# Uniform 7-polytope

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Seven-dimensional geometric object

Graphs of three regular and related uniform polytopes 7-simplex Rectified 7-simplex Truncated 7-simplex Cantellated 7-simplex Runcinated 7-simplex Stericated 7-simplex Pentellated 7-simplex Hexicated 7-simplex 7-orthoplex Truncated 7-orthoplex Rectified 7-orthoplex Cantellated 7-orthoplex Runcinated 7-orthoplex Stericated 7-orthoplex Pentellated 7-orthoplex Hexicated 7-cube Pentellated 7-cube Stericated 7-cube Cantellated 7-cube Runcinated 7-cube 7-cube Truncated 7-cube Rectified 7-cube 7-demicube Cantic 7-cube Runcic 7-cube Steric 7-cube Pentic 7-cube Hexic 7-cube 321 231 132

In [seven-dimensional](/source/Seven-dimensional_space) [geometry](/source/Geometry), a **7-polytope** is a [polytope](/source/Polytope) contained by 6-polytope facets. Each [5-polytope](/source/5-polytope) [ridge](/source/Ridge_(geometry)) being shared by exactly two [6-polytope](/source/6-polytope) [facets](/source/Facet_(mathematics)).

A **uniform 7-polytope** is one whose symmetry group is [transitive on vertices](/source/Vertex-transitive) and whose facets are [uniform 6-polytopes](/source/Uniform_6-polytope).

## Regular 7-polytopes

Regular 7-polytopes are represented by the [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {p,q,r,s,t,u} with **u** {p,q,r,s,t} 6-polytopes [facets](/source/Facet_(mathematics)) around each 4-face.

There are exactly three such [convex regular 7-polytopes](/source/List_of_regular_polytopes#Convex_4):

1. {3,3,3,3,3,3} - [7-simplex](/source/7-simplex)

1. {4,3,3,3,3,3} - [7-cube](/source/7-cube)

1. {3,3,3,3,3,4} - [7-orthoplex](/source/7-orthoplex)

There are no nonconvex regular 7-polytopes.

## Characteristics

The topology of any given 7-polytope is defined by its [Betti numbers](/source/Betti_number) and [torsion coefficients](/source/Torsion_coefficient_(topology)).[1]

The value of the [Euler characteristic](/source/Euler_characteristic) used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]

## Uniform 7-polytopes by fundamental Coxeter groups

Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram):

# Coxeter group Regular and semiregular forms Uniform count 1 A7 [36] 7-simplex - {36}, 71 2 B7 [4,35] 7-cube - {4,35}, 7-orthoplex - {35,4}, 7-demicube - h{4,35}, 127 + 32 3 D7 [33,1,1] 7-demicube, {3,34,1}, 7-orthoplex, {34,31,1}, 95 (0 unique) 4 E7 [33,2,1] 321 - 132 - 231 - 127

Prismatic finite Coxeter groups # Coxeter group Coxeter diagram 6+1 1 A6A1 [35]×[ ] 2 BC6A1 [4,34]×[ ] 3 D6A1 [33,1,1]×[ ] 4 E6A1 [32,2,1]×[ ] 5+2 1 A5I2(p) [3,3,3]×[p] 2 BC5I2(p) [4,3,3]×[p] 3 D5I2(p) [32,1,1]×[p] 5+1+1 1 A5A12 [3,3,3]×[ ]2 2 BC5A12 [4,3,3]×[ ]2 3 D5A12 [32,1,1]×[ ]2 4+3 1 A4A3 [3,3,3]×[3,3] 2 A4B3 [3,3,3]×[4,3] 3 A4H3 [3,3,3]×[5,3] 4 BC4A3 [4,3,3]×[3,3] 5 BC4B3 [4,3,3]×[4,3] 6 BC4H3 [4,3,3]×[5,3] 7 H4A3 [5,3,3]×[3,3] 8 H4B3 [5,3,3]×[4,3] 9 H4H3 [5,3,3]×[5,3] 10 F4A3 [3,4,3]×[3,3] 11 F4B3 [3,4,3]×[4,3] 12 F4H3 [3,4,3]×[5,3] 13 D4A3 [31,1,1]×[3,3] 14 D4B3 [31,1,1]×[4,3] 15 D4H3 [31,1,1]×[5,3] 4+2+1 1 A4I2(p)A1 [3,3,3]×[p]×[ ] 2 BC4I2(p)A1 [4,3,3]×[p]×[ ] 3 F4I2(p)A1 [3,4,3]×[p]×[ ] 4 H4I2(p)A1 [5,3,3]×[p]×[ ] 5 D4I2(p)A1 [31,1,1]×[p]×[ ] 4+1+1+1 1 A4A13 [3,3,3]×[ ]3 2 BC4A13 [4,3,3]×[ ]3 3 F4A13 [3,4,3]×[ ]3 4 H4A13 [5,3,3]×[ ]3 5 D4A13 [31,1,1]×[ ]3 3+3+1 1 A3A3A1 [3,3]×[3,3]×[ ] 2 A3B3A1 [3,3]×[4,3]×[ ] 3 A3H3A1 [3,3]×[5,3]×[ ] 4 BC3B3A1 [4,3]×[4,3]×[ ] 5 BC3H3A1 [4,3]×[5,3]×[ ] 6 H3A3A1 [5,3]×[5,3]×[ ] 3+2+2 1 A3I2(p)I2(q) [3,3]×[p]×[q] 2 BC3I2(p)I2(q) [4,3]×[p]×[q] 3 H3I2(p)I2(q) [5,3]×[p]×[q] 3+2+1+1 1 A3I2(p)A12 [3,3]×[p]×[ ]2 2 BC3I2(p)A12 [4,3]×[p]×[ ]2 3 H3I2(p)A12 [5,3]×[p]×[ ]2 3+1+1+1+1 1 A3A14 [3,3]×[ ]4 2 BC3A14 [4,3]×[ ]4 3 H3A14 [5,3]×[ ]4 2+2+2+1 1 I2(p)I2(q)I2(r)A1 [p]×[q]×[r]×[ ] 2+2+1+1+1 1 I2(p)I2(q)A13 [p]×[q]×[ ]3 2+1+1+1+1+1 1 I2(p)A15 [p]×[ ]5 1+1+1+1+1+1+1 1 A17 [ ]7

