# Uniform 6-polytope

Graphs of three regular and related uniform polytopes :
| 6-simplex | 6-simplex | 6-simplex | 6-simplex | Truncated 6-simplex | Truncated 6-simplex | Truncated 6-simplex | Truncated 6-simplex | Rectified 6-simplex | Rectified 6-simplex | Rectified 6-simplex | Rectified 6-simplex |
| Cantellated 6-simplex | Cantellated 6-simplex | Cantellated 6-simplex | Cantellated 6-simplex | Cantellated 6-simplex | Cantellated 6-simplex | Runcinated 6-simplex | Runcinated 6-simplex | Runcinated 6-simplex | Runcinated 6-simplex | Runcinated 6-simplex | Runcinated 6-simplex |
| Stericated 6-simplex | Stericated 6-simplex | Stericated 6-simplex | Stericated 6-simplex | Stericated 6-simplex | Stericated 6-simplex | Pentellated 6-simplex | Pentellated 6-simplex | Pentellated 6-simplex | Pentellated 6-simplex | Pentellated 6-simplex | Pentellated 6-simplex |
| 6-orthoplex | 6-orthoplex | 6-orthoplex | 6-orthoplex | Truncated 6-orthoplex | Truncated 6-orthoplex | Truncated 6-orthoplex | Truncated 6-orthoplex | Rectified 6-orthoplex | Rectified 6-orthoplex | Rectified 6-orthoplex | Rectified 6-orthoplex |
| Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Runcinated 6-orthoplex | Runcinated 6-orthoplex | Runcinated 6-orthoplex | Runcinated 6-orthoplex | Stericated 6-orthoplex | Stericated 6-orthoplex | Stericated 6-orthoplex | Stericated 6-orthoplex |
| Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Runcinated 6-cube | Runcinated 6-cube | Runcinated 6-cube | Runcinated 6-cube | Runcinated 6-cube | Runcinated 6-cube |
| Stericated 6-cube | Stericated 6-cube | Stericated 6-cube | Stericated 6-cube | Stericated 6-cube | Stericated 6-cube | Pentellated 6-cube | Pentellated 6-cube | Pentellated 6-cube | Pentellated 6-cube | Pentellated 6-cube | Pentellated 6-cube |
| 6-cube | 6-cube | 6-cube | 6-cube | Truncated 6-cube | Truncated 6-cube | Truncated 6-cube | Truncated 6-cube | Rectified 6-cube | Rectified 6-cube | Rectified 6-cube | Rectified 6-cube |
| 6-demicube | 6-demicube | 6-demicube | 6-demicube | Truncated 6-demicube | Truncated 6-demicube | Truncated 6-demicube | Truncated 6-demicube | Cantellated 6-demicube | Cantellated 6-demicube | Cantellated 6-demicube | Cantellated 6-demicube |
| Runcinated 6-demicube | Runcinated 6-demicube | Runcinated 6-demicube | Runcinated 6-demicube | Runcinated 6-demicube | Runcinated 6-demicube | Stericated 6-demicube | Stericated 6-demicube | Stericated 6-demicube | Stericated 6-demicube | Stericated 6-demicube | Stericated 6-demicube |
| 2$_{21}$ | 2$_{21}$ | 2$_{21}$ | 2$_{21}$ | 2$_{21}$ | 2$_{21}$ | 1$_{22}$ | 1$_{22}$ | 1$_{22}$ | 1$_{22}$ | 1$_{22}$ | 1$_{22}$ |
| Truncated 2$_{21}$ | Truncated 2$_{21}$ | Truncated 2$_{21}$ | Truncated 2$_{21}$ | Truncated 2$_{21}$ | Truncated 2$_{21}$ | Truncated 1$_{22}$ | Truncated 1$_{22}$ | Truncated 1$_{22}$ | Truncated 1$_{22}$ | Truncated 1$_{22}$ | Truncated 1$_{22}$ |
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

