graphs of three regular and related uniform polytopes 6 simplex Truncated 6 simplex Rectified 6 simplex Cantellated 6 simplex Runcinated 6 simplex Stericated 6 simplex Pentellated 6 simplex 6 orthoplex Truncated 6 orthoplex Rectified 6 orthoplex Cantellated 6 orthoplex Runcinated 6 orthoplex Stericated 6 orthoplex Cantellated 6 cube Runcinated 6 cube Stericated 6 cube Pentellated 6 cube 6 cube Truncated 6 cube Rectified 6 cube 6 demicube Truncated 6 demicube Cantellated 6 demicube Runcinated 6 demicube Stericated 6 demicube 221 122 Truncated 221 Truncated 122 In six dimensional geometry a uniform 6 polytope is a six dimensional uniform polytope A uniform polypeton is vertex transitive and all facets are uniform 5 polytopes The complete set of convex uniform 6 polytopes has not been determined but most 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 Dynkin diagrams Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6 polytope The simplest uniform polypeta are regular polytopes the 6 simplex 3 3 3 3 3 the 6 cube hexeract 4 3 3 3 3 and the 6 orthoplex hexacross 3 3 3 3 4 Contents 1 History of discovery 2 Uniform 6 polytopes by fundamental Coxeter groups 3 Uniform prismatic families 4 Enumerating the convex uniform 6 polytopes 4 1 The A6 family 4 2 The B6 family 4 3 The D6 family 4 4 The E6 family 4 5 Triaprisms 4 6 Non Wythoffian 6 polytopes 5 Regular and uniform honeycombs 5 1 Regular and uniform hyperbolic honeycombs 6 Notes on the Wythoff construction for the uniform 6 polytopes 7 See also 8 Notes 9 References 10 External links History of discovery edit Regular polytopes convex faces 1852 Ludwig Schlafli proved in his manuscript Theorie der vielfachen Kontinuitat 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 polytera in his publication On the Regular and Semi Regular Figures in Space of n Dimensions 91 1 93 Convex uniform polytopes 1940 The search was expanded systematically by H S M Coxeter in his publication Regular and Semi Regular Polytopes Nonregular uniform star polytopes similar to the nonconvex uniform polyhedra Ongoing Jonathan Bowers and other researchers search for other non convex uniform 6 polytopes with a current count of 41348 known uniform 6 polytopes outside infinite families convex and non convex excluding the prisms of the uniform 5 polytopes The list is not proven complete 91 2 93 91 3 93 Uniform 6 polytopes by fundamental Coxeter groups edit Uniform 6 polytopes with reflective symmetry can be generated by these four Coxeter groups represented by permutations of rings of the Coxeter Dynkin diagrams There are four fundamental reflective symmetry groups which generate 153 unique uniform 6 polytopes Coxeter group Coxeter Dynkin diagram 1 A6 3 3 3 3 3 2 B6 3 3 3 3 4 3 D6 3 3 3 31 1 4 E6 32 2 1 3 32 2 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 prismatic families edit Uniform prismThere are 6 categorical uniform prisms based on the uniform 5 polytopes Coxeter group Notes 1 A5A1 3 3 3 3 2 Prism family based on 