+ Graphs of three regular and related uniform polytopes6-simplexTruncated 6-simplexRectified 6-simplexCantellated 6-simplexRuncinated 6-simplexStericated 6-simplexPentellated 6-simplex6-orthoplexTruncated 6-orthoplexRectified 6-orthoplexCantellated 6-orthoplexRuncinated 6-orthoplexStericated 6-orthoplexCantellated 6-cubeRuncinated 6-cubeStericated 6-cubePentellated 6-cube6-cubeTruncated 6-cubeRectified 6-cube6-demicubeTruncated 6-demicubeCantellated 6-demicubeRuncinated 6-demicubeStericated 6-demicube221122Truncated 221Truncated 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 polypeta: 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}.
History of discovery Regular polytopes: (convex faces) 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 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.Uniform Polypeta, Jonathan BowersUniform polytope
Uniform 6-polytopes by fundamental Coxeter groups 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 groupCoxeter-Dynkin diagram1A6 [3,3,3,3,3]2B6[3,3,3,3,4]3D6[3,3,3,31,1]4E6[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 Uniform prism
There are 6 categorical uniform prisms based on the uniform 5-polytopes.
#Coxeter groupNotes1A5A1 [3,3,3,3,2]Prism family based on 5-simplex2B5A1[4,3,3,3,2]Prism family based on 5-cube3aD5A1 [32,1,1,2]Prism family based on 5-demicube
#Coxeter groupNotes4A3I2(p)A1 [3,3,2,p,2]Prism family based on tetrahedral-p-gonal duoprisms5B3I2(p)A1 [4,3,2,p,2]Prism family based on cubic-p-gonal duoprisms6H3I2(p)A1 [5,3,2,p,2]Prism family based on dodecahedral-p-gonal duoprisms
Uniform duoprism
There 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 groupNotes1A4I2(p) [3,3,3,2,p]Family based on 5-cell-p-gonal duoprisms.2B4I2(p) [4,3,3,2,p]Family based on tesseract-p-gonal duoprisms.3F4I2(p) [3,4,3,2,p]Family based on 24-cell-p-gonal duoprisms.4H4I2(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 groupNotes6A32 [3,3,2,3,3]Family based on tetrahedral duoprisms.7A3B3 [3,3,2,4,3]Family based on tetrahedral-cubic duoprisms.8A3H3 [3,3,2,5,3]Family based on tetrahedral-dodecahedral duoprisms.9B32 [4,3,2,4,3]Family based on cubic duoprisms.10B3H3 [4,3,2,5,3]Family based on cubic-dodecahedral duoprisms.11H32 [5,3,2,5,3]Family based on dodecahedral duoprisms.
Uniform triaprism
There 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 groupNotes1I2(p)I2(q)I2(r) [p,2,q,2,r]Family based on p,q,r-gonal triprisms
Enumerating the convex uniform 6-polytopes Simplex family: A6 - 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
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-DynkinJohnson naming systemBowers name and (acronym)Base pointElement counts 5 4 3 2 1 01 6-simplexheptapeton (hop)(0,0,0,0,0,0,1)72135352172 Rectified 6-simplexrectified heptapeton (ril)(0,0,0,0,0,1,1) 14 63 140 175 105 213 Truncated 6-simplextruncated heptapeton (til)(0,0,0,0,0,1,2) 14 63 140 175 126 424 Birectified 6-simplexbirectified heptapeton (bril)(0,0,0,0,1,1,1) 14 84 245 350 210 355 Cantellated 6-simplexsmall rhombated heptapeton (sril)(0,0,0,0,1,1,2) 35 210 560 805 525 1056 Bitruncated 6-simplexbitruncated heptapeton (batal)(0,0,0,0,1,2,2) 14 84 245 385 315 