* 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.
* 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 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 group | Coxeter group |
| 1 | A$_6$ | [3,3,3,3,3] |
| 2 | B$_6$ | [3,3,3,3,4] |
| 3 | D$_6$ | [3,3,3,3$^{1,1}$] |
| 4 | E$_6$ | [3$^{2,2,1}$] |
| 4 | E$_6$ | [3,3$^{2,2}$] |
* 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 group | Coxeter group | Notes |
| 1 | A$_5$A$_1$ | [3,3,3,3,2] | Prism family based on 5-simplex |
| 2 | B$_5$A$_1$ | [4,3,3,3,2] | Prism family based on 5-cube |
| 3a | D$_5$A$_1$ | [3$^{2,1,1}$,2] | Prism family based on 5-demicube |
| # | Coxeter group | Coxeter group | Notes |
| 4 | A$_3$I$_2$(p)A$_1$ | [3,3,2,p,2] | Prism family based on tetrahedral-p-gonal duoprisms |
| 5 | B$_3$I$_2$(p)A$_1$ | [4,3,2,p,2] | Prism family based on cubic-p-gonal duoprisms |
| 6 | H$_3$I$_2$(p)A$_1$ | [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 | Coxeter group | Notes |
| 1 | A$_4$I$_2$(p) | [3,3,3,2,p] | Family based on 5-cell-p-gonal duoprisms. |
| 2 | B$_4$I$_2$(p) | [4,3,3,2,p] | Family based on tesseract-p-gonal duoprisms. |
| 3 | F$_4$I$_2$(p) | [3,4,3,2,p] | Family based on 24-cell-p-gonal duoprisms. |
| 4 | H$_4$I$_2$(p) | [5,3,3,2,p] | Family based on 120-cell-p-gonal duoprisms. |
| 5 | D$_4$I$_2$(p) | [3$^{1,1,1}$,2,p] | Family based on demitesseract-p-gonal duoprisms. |
| # | Coxeter group | Coxeter group | Notes |
| 6 | A$_3$$^2$ | [3,3,2,3,3] | Family based on tetrahedral duoprisms. |
| 7 | A$_3$B$_3$ | [3,3,2,4,3] | Family based on tetrahedral-cubic duoprisms. |
| 8 | A$_3$H$_3$ | [3,3,2,5,3] | Family based on tetrahedral-dodecahedral duoprisms. |
| 9 | B$_3$$^2$ | [4,3,2,4,3] | Family based on cubic duoprisms. |
| 10 | B$_3$H$_3$ | [4,3,2,5,3] | Family based on cubic-dodecahedral duoprisms. |
| 11 | H$_3$$^2$ | [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 | Coxeter group | Coxeter group | Notes |
| 1 | I$_2$(p)I$_2$(q)I$_2$(r) | [p,2,q,2,r] | | Family based on p,q,r-gonal triprisms |

## Enumerating the convex uniform 6-polytopes

* Simplex family: A$_6$ [3$^4$] -
** 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
*** {3$^4$} - 6-simplex -
* Hypercube/orthoplex family: B$_6$ [4,3$^4$] -
** 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
*** {4,3$^3$} — 6-cube (hexeract) -
*** {3$^3$,4} — 6-orthoplex, (hexacross) -
* Demihypercube D$_6$ family: [3$^{3,1,1}$] -
** 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
*** {3,3$^{2,1}$}, 1$_{21}$ 6-demicube (demihexeract) - ; also as h{4,3$^3$},
*** {3,3,3$^{1,1}$}, 2$_{11}$ 6-orthoplex - , a half symmetry form of .
* E$_6$ family: [3$^{3,1,1}$] -
** 39 uniform 6-polytopes as permutations of rings in the group diagram, including:
*** {3,3,3$^{2,1}$}, 2$_{21}$ -
*** {3,3$^{2,2}$}, 1$_{22}$ -
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], [3$^{2,1,1}$,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 A$_6$ 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 A$_6$ 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 | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin | Johnson naming system Bowers name and (acronym) | Base point | 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 B$_6$ family