5 simplex 2 B5A1 4 3 3 3 2 Prism family based on 5 cube 3a D5A1 32 1 1 2 Prism family based on 5 demicube Coxeter group Notes 4 A3I2 p A1 3 3 2 p 2 Prism family based on tetrahedral p gonal duoprisms 5 B3I2 p A1 4 3 2 p 2 Prism family based on cubic p gonal duoprisms 6 H3I2 p A1 5 3 2 p 2 Prism family based on dodecahedral p gonal duoprisms Uniform duoprismThere are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower dimensional uniform polytopes Five are formed as the product of a uniform 4 polytope with a regular polygon and six are formed by the product of two uniform polyhedra Coxeter group Notes 1 A4I2 p 3 3 3 2 p Family based on 5 cell p gonal duoprisms 2 B4I2 p 4 3 3 2 p Family based on tesseract p gonal duoprisms 3 F4I2 p 3 4 3 2 p Family based on 24 cell p gonal duoprisms 4 H4I2 p 5 3 3 2 p Family based on 120 cell p gonal duoprisms 5 D4I2 p 31 1 1 2 p Family based on demitesseract p gonal duoprisms Coxeter group Notes 6 A32 3 3 2 3 3 Family based on tetrahedral duoprisms 7 A3B3 3 3 2 4 3 Family based on tetrahedral cubic duoprisms 8 A3H3 3 3 2 5 3 Family based on tetrahedral dodecahedral duoprisms 9 B32 4 3 2 4 3 Family based on cubic duoprisms 10 B3H3 4 3 2 5 3 Family based on cubic dodecahedral duoprisms 11 H32 5 3 2 5 3 Family based on dodecahedral duoprisms Uniform triaprismThere is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons Each combination of at least one ring on every connected group produces a uniform prismatic 6 polytope Coxeter group Notes 1 I2 p I2 q I2 r p 2 q 2 r Family based on p q r gonal triprisms Enumerating the convex uniform 6 polytopes edit Simplex family A6 34 35 uniform 6 polytopes as permutations of rings in the group diagram including one regular 34 6 simplex Hypercube orthoplex family B6 4 34 63 uniform 6 polytopes as permutations of rings in the group diagram including two regular forms 4 33 6 cube hexeract 33 4 6 orthoplex hexacross Demihypercube D6 family 33 1 1 47 uniform 6 polytopes 16 unique as permutations of rings in the group diagram including 3 32 1 121 6 demicube demihexeract also as h 4 33 3 3 31 1 211 6 orthoplex a half symmetry form of E6 family 33 1 1 39 uniform 6 polytopes as permutations of rings in the group diagram including 3 3 32 1 221 3 32 2 122 These fundamental families generate 153 nonprismatic convex uniform polypeta In addition there are 57 uniform 6 polytope constructions based on prisms of the uniform 5 polytopes 3 3 3 3 2 4 3 3 3 2 32 1 1 2 excluding the penteract prism as a duplicate of the hexeract In addition there are infinitely many uniform 6 polytope based on Duoprism prism families 3 3 2 p 2 4 3 2 p 2 5 3 2 p 2 Duoprism families 3 3 3 2 p 4 3 3 2 p 5 3 3 2 p Triaprism family p 2 q 2 r The A6 family edit Further information list of A6 polytopes There are 32 4 1 35 forms derived by marking one or more nodes of the Coxeter Dynkin diagram All 35 are enumerated below They are named by Norman Johnson from the Wythoff construction operations upon regular 6 simplex