Uniform 6-polytope
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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}.
History of discovery[edit]
- 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.[1]
- 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.[2][3]
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 prism
There 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 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 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 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 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 -
- {34} - 6-simplex -
- 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
- 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) -
- {4,33} — 6-cube (hexeract) -
- 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
- 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 .
- {3,32,1}, 121 6-demicube (demihexeract) -
- 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
- E6 family: [33,1,1] -
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]
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 system Bowers name and (acronym) | Base point | Element counts | |||||
|---|---|---|---|---|---|---|---|---|---|
| 5 | 4 | 3 | 2 | 1 | 0 | ||||
| 1 |  | 6-simplex heptapeton (hop) | (0,0,0,0,0,0,1) | 7 | 21 | 35 | 35 | 21 | 7 |
| 2 | | Rectified 6-simplex rectified heptapeton (ril) | (0,0,0,0,0,1,1) | 14 | 63 | 140 | 175 | 105 | 21 |
| 3 | | Truncated 6-simplex truncated heptapeton (til) | (0,0,0,0,0,1,2) | 14 | 63 | 140 | 175 | 126 | 42 |
| 4 | | Birectified 6-simplex birectified heptapeton (bril) | (0,0,0,0,1,1,1) | 14 | 84 | 245 | 350 | 210 | 35 |
| 5 | | Cantellated 6-simplex small rhombated heptapeton (sril) | (0,0,0,0,1,1,2) | 35 | 210 | 560 | 805 | 525 | 105 |
| 6 | | Bitruncated 6-simplex bitruncated heptapeton (batal) | (0,0,0,0,1,2,2) | 14 | 84 | 245 | 385 | 315 | 105 |
| 7 | | Cantitruncated 6-simplex great rhombated heptapeton (gril) | (0,0,0,0,1,2,3) | 35 | 210 | 560 | 805 | 630 | 210 |
| 8 | | Runcinated 6-simplex small prismated heptapeton (spil) | (0,0,0,1,1,1,2) | 70 | 455 | 1330 | 1610 | 840 | 140 |
| 9 | | Bicantellated 6-simplex small birhombated heptapeton (sabril) | (0,0,0,1,1,2,2) | 70 | 455 | 1295 | 1610 | 840 | 140 |
| 10 | | Runcitruncated 6-simplex prismatotruncated heptapeton (patal) | (0,0,0,1,1,2,3) | 70 | 560 | 1820 | 2800 | 1890 | 420 |
| 11 | | Tritruncated 6-simplex tetradecapeton (fe) | (0,0,0,1,2,2,2) | 14 | 84 | 280 | 490 | 420 | 140 |
| 12 | | Runcicantellated 6-simplex prismatorhombated heptapeton (pril) | (0,0,0,1,2,2,3) | 70 | 455 | 1295 | 1960 | 1470 | 420 |
| 13 | | Bicantitruncated 6-simplex great birhombated heptapeton (gabril) | (0,0,0,1,2,3,3) | 49 | 329 | 980 | 1540 | 1260 | 420 |
| 14 | | Runcicantitruncated 6-simplex great prismated heptapeton (gapil) | (0,0,0,1,2,3,4) | 70 | 560 | 1820 | 3010 | 2520 | 840 |
| 15 | | Stericated 6-simplex small cellated heptapeton (scal) | (0,0,1,1,1,1,2) | 105 | 700 | 1470 | 1400 | 630 | 105 |
| 16 | | Biruncinated 6-simplex small biprismato-tetradecapeton (sibpof) | (0,0,1,1,1,2,2) | 84 | 714 | 2100 | 2520 | 1260 | 210 |
| 17 | | Steritruncated 6-simplex cellitruncated heptapeton (catal) | (0,0,1,1,1,2,3) | 105 | 945 | 2940 | 3780 | 2100 | 420 |
| 18 | | Stericantellated 6-simplex cellirhombated heptapeton (cral) | (0,0,1,1,2,2,3) | 105 | 1050 | 3465 | 5040 | 3150 | 630 |