There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
The B$_6$ family has symmetry of order 46080 (6 factorial x 2$^6$).
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 | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram | Schläfli symbol | Names | 5 | 4 | 3 | 2 | 1 | 0 |
| 36 | | t$_0${3,3,3,3,4} | 6-orthoplex Hexacontatetrapeton (gee) | 64 | 192 | 240 | 160 | 60 | 12 |
| 37 | | t$_1${3,3,3,3,4} | Rectified 6-orthoplex Rectified hexacontatetrapeton (rag) | 76 | 576 | 1200 | 1120 | 480 | 60 |
| 38 | | t$_2${3,3,3,3,4} | Birectified 6-orthoplex Birectified hexacontatetrapeton (brag) | 76 | 636 | 2160 | 2880 | 1440 | 160 |
| 39 | | t$_2${4,3,3,3,3} | Birectified 6-cube Birectified hexeract (brox) | 76 | 636 | 2080 | 3200 | 1920 | 240 |
| 40 | | t$_1${4,3,3,3,3} | Rectified 6-cube Rectified hexeract (rax) | 76 | 444 | 1120 | 1520 | 960 | 192 |
| 41 | | t$_0${4,3,3,3,3} | 6-cube Hexeract (ax) | 12 | 60 | 160 | 240 | 192 | 64 |
| 42 | | t$_{0,1}${3,3,3,3,4} | Truncated 6-orthoplex Truncated hexacontatetrapeton (tag) | 76 | 576 | 1200 | 1120 | 540 | 120 |
| 43 | | t$_{0,2}${3,3,3,3,4} | Cantellated 6-orthoplex Small rhombated hexacontatetrapeton (srog) | 136 | 1656 | 5040 | 6400 | 3360 | 480 |
| 44 | | t$_{1,2}${3,3,3,3,4} | Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag) | | | | | 1920 | 480 |
| 45 | | t$_{0,3}${3,3,3,3,4} | Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog) | | | | | 7200 | 960 |
| 46 | | t$_{1,3}${3,3,3,3,4} | Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg) | | | | | 8640 | 1440 |
| 47 | | t$_{2,3}${4,3,3,3,3} | Tritruncated 6-cube Hexeractihexacontitetrapeton (xog) | | | | | 3360 | 960 |
| 48 | | t$_{0,4}${3,3,3,3,4} | Stericated 6-orthoplex Small cellated hexacontatetrapeton (scag) | | | | | 5760 | 960 |
| 49 | | t$_{1,4}${4,3,3,3,3} | Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog) | | | | | 11520 | 1920 |
| 50 | | t$_{1,3}${4,3,3,3,3} | Bicantellated 6-cube Small birhombated hexeract (saborx) | | | | | 9600 | 1920 |
| 51 | | t$_{1,2}${4,3,3,3,3} | Bitruncated 6-cube Bitruncated hexeract (botox) | | | | | 2880 | 960 |
| 52 | | t$_{0,5}${4,3,3,3,3} | Pentellated 6-cube Small teri-hexeractihexacontitetrapeton (stoxog) | | | | | 1920 | 384 |
| 53 | | t$_{0,4}${4,3,3,3,3} | Stericated 6-cube Small cellated hexeract (scox) | | | | | 5760 | 960 |
| 54 | | t$_{0,3}${4,3,3,3,3} | Runcinated 6-cube Small prismated hexeract (spox) | | | | | 7680 | 1280 |
| 55 | | t$_{0,2}${4,3,3,3,3} | Cantellated 6-cube Small rhombated hexeract (srox) | | | | | 4800 | 960 |
| 56 | | t$_{0,1}${4,3,3,3,3} | Truncated 6-cube Truncated hexeract (tox) | 76 | 444 | 1120 | 1520 | 1152 | 384 |
| 57 | | t$_{0,1,2}${3,3,3,3,4} | Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog) | | | | | 3840 | 960 |
| 58 | | t$_{0,1,3}${3,3,3,3,4} | Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag) | | | | | 15840 | 2880 |
| 59 | | t$_{0,2,3}${3,3,3,3,4} | Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog) | | | | | 11520 | 2880 |
| 60 | | t$_{1,2,3}${3,3,3,3,4} | Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg) | | | | | 10080 | 2880 |
| 61 | | t$_{0,1,4}${3,3,3,3,4} | Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog) | | | | | 19200 | 3840 |
| 62 | | t$_{0,2,4}${3,3,3,3,4} | Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag) | | | | | 28800 | 5760 |
| 63 | | t$_{1,2,4}${3,3,3,3,4} | Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax) | | | | | 23040 | 5760 |
| 64 | | t$_{0,3,4}${3,3,3,3,4} | Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog) | | | | | 15360 | 3840 |
| 65 | | t$_{1,2,4}${4,3,3,3,3} | Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag) | | | | | 23040 | 5760 |
| 66 | | t$_{1,2,3}${4,3,3,3,3} | Bicantitruncated 6-cube Great birhombated hexeract (gaborx) | | | | | 11520 | 3840 |