heptapeton Bowers style acronym names are given in parentheses for cross referencing The A6 family has symmetry of order 5040 7 factorial The coordinates of uniform 6 polytopes with 6 simplex symmetry can be generated as permutations of simple integers in 7 space all in hyperplanes with normal vector 1 1 1 1 1 1 1 Coxeter Dynkin Johnson naming systemBowers name and acronym Base point Element counts 5 4 3 2 1 0 1 6 simplexheptapeton hop 0 0 0 0 0 0 1 7 21 35 35 21 7 2 Rectified 6 simplexrectified heptapeton ril 0 0 0 0 0 1 1 14 63 140 175 105 21 3 Truncated 6 simplextruncated heptapeton til 0 0 0 0 0 1 2 14 63 140 175 126 42 4 Birectified 6 simplexbirectified heptapeton bril 0 0 0 0 1 1 1 14 84 245 350 210 35 5 Cantellated 6 simplexsmall rhombated heptapeton sril 0 0 0 0 1 1 2 35 210 560 805 525 105 6 Bitruncated 6 simplexbitruncated heptapeton batal 0 0 0 0 1 2 2 14 84 245 385 315 105 7 Cantitruncated 6 simplexgreat rhombated heptapeton gril 0 0 0 0 1 2 3 35 210 560 805 630 210 8 Runcinated 6 simplexsmall prismated heptapeton spil 0 0 0 1 1 1 2 70 455 1330 1610 840 140 9 Bicantellated 6 simplexsmall birhombated heptapeton sabril 0 0 0 1 1 2 2 70 455 1295 1610 840 140 10 Runcitruncated 6 simplexprismatotruncated heptapeton patal 0 0 0 1 1 2 3 70 560 1820 2800 1890 420 11 Tritruncated 6 simplextetradecapeton fe 0 0 0 1 2 2 2 14 84 280 490 420 140 12 Runcicantellated 6 simplexprismatorhombated heptapeton pril 0 0 0 1 2 2 3 70 455 1295 1960 1470 420 13 Bicantitruncated 6 simplexgreat birhombated heptapeton gabril 0 0 0 1 2 3 3 49 329 980 1540 1260 420 14 Runcicantitruncated 6 simplexgreat prismated heptapeton gapil 0 0 0 1 2 3 4 70 560 1820 3010 2520 840 15 Stericated 6 simplexsmall cellated heptapeton scal 0 0 1 1 1 1 2 105 700 1470 1400 630 105 16 Biruncinated 6 simplexsmall biprismato tetradecapeton sibpof 0 0 1 1 1 2 2 84 714 2100 2520 1260 210 17 Steritruncated 6 simplexcellitruncated heptapeton catal 0 0 1 1 1 2 3 105 945 2940 3780 2100 420 18 Stericantellated 6 simplexcellirhombated heptapeton cral 0 0 1 1 2 2 3 105 1050 3465 5040 3150 630 19 Biruncitruncated 6 simplexbiprismatorhombated heptapeton bapril 0 0 1 1 2 3 3 84 714 2310 3570 2520 630 20 Stericantitruncated 6 simplexcelligreatorhombated heptapeton cagral 0 0 1 1 2 3 4 105 1155 4410 7140 5040 1260 21 Steriruncinated 6 simplexcelliprismated heptapeton copal 0 0 1 2 2 2 3 105 700 1995 2660 1680 420 22 Steriruncitruncated 6 simplexcelliprismatotruncated heptapeton captal 0 0 1 2 2 3 4 105 945 3360 5670 4410 1260 23 Steriruncicantellated 6 simplexcelliprismatorhombated heptapeton copril 0 0 1 2 3 3 4 105 1050 3675 5880 4410 1260 24 Biruncicantitruncated 6 simplexgreat biprismato tetradecapeton gibpof 0 0 1 2 3 4 4 84 714 2520 4410 3780 1260 25 Steriruncicantitruncated 6 simplexgreat cellated heptapeton gacal 0 0 1 2 3 4 5 105 1155 4620 8610 7560 2520 26 Pentellated 6 simplexsmall teri tetradecapeton staff 0 1 1 1 1 1 2 126 434 630 490 210 42 27 Pentitruncated 6 simplexteracellated heptapeton