| 19 | | Biruncitruncated 6-simplex biprismatorhombated heptapeton (bapril) | (0,0,1,1,2,3,3) | 84 | 714 | 2310 | 3570 | 2520 | 630 |
| 20 | | Stericantitruncated 6-simplex celligreatorhombated heptapeton (cagral) | (0,0,1,1,2,3,4) | 105 | 1155 | 4410 | 7140 | 5040 | 1260 |
| 21 | | Steriruncinated 6-simplex celliprismated heptapeton (copal) | (0,0,1,2,2,2,3) | 105 | 700 | 1995 | 2660 | 1680 | 420 |
| 22 | | Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal) | (0,0,1,2,2,3,4) | 105 | 945 | 3360 | 5670 | 4410 | 1260 |
| 23 | | Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril) | (0,0,1,2,3,3,4) | 105 | 1050 | 3675 | 5880 | 4410 | 1260 |
| 24 | | Biruncicantitruncated 6-simplex great biprismato-tetradecapeton (gibpof) | (0,0,1,2,3,4,4) | 84 | 714 | 2520 | 4410 | 3780 | 1260 |
| 25 | | Steriruncicantitruncated 6-simplex great cellated heptapeton (gacal) | (0,0,1,2,3,4,5) | 105 | 1155 | 4620 | 8610 | 7560 | 2520 |
| 26 | | Pentellated 6-simplex small teri-tetradecapeton (staff) | (0,1,1,1,1,1,2) | 126 | 434 | 630 | 490 | 210 | 42 |
| 27 | | Pentitruncated 6-simplex teracellated heptapeton (tocal) | (0,1,1,1,1,2,3) | 126 | 826 | 1785 | 1820 | 945 | 210 |
| 28 | | Penticantellated 6-simplex teriprismated heptapeton (topal) | (0,1,1,1,2,2,3) | 126 | 1246 | 3570 | 4340 | 2310 | 420 |
| 29 | | Penticantitruncated 6-simplex terigreatorhombated heptapeton (togral) | (0,1,1,1,2,3,4) | 126 | 1351 | 4095 | 5390 | 3360 | 840 |
| 30 | | Pentiruncitruncated 6-simplex tericellirhombated heptapeton (tocral) | (0,1,1,2,2,3,4) | 126 | 1491 | 5565 | 8610 | 5670 | 1260 |
| 31 | | Pentiruncicantellated 6-simplex teriprismatorhombi-tetradecapeton (taporf) | (0,1,1,2,3,3,4) | 126 | 1596 | 5250 | 7560 | 5040 | 1260 |
| 32 | | Pentiruncicantitruncated 6-simplex terigreatoprismated heptapeton (tagopal) | (0,1,1,2,3,4,5) | 126 | 1701 | 6825 | 11550 | 8820 | 2520 |
| 33 | | Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf) | (0,1,2,2,2,3,4) | 126 | 1176 | 3780 | 5250 | 3360 | 840 |
| 34 | | Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral) | (0,1,2,2,3,4,5) | 126 | 1596 | 6510 | 11340 | 8820 | 2520 |
| 35 | | Omnitruncated 6-simplex great teri-tetradecapeton (gotaf) | (0,1,2,3,4,5,6) | 126 | 1806 | 8400 | 16800 | 15120 | 5040 |
The B6 family[edit]
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 | Schläfli symbol | Names | Element counts | |||||
|---|---|---|---|---|---|---|---|---|---|
| 5 | 4 | 3 | 2 | 1 | 0 | ||||
| 36 | | t0{3,3,3,3,4} | 6-orthoplex Hexacontatetrapeton (gee) | 64 | 192 | 240 | 160 | 60 | 12 |
| 37 | | t1{3,3,3,3,4} | Rectified 6-orthoplex Rectified hexacontatetrapeton (rag) | 76 | 576 | 1200 | 1120 | 480 | 60 |
| 38 | | t2{3,3,3,3,4} | Birectified 6-orthoplex Birectified hexacontatetrapeton (brag) | 76 | 636 | 2160 | 2880 | 1440 | 160 |
| 39 | | t2{4,3,3,3,3} | Birectified 6-cube Birectified hexeract (brox) | 76 | 636 | 2080 | 3200 | 1920 | 240 |
| 40 | | t1{4,3,3,3,3} | Rectified 6-cube Rectified hexeract (rax) | 76 | 444 | 1120 | 1520 | 960 | 192 |
| 41 | | t0{4,3,3,3,3} | 6-cube Hexeract (ax) | 12 | 60 | 160 | 240 | 192 | 64 |
| 42 | | t0,1{3,3,3,3,4} | Truncated 6-orthoplex Truncated hexacontatetrapeton (tag) | 76 | 576 | 1200 | 1120 | 540 | 120 |
| 43 | | t0,2{3,3,3,3,4} | Cantellated 6-orthoplex Small rhombated hexacontatetrapeton (srog) | 136 | 1656 | 5040 | 6400 | 3360 | 480 |
| 44 | | t1,2{3,3,3,3,4} | Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag) | 1920 | 480 | ||||