| 67 | | t$_{0,1,5}${3,3,3,3,4} | Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox) | | | | | 8640 | 1920 |
| 68 | | t$_{0,2,5}${3,3,3,3,4} | Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox) | | | | | 21120 | 3840 |
| 69 | | t$_{0,3,4}${4,3,3,3,3} | Steriruncinated 6-cube Celliprismated hexeract (copox) | | | | | 15360 | 3840 |
| 70 | | t$_{0,2,5}${4,3,3,3,3} | Penticantellated 6-cube Terirhombated hexeract (topag) | | | | | 21120 | 3840 |
| 71 | | t$_{0,2,4}${4,3,3,3,3} | Stericantellated 6-cube Cellirhombated hexeract (crax) | | | | | 28800 | 5760 |
| 72 | | t$_{0,2,3}${4,3,3,3,3} | Runcicantellated 6-cube Prismatorhombated hexeract (prox) | | | | | 13440 | 3840 |
| 73 | | t$_{0,1,5}${4,3,3,3,3} | Pentitruncated 6-cube Teritruncated hexeract (tacog) | | | | | 8640 | 1920 |
| 74 | | t$_{0,1,4}${4,3,3,3,3} | Steritruncated 6-cube Cellitruncated hexeract (catax) | | | | | 19200 | 3840 |
| 75 | | t$_{0,1,3}${4,3,3,3,3} | Runcitruncated 6-cube Prismatotruncated hexeract (potax) | | | | | 17280 | 3840 |
| 76 | | t$_{0,1,2}${4,3,3,3,3} | Cantitruncated 6-cube Great rhombated hexeract (grox) | | | | | 5760 | 1920 |
| 77 | | t$_{0,1,2,3}${3,3,3,3,4} | Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog) | | | | | 20160 | 5760 |
| 78 | | t$_{0,1,2,4}${3,3,3,3,4} | Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg) | | | | | 46080 | 11520 |
| 79 | | t$_{0,1,3,4}${3,3,3,3,4} | Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog) | | | | | 40320 | 11520 |
| 80 | | t$_{0,2,3,4}${3,3,3,3,4} | Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag) | | | | | 40320 | 11520 |
| 81 | | t$_{1,2,3,4}${4,3,3,3,3} | Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog) | | | | | 34560 | 11520 |
| 82 | | t$_{0,1,2,5}${3,3,3,3,4} | Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig) | | | | | 30720 | 7680 |
| 83 | | t$_{0,1,3,5}${3,3,3,3,4} | Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax) | | | | | 51840 | 11520 |
| 84 | | t$_{0,2,3,5}${4,3,3,3,3} | Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog) | | | | | 46080 | 11520 |
| 85 | | t$_{0,2,3,4}${4,3,3,3,3} | Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix) | | | | | 40320 | 11520 |
| 86 | | t$_{0,1,4,5}${4,3,3,3,3} | Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog) | | | | | 30720 | 7680 |
| 87 | | t$_{0,1,3,5}${4,3,3,3,3} | Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag) | | | | | 51840 | 11520 |
| 88 | | t$_{0,1,3,4}${4,3,3,3,3} | Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix) | | | | | 40320 | 11520 |
| 89 | | t$_{0,1,2,5}${4,3,3,3,3} | Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix) | | | | | 30720 | 7680 |
| 90 | | t$_{0,1,2,4}${4,3,3,3,3} | Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx) | | | | | 46080 | 11520 |
| 91 | | t$_{0,1,2,3}${4,3,3,3,3} | Runcicantitruncated 6-cube Great prismated hexeract (gippox) | | | | | 23040 | 7680 |
| 92 | | t$_{0,1,2,3,4}${3,3,3,3,4} | Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog) | | | | | 69120 | 23040 |
| 93 | | t$_{0,1,2,3,5}${3,3,3,3,4} | Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog) | | | | | 80640 | 23040 |
| 94 | | t$_{0,1,2,4,5}${3,3,3,3,4} | Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg) | | | | | 80640 | 23040 |
| 95 | | t$_{0,1,2,4,5}${4,3,3,3,3} | Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax) | | | | | 80640 | 23040 |
| 96 | | t$_{0,1,2,3,5}${4,3,3,3,3} | Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox) | | | | | 80640 | 23040 |
| 97 | | t$_{0,1,2,3,4}${4,3,3,3,3} | Steriruncicantitruncated 6-cube Great cellated hexeract (gocax) | | | | | 69120 | 23040 |
| 98 | | t$_{0,1,2,3,4,5}${4,3,3,3,3} | Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog) | | | | | 138240 | 46080 |