tocal 0 1 1 1 1 2 3 126 826 1785 1820 945 210 28 Penticantellated 6 simplexteriprismated heptapeton topal 0 1 1 1 2 2 3 126 1246 3570 4340 2310 420 29 Penticantitruncated 6 simplexterigreatorhombated heptapeton togral 0 1 1 1 2 3 4 126 1351 4095 5390 3360 840 30 Pentiruncitruncated 6 simplextericellirhombated heptapeton tocral 0 1 1 2 2 3 4 126 1491 5565 8610 5670 1260 31 Pentiruncicantellated 6 simplexteriprismatorhombi tetradecapeton taporf 0 1 1 2 3 3 4 126 1596 5250 7560 5040 1260 32 Pentiruncicantitruncated 6 simplexterigreatoprismated heptapeton tagopal 0 1 1 2 3 4 5 126 1701 6825 11550 8820 2520 33 Pentisteritruncated 6 simplextericellitrunki tetradecapeton tactaf 0 1 2 2 2 3 4 126 1176 3780 5250 3360 840 34 Pentistericantitruncated 6 simplextericelligreatorhombated heptapeton tacogral 0 1 2 2 3 4 5 126 1596 6510 11340 8820 2520 35 Omnitruncated 6 simplexgreat teri tetradecapeton gotaf 0 1 2 3 4 5 6 126 1806 8400 16800 15120 5040 The B6 family edit Further information list of B6 polytopes There are 63 forms based on all permutations of the Coxeter Dynkin diagrams with one or more rings The B6 family has symmetry of order 46080 6 factorial x 26 They are named by Norman Johnson from the Wythoff construction operations upon the regular 6 cube and 6 orthoplex Bowers names and acronym names are given for cross referencing Coxeter Dynkin diagram Schlafli symbol Names Element counts 5 4 3 2 1 0 36 t0 3 3 3 3 4 6 orthoplexHexacontatetrapeton gee 64 192 240 160 60 12 37 t1 3 3 3 3 4 Rectified 6 orthoplexRectified hexacontatetrapeton rag 76 576 1200 1120 480 60 38 t2 3 3 3 3 4 Birectified 6 orthoplexBirectified hexacontatetrapeton brag 76 636 2160 2880 1440 160 39 t2 4 3 3 3 3 Birectified 6 cubeBirectified hexeract brox 76 636 2080 3200 1920 240 40 t1 4 3 3 3 3 Rectified 6 cubeRectified hexeract rax 76 444 1120 1520 960 192 41 t0 4 3 3 3 3 6 cubeHexeract ax 12 60 160 240 192 64 42 t0 1 3 3 3 3 4 Truncated 6 orthoplexTruncated hexacontatetrapeton tag 76 576 1200 1120 540 120 43 t0 2 3 3 3 3 4 Cantellated 6 orthoplexSmall rhombated hexacontatetrapeton srog 136 1656 5040 6400 3360 480 44 t1 2 3 3 3 3 4 Bitruncated 6 orthoplexBitruncated hexacontatetrapeton botag 1920 480 45 t0 3 3 3 3 3 4 Runcinated 6 orthoplexSmall prismated hexacontatetrapeton spog 7200 960 46 t1 3 3 3 3 3 4 Bicantellated 6 orthoplexSmall birhombated hexacontatetrapeton siborg 8640 1440 47 t2 3 4 3 3 3 3 Tritruncated 6 cubeHexeractihexacontitetrapeton xog 3360 960 48 t0 4 3 3 3 3 4 Stericated 6 orthoplexSmall cellated hexacontatetrapeton scag 5760 960 49 t1 4 4 3 3 3 3 Biruncinated 6 cubeSmall biprismato hexeractihexacontitetrapeton sobpoxog 11520 1920 50 t1 3 4 3 3 3 3 Bicantellated 6 cubeSmall birhombated hexeract saborx 9600 1920 51 t1 2 4 3 3 3 3 Bitruncated 6 cubeBitruncated hexeract botox 2880 960 52 t0 5 4 3 3 3 3 Pentellated 6 cubeSmall teri hexeractihexacontitetrapeton stoxog 1920 384 53 t0 4 4 3 3 3 3 Stericated 6 cubeSmall cellated hexeract scox 5760 960 54 t0 3 4 3 3 3 3 