| 45 | | t0,3{3,3,3,3,4} | Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog) | 7200 | 960 | ||||
| 46 | | t1,3{3,3,3,3,4} | Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg) | 8640 | 1440 | ||||
| 47 | | t2,3{4,3,3,3,3} | Tritruncated 6-cube Hexeractihexacontitetrapeton (xog) | 3360 | 960 | ||||
| 48 | | t0,4{3,3,3,3,4} | Stericated 6-orthoplex Small cellated hexacontatetrapeton (scag) | 5760 | 960 | ||||
| 49 | | t1,4{4,3,3,3,3} | Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog) | 11520 | 1920 | ||||
| 50 | | t1,3{4,3,3,3,3} | Bicantellated 6-cube Small birhombated hexeract (saborx) | 9600 | 1920 | ||||
| 51 | | t1,2{4,3,3,3,3} | Bitruncated 6-cube Bitruncated hexeract (botox) | 2880 | 960 | ||||
| 52 | | t0,5{4,3,3,3,3} | Pentellated 6-cube Small teri-hexeractihexacontitetrapeton (stoxog) | 1920 | 384 | ||||
| 53 | | t0,4{4,3,3,3,3} | Stericated 6-cube Small cellated hexeract (scox) | 5760 | 960 | ||||
| 54 | | t0,3{4,3,3,3,3} | Runcinated 6-cube Small prismated hexeract (spox) | 7680 | 1280 | ||||
| 55 | | t0,2{4,3,3,3,3} | Cantellated 6-cube Small rhombated hexeract (srox) | 4800 | 960 | ||||
| 56 | | t0,1{4,3,3,3,3} | Truncated 6-cube Truncated hexeract (tox) | 76 | 444 | 1120 | 1520 | 1152 | 384 |
| 57 | | t0,1,2{3,3,3,3,4} | Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog) | 3840 | 960 | ||||
| 58 | | t0,1,3{3,3,3,3,4} | Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag) | 15840 | 2880 | ||||
| 59 | | t0,2,3{3,3,3,3,4} | Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog) | 11520 | 2880 | ||||
| 60 | | t1,2,3{3,3,3,3,4} | Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg) | 10080 | 2880 | ||||
| 61 | | t0,1,4{3,3,3,3,4} | Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog) | 19200 | 3840 | ||||
| 62 | | t0,2,4{3,3,3,3,4} | Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag) | 28800 | 5760 | ||||
| 63 | | t1,2,4{3,3,3,3,4} | Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax) | 23040 | 5760 | ||||
| 64 | | t0,3,4{3,3,3,3,4} | Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog) | 15360 | 3840 | ||||
| 65 | | t1,2,4{4,3,3,3,3} | Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag) | 23040 | 5760 | ||||
| 66 | | t1,2,3{4,3,3,3,3} | Bicantitruncated 6-cube Great birhombated hexeract (gaborx) | 11520 | 3840 | ||||
| 67 | | t0,1,5{3,3,3,3,4} | Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox) | 8640 | 1920 | ||||
| 68 | | t0,2,5{3,3,3,3,4} | Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox) | 21120 | 3840 | ||||
| 69 | | t0,3,4{4,3,3,3,3} | Steriruncinated 6-cube Celliprismated hexeract (copox) | 15360 | 3840 | ||||
| 70 | | t0,2,5{4,3,3,3,3} | Penticantellated 6-cube Terirhombated hexeract (topag) | 21120 | 3840 | ||||
| 71 | | t0,2,4{4,3,3,3,3} | Stericantellated 6-cube Cellirhombated hexeract (crax) | 28800 | 5760 | ||||
| 72 | | t0,2,3{4,3,3,3,3} | Runcicantellated 6-cube Prismatorhombated hexeract (prox) | 13440 | 3840 | ||||
| 73 | | t0,1,5{4,3,3,3,3} | Pentitruncated 6-cube Teritruncated hexeract (tacog) | 8640 | 1920 | ||||
| 74 | | t0,1,4{4,3,3,3,3} | Steritruncated 6-cube Cellitruncated hexeract (catax) | 19200 | 3840 | ||||
| 75 | | t0,1,3{4,3,3,3,3} | Runcitruncated 6-cube Prismatotruncated hexeract (potax) | 17280 | 3840 | ||||
| 76 | | t0,1,2{4,3,3,3,3} | Cantitruncated 6-cube Great rhombated hexeract (grox) | 5760 | 1920 | ||||
| 77 | | t0,1,2,3{3,3,3,3,4} | Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog) | 20160 | 5760 | ||||