### The D$_6$ family

The D$_6$ family has symmetry of order 23040 (6 factorial x 2$^5$).
This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D$_6$ Coxeter-Dynkin diagram. Of these, 31 (2×16−1) are repeated from the B$_6$ 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 | Element counts | Element counts | Element counts | Element counts | Element counts | Circumrad |
| # | Coxeter diagram | Names | Base point (Alternately signed) | 5 | 4 | 3 | 2 | 1 | 0 | Circumrad |
| 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 | = | Pentiruncic 6-cube Cellirhombated hemihexeract (crohax) | (1,1,1,3,3,5) | | | | | 10560 | 1920 | 2.3979158 |
| 109 | = | Pentisteric 6-cube Celliprismated hemihexeract (cophix) | (1,1,1,1,3,5) | | | | | 5280 | 960 | 2.1794496 |
| 110 | = | Steriruncicantic 6-cube Great prismated hemihexeract (gophax) | (1,1,3,5,7,7) | | | | | 17280 | 5760 | 4.0926762 |
| 111 | = | Pentiruncicantic 6-cube Celligreatorhombated hemihexeract (cagrohax) | (1,1,3,5,5,7) | | | | | 20160 | 5760 | 3.7080991 |
| 112 | = | Pentistericantic 6-cube Celliprismatotruncated hemihexeract (capthix) | (1,1,3,3,5,7) | | | | | 23040 | 5760 | 3.4278274 |
| 113 | = | Pentisteriruncic 6-cube Celliprismatorhombated hemihexeract (caprohax) | (1,1,1,3,5,7) | | | | | 15360 | 3840 | 3.2787192 |
| 114 | = | Pentisteriruncicantic 6-cube Great cellated hemihexeract (gochax) | (1,1,3,5,7,9) | | | | | 34560 | 11520 | 4.5552168 |