1057 Cantitruncated 6-simplexgreat rhombated heptapeton (gril)(0,0,0,0,1,2,3) 35 210 560 805 630 2108 Runcinated 6-simplexsmall prismated heptapeton (spil)(0,0,0,1,1,1,2) 70 455 1330 1610 840 1409 Bicantellated 6-simplexsmall birhombated heptapeton (sabril)(0,0,0,1,1,2,2) 70 455 1295 1610 840 14010 Runcitruncated 6-simplexprismatotruncated heptapeton (patal)(0,0,0,1,1,2,3) 70 560 1820 2800 1890 42011 Tritruncated 6-simplextetradecapeton (fe)(0,0,0,1,2,2,2) 14 84 280 490 420 14012 Runcicantellated 6-simplexprismatorhombated heptapeton (pril)(0,0,0,1,2,2,3) 70 455 1295 1960 1470 42013 Bicantitruncated 6-simplexgreat birhombated heptapeton (gabril)(0,0,0,1,2,3,3) 49 329 980 1540 1260 42014 Runcicantitruncated 6-simplexgreat prismated heptapeton (gapil)(0,0,0,1,2,3,4) 70 560 1820 3010 2520 84015 Stericated 6-simplexsmall cellated heptapeton (scal)(0,0,1,1,1,1,2)1057001470140063010516 Biruncinated 6-simplexsmall biprismato-tetradecapeton (sibpof)(0,0,1,1,1,2,2) 84 714 2100 2520 1260 21017 Steritruncated 6-simplexcellitruncated heptapeton (catal)(0,0,1,1,1,2,3) 105 945 2940 3780 2100 42018 Stericantellated 6-simplexcellirhombated heptapeton (cral)(0,0,1,1,2,2,3) 105 1050 3465 5040 3150 63019 Biruncitruncated 6-simplexbiprismatorhombated heptapeton (bapril)(0,0,1,1,2,3,3) 84 714 2310 3570 2520 63020 Stericantitruncated 6-simplexcelligreatorhombated heptapeton (cagral)(0,0,1,1,2,3,4) 105 1155 4410 7140 5040 126021 Steriruncinated 6-simplexcelliprismated heptapeton (copal)(0,0,1,2,2,2,3) 105 700 1995 2660 1680 42022Steriruncitruncated 6-simplexcelliprismatotruncated heptapeton (captal) (0,0,1,2,2,3,4) 105 945 3360 5670 4410 126023 Steriruncicantellated 6-simplexcelliprismatorhombated heptapeton (copril)(0,0,1,2,3,3,4) 105 1050 3675 5880 4410 126024 Biruncicantitruncated 6-simplexgreat biprismato-tetradecapeton (gibpof)(0,0,1,2,3,4,4) 84 714 2520 4410 3780 126025 Steriruncicantitruncated 6-simplexgreat cellated heptapeton (gacal)(0,0,1,2,3,4,5) 105 1155 4620 8610 7560 252026 Pentellated 6-simplexsmall teri-tetradecapeton (staff)(0,1,1,1,1,1,2) 126 434 630 490 210 4227 Pentitruncated 6-simplexteracellated heptapeton (tocal)(0,1,1,1,1,2,3) 126 826 1785 1820 945 21028 Penticantellated 6-simplexteriprismated heptapeton (topal)(0,1,1,1,2,2,3) 126 1246 3570 4340 2310 42029 Penticantitruncated 6-simplexterigreatorhombated heptapeton (togral)(0,1,1,1,2,3,4) 126 1351 4095 5390 3360 84030 Pentiruncitruncated 6-simplextericellirhombated heptapeton (tocral)(0,1,1,2,2,3,4) 126 1491 5565 8610 5670 126031 Pentiruncicantellated 6-simplexteriprismatorhombi-tetradecapeton (taporf)(0,1,1,2,3,3,4) 126 1596 5250 7560 5040 126032 Pentiruncicantitruncated 6-simplexterigreatoprismated heptapeton (tagopal)(0,1,1,2,3,4,5) 126 1701 6825 11550 8820 252033 Pentisteritruncated 6-simplextericellitrunki-tetradecapeton (tactaf)(0,1,2,2,2,3,4) 126 1176 3780 5250 3360 84034 Pentistericantitruncated 6-simplextericelligreatorhombated heptapeton (tacogral)(0,1,2,2,3,4,5) 126 1596 6510 11340 8820 252035 Omnitruncated 6-simplexgreat teri-tetradecapeton (gotaf)(0,1,2,3,4,5,6) 126 1806 8400 16800 15120 5040
The B6 family
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.