| 78 | | t0,1,2,4{3,3,3,3,4} | Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg) | 46080 | 11520 | ||||
| 79 | | t0,1,3,4{3,3,3,3,4} | Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog) | 40320 | 11520 | ||||
| 80 | | t0,2,3,4{3,3,3,3,4} | Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag) | 40320 | 11520 | ||||
| 81 | | t1,2,3,4{4,3,3,3,3} | Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog) | 34560 | 11520 | ||||
| 82 | | t0,1,2,5{3,3,3,3,4} | Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig) | 30720 | 7680 | ||||
| 83 | | t0,1,3,5{3,3,3,3,4} | Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax) | 51840 | 11520 | ||||
| 84 | | t0,2,3,5{4,3,3,3,3} | Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog) | 46080 | 11520 | ||||
| 85 | | t0,2,3,4{4,3,3,3,3} | Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix) | 40320 | 11520 | ||||
| 86 | | t0,1,4,5{4,3,3,3,3} | Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog) | 30720 | 7680 | ||||
| 87 | | t0,1,3,5{4,3,3,3,3} | Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag) | 51840 | 11520 | ||||
| 88 | | t0,1,3,4{4,3,3,3,3} | Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix) | 40320 | 11520 | ||||
| 89 | | t0,1,2,5{4,3,3,3,3} | Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix) | 30720 | 7680 | ||||
| 90 | | t0,1,2,4{4,3,3,3,3} | Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx) | 46080 | 11520 | ||||
| 91 | | t0,1,2,3{4,3,3,3,3} | Runcicantitruncated 6-cube Great prismated hexeract (gippox) | 23040 | 7680 | ||||
| 92 | | t0,1,2,3,4{3,3,3,3,4} | Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog) | 69120 | 23040 | ||||
| 93 | | t0,1,2,3,5{3,3,3,3,4} | Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog) | 80640 | 23040 | ||||
| 94 | | t0,1,2,4,5{3,3,3,3,4} | Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg) | 80640 | 23040 | ||||
| 95 | | t0,1,2,4,5{4,3,3,3,3} | Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax) | 80640 | 23040 | ||||
| 96 | | t0,1,2,3,5{4,3,3,3,3} | Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox) | 80640 | 23040 | ||||
| 97 | | t0,1,2,3,4{4,3,3,3,3} | Steriruncicantitruncated 6-cube Great cellated hexeract (gocax) | 69120 | 23040 | ||||
| 98 | | t0,1,2,3,4,5{4,3,3,3,3} | Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog) | 138240 | 46080 | ||||
The D6 family[edit]
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-demicube Hemihexeract (hax) | (1,1,1,1,1,1) | 44 | 252 | 640 | 640 | 240 | 32 | 0.8660254 |
| 100 | = | Cantic 6-cube Truncated hemihexeract (thax) | (1,1,3,3,3,3) | 76 | 636 | 2080 | 3200 | 2160 | 480 | 2.1794493 |
| 101 | = | Runcic 6-cube Small rhombated hemihexeract (sirhax) | (1,1,1,3,3,3) | 3840 | 640 | 1.9364916 | ||||
| 102 | = | Steric 6-cube Small prismated hemihexeract (sophax) | (1,1,1,1,3,3) | 3360 | 480 | 1.6583123 | ||||
| 103 | = | Pentic 6-cube Small cellated demihexeract (sochax) | (1,1,1,1,1,3) | 1440 | 192 | 1.3228756 | ||||
| 104 | = | Runcicantic 6-cube Great rhombated hemihexeract (girhax) | (1,1,3,5,5,5) | 5760 | 1920 | 3.2787192 | ||||
| 105 | = | Stericantic 6-cube Prismatotruncated hemihexeract (pithax) | (1,1,3,3,5,5) | 12960 | 2880 | 2.95804 | ||||
| 106 | = | Steriruncic 6-cube Prismatorhombated hemihexeract (prohax) | (1,1,1,3,5,5) | 7680 | 1920 | 2.7838821 | ||||
| 107 | = | Penticantic 6-cube Cellitruncated hemihexeract (cathix) | (1,1,3,3,3,5) | 9600 | 1920 | 2.5980761 | ||||
| 108 | | |||||||||