### The E$_6$ family

There are 39 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Bowers-style acronym names are given for cross-referencing. The E$_6$ family has symmetry of order 51,840.
| # | Coxeter diagram | Names | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter diagram | Names | 5-faces | 4-faces | Cells | Faces | Edges | Vertices |
| 115 | | 2$_{21}$ Icosiheptaheptacontidipeton (jak) | 99 | 648 | 1080 | 720 | 216 | 27 |
| 116 | | Rectified 2$_{21}$ Rectified icosiheptaheptacontidipeton (rojak) | 126 | 1350 | 4320 | 5040 | 2160 | 216 |
| 117 | | Truncated 2$_{21}$ Truncated icosiheptaheptacontidipeton (tojak) | 126 | 1350 | 4320 | 5040 | 2376 | 432 |
| 118 | | Cantellated 2$_{21}$ Small rhombated icosiheptaheptacontidipeton (sirjak) | 342 | 3942 | 15120 | 24480 | 15120 | 2160 |
| 119 | | Runcinated 2$_{21}$ Small demiprismated icosiheptaheptacontidipeton (shopjak) | 342 | 4662 | 16200 | 19440 | 8640 | 1080 |
| 120 | | Demified icosiheptaheptacontidipeton (hejak) | 342 | 2430 | 7200 | 7920 | 3240 | 432 |
| 121 | | Bitruncated 2$_{21}$ Bitruncated icosiheptaheptacontidipeton (botajik) | | | | | | 2160 |
| 122 | | Demirectified icosiheptaheptacontidipeton (harjak) | | | | | | 1080 |
| 123 | | Cantitruncated 2$_{21}$ Great rhombated icosiheptaheptacontidipeton (girjak) | | | | | | 4320 |
| 124 | | Runcitruncated 2$_{21}$ Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak) | | | | | | 4320 |
| 125 | | Steritruncated 2$_{21}$ Cellitruncated icosiheptaheptacontidipeton (catjak) | | | | | | 2160 |
| 126 | | Demitruncated icosiheptaheptacontidipeton (hotjak) | | | | | | 2160 |
| 127 | | Runcicantellated 2$_{21}$ Demiprismatorhombated icosiheptaheptacontidipeton (haprojak) | | | | | | 6480 |
| 128 | | Small demirhombated icosiheptaheptacontidipeton (shorjak) | | | | | | 4320 |
| 129 | | Small prismated icosiheptaheptacontidipeton (spojak) | | | | | | 4320 |
| 130 | | Tritruncated icosiheptaheptacontidipeton (titajak) | | | | | | 4320 |
| 131 | | Runcicantitruncated 2$_{21}$ Great demiprismated icosiheptaheptacontidipeton (ghopjak) | | | | | | 12960 |
| 132 | | Stericantitruncated 2$_{21}$ Celligreatorhombated icosiheptaheptacontidipeton (cograjik) | | | | | | 12960 |
| 133 | | Great demirhombated icosiheptaheptacontidipeton (ghorjak) | | | | | | 8640 |
| 134 | | Prismatotruncated icosiheptaheptacontidipeton (potjak) | | | | | | 12960 |
| 135 | | Demicellitruncated icosiheptaheptacontidipeton (hictijik) | | | | | | 8640 |
| 136 | | Prismatorhombated icosiheptaheptacontidipeton (projak) | | | | | | 12960 |
| 137 | | Great prismated icosiheptaheptacontidipeton (gapjak) | | | | | | 25920 |
| 138 | | Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik) | | | | | | 25920 |
| # | Coxeter diagram | Names | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter diagram | Names | 5-faces | 4-faces | Cells | Faces | Edges | Vertices |
| 139 | = | 1$_{22}$ Pentacontatetrapeton (mo) | 54 | 702 | 2160 | 2160 | 720 | 72 |
| 140 | = | Rectified 1$_{22}$ Rectified pentacontatetrapeton (ram) | 126 | 1566 | 6480 | 10800 | 6480 | 720 |
| 141 | = | Birectified 1$_{22}$ Birectified pentacontatetrapeton (barm) | 126 | 2286 | 10800 | 19440 | 12960 | 2160 |
| 142 | = | Trirectified 1$_{22}$ Trirectified pentacontatetrapeton (trim) | 558 | 4608 | 8640 | 6480 | 2160 | 270 |
| 143 | = | Truncated 1$_{22}$ Truncated pentacontatetrapeton (tim) | | | | | 13680 | 1440 |
| 144 | = | Bitruncated 1$_{22}$ Bitruncated pentacontatetrapeton (bitem) | | | | | | 6480 |
| 145 | = | Tritruncated 1$_{22}$ Tritruncated pentacontatetrapeton (titam) | | | | | | 8640 |
| 146 | = | Cantellated 1$_{22}$ Small rhombated pentacontatetrapeton (sram) | | | | | | 6480 |
| 147 | = | Cantitruncated 1$_{22}$ Great rhombated pentacontatetrapeton (gram) | | | | | | 12960 |
| 148 | = | Runcinated 1$_{22}$ Small prismated pentacontatetrapeton (spam) | | | | | | 2160 |
| 149 | = | Bicantellated 1$_{22}$ Small birhombated pentacontatetrapeton (sabrim) | | | | | | 6480 |
| 150 | = | Bicantitruncated 1$_{22}$ Great birhombated pentacontatetrapeton (gabrim) | | | | | | 12960 |
| 151 | = | Runcitruncated 1$_{22}$ Prismatotruncated pentacontatetrapeton (patom) | | | | | | 12960 |
| 152 | = | Runcicantellated 1$_{22}$ Prismatorhombated pentacontatetrapeton (prom) | | | | | | 25920 |
| 153 | = | Omnitruncated 1$_{22}$ Great prismated pentacontatetrapeton (gopam) | | | | | | 51840 |