Runcinated 6 cubeSmall prismated hexeract spox 7680 1280 55 t0 2 4 3 3 3 3 Cantellated 6 cubeSmall rhombated hexeract srox 4800 960 56 t0 1 4 3 3 3 3 Truncated 6 cubeTruncated hexeract tox 76 444 1120 1520 1152 384 57 t0 1 2 3 3 3 3 4 Cantitruncated 6 orthoplexGreat rhombated hexacontatetrapeton grog 3840 960 58 t0 1 3 3 3 3 3 4 Runcitruncated 6 orthoplexPrismatotruncated hexacontatetrapeton potag 15840 2880 59 t0 2 3 3 3 3 3 4 Runcicantellated 6 orthoplexPrismatorhombated hexacontatetrapeton prog 11520 2880 60 t1 2 3 3 3 3 3 4 Bicantitruncated 6 orthoplexGreat birhombated hexacontatetrapeton gaborg 10080 2880 61 t0 1 4 3 3 3 3 4 Steritruncated 6 orthoplexCellitruncated hexacontatetrapeton catog 19200 3840 62 t0 2 4 3 3 3 3 4 Stericantellated 6 orthoplexCellirhombated hexacontatetrapeton crag 28800 5760 63 t1 2 4 3 3 3 3 4 Biruncitruncated 6 orthoplexBiprismatotruncated hexacontatetrapeton boprax 23040 5760 64 t0 3 4 3 3 3 3 4 Steriruncinated 6 orthoplexCelliprismated hexacontatetrapeton copog 15360 3840 65 t1 2 4 4 3 3 3 3 Biruncitruncated 6 cubeBiprismatotruncated hexeract boprag 23040 5760 66 t1 2 3 4 3 3 3 3 Bicantitruncated 6 cubeGreat birhombated hexeract gaborx 11520 3840 67 t0 1 5 3 3 3 3 4 Pentitruncated 6 orthoplexTeritruncated hexacontatetrapeton tacox 8640 1920 68 t0 2 5 3 3 3 3 4 Penticantellated 6 orthoplexTerirhombated hexacontatetrapeton tapox 21120 3840 69 t0 3 4 4 3 3 3 3 Steriruncinated 6 cubeCelliprismated hexeract copox 15360 3840 70 t0 2 5 4 3 3 3 3 Penticantellated 6 cubeTerirhombated hexeract topag 21120 3840 71 t0 2 4 4 3 3 3 3 Stericantellated 6 cubeCellirhombated hexeract crax 28800 5760 72 t0 2 3 4 3 3 3 3 Runcicantellated 6 cubePrismatorhombated hexeract prox 13440 3840 73 t0 1 5 4 3 3 3 3 Pentitruncated 6 cubeTeritruncated hexeract tacog 8640 1920 74 t0 1 4 4 3 3 3 3 Steritruncated 6 cubeCellitruncated hexeract catax 19200 3840 75 t0 1 3 4 3 3 3 3 Runcitruncated 6 cubePrismatotruncated hexeract potax 17280 3840 76 t0 1 2 4 3 3 3 3 Cantitruncated 6 cubeGreat rhombated hexeract grox 5760 1920 77 t0 1 2 3 3 3 3 3 4 Runcicantitruncated 6 orthoplexGreat prismated hexacontatetrapeton gopog 20160 5760 78 t0 1 2 4 3 3 3 3 4 Stericantitruncated 6 orthoplexCelligreatorhombated hexacontatetrapeton cagorg 46080 11520 79 t0 1 3 4 3 3 3 3 4 Steriruncitruncated 6 orthoplexCelliprismatotruncated hexacontatetrapeton captog 40320 11520 80 t0 2 3 4 3 3 3 3 4 Steriruncicantellated 6 orthoplexCelliprismatorhombated hexacontatetrapeton coprag 40320 11520 81 t1 2 3 4 4 3 3 3 3 Biruncicantitruncated 6 cubeGreat biprismato hexeractihexacontitetrapeton gobpoxog 34560 11520 82 t0 1 2 5 3 3 3 3 4 Penticantitruncated 6 orthoplexTerigreatorhombated hexacontatetrapeton togrig 30720 7680 83 t0 1 3 5 3 3 3 3 4 Pentiruncitruncated 6 orthoplexTeriprismatotruncated hexacontatetrapeton tocrax 51840 11520 84 t0 2 3 5 4 3 3 3 3 Pentiruncicantellated 6 cubeTeriprismatorhombi hexeractihexacontitetrapeton tiprixog 46080 11520 85 t0 2 3 4 4 3 3 3 3 Steriruncicantellated 