### Triaprisms

Uniform triaprisms, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2. {4}×{4}×{4} makes a lower symmetry form of the 6-cube.
The extended f-vector is (p,p,1)*(q,q,1)*(r,r,1)=(pqr,3pqr,3pqr+pq+pr+qr,3p(p+1),3p,1).
| Coxeter diagram | Names | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| Coxeter diagram | Names | 5-faces | 4-faces | Cells | Faces | Edges | Vertices |
| | {p}×{q}×{r} | p+q+r | pq+pr+qr+p+q+r | pqr+2(pq+pr+qr) | 3pqr+pq+pr+qr | 3pqr | pqr |
| | {p}×{p}×{p} | 3p | 3p(p+1) | p$^2$(p+6) | 3p$^2$(p+1) | 3p$^3$ | p$^3$ |
| | {3}×{3}×{3} (trittip) | 9 | 36 | 81 | 99 | 81 | 27 |
| | {4}×{4}×{4} = 6-cube | 12 | 60 | 160 | 240 | 192 | 64 |

### Non-Wythoffian 6-polytopes

In 6 dimensions and above, there are an infinite amount of non-Wythoffian convex uniform polytopes: the Cartesian product of the grand antiprism in 4 dimensions and any regular polygon in 2 dimensions. It is not yet proven whether or not there are more.

## Regular and uniform honeycombs

There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space:
| # | Coxeter group | Coxeter group | Forms |
| 1 | ${\tilde {A}}_{5}$ | [3$^{[6]}$] | 12 |
| 2 | ${\tilde {C}}_{5}$ | [4,3$^3$,4] | 35 |
| 3 | ${\tilde {B}}_{5}$ | [4,3,3$^{1,1}$] [4,3$^3$,4,1$^+$] | 47 (16 new) |
| 4 | ${\tilde {D}}_{5}$ | [3$^{1,1}$,3,3$^{1,1}$] [1$^+$,4,3$^3$,4,1$^+$] | 20 (3 new) |
Regular and uniform honeycombs include:
* ${\tilde {A}}_{5}$ There are 12 unique uniform honeycombs, including:
** 5-simplex honeycomb
** Truncated 5-simplex honeycomb
** Omnitruncated 5-simplex honeycomb
* ${\tilde {C}}_{5}$ There are 35 uniform honeycombs, including:
** Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,3$^3$,4}, =
* ${\tilde {B}}_{5}$ There are 47 uniform honeycombs, 16 new, including:
** The uniform alternated hypercube honeycomb, 5-demicubic honeycomb, with symbols h{4,3$^3$,4}, = =
* ${\tilde {D}}_{5}$, [3$^{1,1}$,3,3$^{1,1}$]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,3$^3$,4}, = . The other two new ones are = , = .
Prismatic groups :
| # | Coxeter group | Coxeter group |
| 1 | ${\tilde {A}}_{4}$x${\tilde {I}}_{1}$ | [3$^{[5]}$,2,∞] |
| 2 | ${\tilde {B}}_{4}$x${\tilde {I}}_{1}$ | [4,3,3$^{1,1}$,2,∞] |
| 3 | ${\tilde {C}}_{4}$x${\tilde {I}}_{1}$ | [4,3,3,4,2,∞] |
| 4 | ${\tilde {D}}_{4}$x${\tilde {I}}_{1}$ | [3$^{1,1,1,1}$,2,∞] |
| 5 | ${\tilde {F}}_{4}$x${\tilde {I}}_{1}$ | [3,4,3,3,2,∞] |
| 6 | ${\tilde {C}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3,4,2,∞,2,∞] |
| 7 | ${\tilde {B}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3$^{1,1}$,2,∞,2,∞] |
| 8 | ${\tilde {A}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{[4]}$,2,∞,2,∞] |
| 9 | ${\tilde {C}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,4,2,∞,2,∞,2,∞] |
| 10 | ${\tilde {H}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [6,3,2,∞,2,∞,2,∞] |
| 11 | ${\tilde {A}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{[3]}$,2,∞,2,∞,2,∞] |
| 12 | ${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [∞,2,∞,2,∞,2,∞,2,∞] |
| 13 | ${\tilde {A}}_{2}$x${\tilde {A}}_{2}$x${\tilde {I}}_{1}$ | [3$^{[3]}$,2,3$^{[3]}$,2,∞] |
| 14 | ${\tilde {A}}_{2}$x${\tilde {B}}_{2}$x${\tilde {I}}_{1}$ | [3$^{[3]}$,2,4,4,2,∞] |
| 15 | ${\tilde {A}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$ | [3$^{[3]}$,2,6,3,2,∞] |
| 16 | ${\tilde {B}}_{2}$x${\tilde {B}}_{2}$x${\tilde {I}}_{1}$ | [4,4,2,4,4,2,∞] |
| 17 | ${\tilde {B}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$ | [4,4,2,6,3,2,∞] |
| 18 | ${\tilde {G}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$ | [6,3,2,6,3,2,∞] |
| 19 | ${\tilde {A}}_{3}$x${\tilde {A}}_{2}$ | [3$^{[4]}$,2,3$^{[3]}$] |
| 20 | ${\tilde {B}}_{3}$x${\tilde {A}}_{2}$ | [4,3$^{1,1}$,2,3$^{[3]}$] |
| 21 | ${\tilde {C}}_{3}$x${\tilde {A}}_{2}$ | [4,3,4,2,3$^{[3]}$] |
| 22 | ${\tilde {A}}_{3}$x${\tilde {B}}_{2}$ | [3$^{[4]}$,2,4,4] |
| 23 | ${\tilde {B}}_{3}$x${\tilde {B}}_{2}$ | [4,3$^{1,1}$,2,4,4] |
| 24 | ${\tilde {C}}_{3}$x${\tilde {B}}_{2}$ | [4,3,4,2,4,4] |
| 25 | ${\tilde {A}}_{3}$x${\tilde {G}}_{2}$ | [3$^{[4]}$,2,6,3] |
| 26 | ${\tilde {B}}_{3}$x${\tilde {G}}_{2}$ | [4,3$^{1,1}$,2,6,3] |
| 27 | ${\tilde {C}}_{3}$x${\tilde {G}}_{2}$ | [4,3,4,2,6,3] |

### Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 12 paracompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.
Hyperbolic paracompact groups :
| ${\bar {P}}_{5}$ = [3,3$^{[5]}$]: ${\widehat {AU}}_{5}$ = [(3,3,3,3,3,4)]: ${\widehat {AR}}_{5}$ = [(3,3,4,3,3,4)]: | ${\bar {S}}_{5}$ = [4,3,3$^{2,1}$]: ${\bar {O}}_{5}$ = [3,4,3$^{1,1}$]: ${\bar {N}}_{5}$ = [3,(3,4)$^{1,1}$]: | ${\bar {U}}_{5}$ = [3,3,3,4,3]: ${\bar {X}}_{5}$ = [3,3,4,3,3]: ${\bar {R}}_{5}$ = [3,4,3,3,4]: | ${\bar {Q}}_{5}$ = [3$^{2,1,1,1}$]: ${\bar {M}}_{5}$ = [4,3,3$^{1,1,1}$]: ${\bar {L}}_{5}$ = [3$^{1,1,1,1,1}$]: |

## Notes on the Wythoff construction for the uniform 6-polytopes

Construction of the reflective 6-dimensional uniform polytopes 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 6-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.
Here's the primary operators available for constructing and naming the uniform 6-polytopes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
| Operation | Extended Schläfli symbol | Description |
| Parent | t$_0${p,q,r,s,t} | Any regular 6-polytope |
| Rectified | t$_1${p,q,r,s,t} | The edges are fully truncated into single points. The 6-polytope now has the combined faces of the parent and dual. |
| Birectified | t$_2${p,q,r,s,t} | Birectification reduces cells to their duals. |
| Truncated | t$_{0,1}${p,q,r,s,t} | 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 6-polytope. The 6-polytope has its original faces doubled in sides, and contains the faces of the dual. |
| Bitruncated | t$_{1,2}${p,q,r,s,t} | Bitrunction transforms cells to their dual truncation. |
| Tritruncated | t$_{2,3}${p,q,r,s,t} | Tritruncation transforms 4-faces to their dual truncation. |
| Cantellated | t$_{0,2}${p,q,r,s,t} | In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. |
| Bicantellated | t$_{1,3}${p,q,r,s,t} | In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. |
| Runcinated | t$_{0,3}${p,q,r,s,t} | Runcination reduces cells and creates new cells at the vertices and edges. |
| Biruncinated | t$_{1,4}${p,q,r,s,t} | Runcination reduces cells and creates new cells at the vertices and edges. |
| Stericated | t$_{0,4}${p,q,r,s,t} | Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps. |
| Pentellated | t$_{0,5}${p,q,r,s,t} | Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. (expansion operation for polypeta) |
| Omnitruncated | t$_{0,1,2,3,4,5}${p,q,r,s,t} | All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied. |