6 cubeCelliprismatorhombated hexeract coprix 40320 11520 86 t0 1 4 5 4 3 3 3 3 Pentisteritruncated 6 cubeTericelli hexeractihexacontitetrapeton tactaxog 30720 7680 87 t0 1 3 5 4 3 3 3 3 Pentiruncitruncated 6 cubeTeriprismatotruncated hexeract tocrag 51840 11520 88 t0 1 3 4 4 3 3 3 3 Steriruncitruncated 6 cubeCelliprismatotruncated hexeract captix 40320 11520 89 t0 1 2 5 4 3 3 3 3 Penticantitruncated 6 cubeTerigreatorhombated hexeract togrix 30720 7680 90 t0 1 2 4 4 3 3 3 3 Stericantitruncated 6 cubeCelligreatorhombated hexeract cagorx 46080 11520 91 t0 1 2 3 4 3 3 3 3 Runcicantitruncated 6 cubeGreat prismated hexeract gippox 23040 7680 92 t0 1 2 3 4 3 3 3 3 4 Steriruncicantitruncated 6 orthoplexGreat cellated hexacontatetrapeton gocog 69120 23040 93 t0 1 2 3 5 3 3 3 3 4 Pentiruncicantitruncated 6 orthoplexTerigreatoprismated hexacontatetrapeton tagpog 80640 23040 94 t0 1 2 4 5 3 3 3 3 4 Pentistericantitruncated 6 orthoplexTericelligreatorhombated hexacontatetrapeton tecagorg 80640 23040 95 t0 1 2 4 5 4 3 3 3 3 Pentistericantitruncated 6 cubeTericelligreatorhombated hexeract tocagrax 80640 23040 96 t0 1 2 3 5 4 3 3 3 3 Pentiruncicantitruncated 6 cubeTerigreatoprismated hexeract tagpox 80640 23040 97 t0 1 2 3 4 4 3 3 3 3 Steriruncicantitruncated 6 cubeGreat cellated hexeract gocax 69120 23040 98 t0 1 2 3 4 5 4 3 3 3 3 Omnitruncated 6 cubeGreat teri hexeractihexacontitetrapeton gotaxog 138240 46080 The D6 family edit Further information list of D6 polytopes The D6 family has symmetry of order 23040 6 factorial x 25 This family has 3 16 1 47 Wythoffian uniform polytopes generated by marking one or more nodes of the D6 Coxeter Dynkin diagram Of these 31 2 16 1 are repeated from the B6 family and 16 are unique to this family The 16 unique forms are enumerated below Bowers style acronym names are given for cross referencing Coxeter diagram Names Base point Alternately signed Element counts Circumrad 5 4 3 2 1 0 99 6 demicubeHemihexeract hax 1 1 1 1 1 1 44 252 640 640 240 32 0 8660254 100 Cantic 6 cubeTruncated hemihexeract thax 1 1 3 3 3 3 76 636 2080 3200 2160 480 2 1794493 101 Runcic 6 cubeSmall rhombated hemihexeract sirhax 1 1 1 3 3 3 3840 640 1 9364916 102 Steric 6 cubeSmall prismated hemihexeract sophax 1 1 1 1 3 3 3360 480 1 6583123 103 Pentic 6 cubeSmall cellated demihexeract sochax 1 1 1 1 1 3 1440 192 1 3228756 104 Runcicantic 6 cubeGreat rhombated hemihexeract girhax 1 1 3 5 5 5 5760 1920 3 2787192 105 Stericantic 6 cubePrismatotruncated hemihexeract pithax 1 1 3 3 5 5 12960 2880 2 95804 106 Steriruncic 6 cubePrismatorhombated hemihexeract prohax 1 1 1 3 5 5 7680 1920 2 7838821 107 Penticantic 6 cubeCellitruncated hemihexeract cathix 1 1 3 3 3 5 9600 1920 2 5980761 108 Pentiruncic 6 cubeCellirhombated hemihexeract crohax 1 1 1 3 3 5 10560 1920 2 3979158 109 Pentisteric 6 cubeCelliprismated hemihexeract cophix 1 1 1 1 3 5 5280 960 2 1794496 110 span, wikipedia, wiki, book, books, library,

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