## The A7 family

The A7 family has symmetry of order 40320 (8 [factorial](/source/Factorial)).

There are 71 (64 + 8 − 1) forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) with one or more rings. All 71 are enumerated below. [Norman Johnson](/source/Norman_Johnson_(mathematician))'s truncation names are given. Bowers names and acronym are also given for cross-referencing.

See also a [list of A7 polytopes](/source/List_of_A7_polytopes) for symmetric [Coxeter plane](/source/Coxeter_plane) graphs of these polytopes.

A7 uniform polytopes # Coxeter-Dynkin diagram Truncation indices Johnson name Bowers name (and acronym) Basepoint Element counts 6 5 4 3 2 1 0 1 t0 7-simplex (oca) (0,0,0,0,0,0,0,1) 8 28 56 70 56 28 8 2 t1 Rectified 7-simplex (roc) (0,0,0,0,0,0,1,1) 16 84 224 350 336 168 28 3 t2 Birectified 7-simplex (broc) (0,0,0,0,0,1,1,1) 16 112 392 770 840 420 56 4 t3 Trirectified 7-simplex (he) (0,0,0,0,1,1,1,1) 16 112 448 980 1120 560 70 5 t0,1 Truncated 7-simplex (toc) (0,0,0,0,0,0,1,2) 16 84 224 350 336 196 56 6 t0,2 Cantellated 7-simplex (saro) (0,0,0,0,0,1,1,2) 44 308 980 1750 1876 1008 168 7 t1,2 Bitruncated 7-simplex (bittoc) (0,0,0,0,0,1,2,2) 588 168 8 t0,3 Runcinated 7-simplex (spo) (0,0,0,0,1,1,1,2) 100 756 2548 4830 4760 2100 280 9 t1,3 Bicantellated 7-simplex (sabro) (0,0,0,0,1,1,2,2) 2520 420 10 t2,3 Tritruncated 7-simplex (tattoc) (0,0,0,0,1,2,2,2) 980 280 11 t0,4 Stericated 7-simplex (sco) (0,0,0,1,1,1,1,2) 2240 280 12 t1,4 Biruncinated 7-simplex (sibpo) (0,0,0,1,1,1,2,2) 4200 560 13 t2,4 Tricantellated 7-simplex (stiroh) (0,0,0,1,1,2,2,2) 3360 560 14 t0,5 Pentellated 7-simplex (seto) (0,0,1,1,1,1,1,2) 1260 168 15 t1,5 Bistericated 7-simplex (sabach) (0,0,1,1,1,1,2,2) 3360 420 16 t0,6 Hexicated 7-simplex (suph) (0,1,1,1,1,1,1,2) 336 56 17 t0,1,2 Cantitruncated 7-simplex (garo) (0,0,0,0,0,1,2,3) 1176 336 18 t0,1,3 Runcitruncated 7-simplex (patto) (0,0,0,0,1,1,2,3) 4620 840 19 t0,2,3 Runcicantellated 7-simplex (paro) (0,0,0,0,1,2,2,3) 3360 840 20 t1,2,3 Bicantitruncated 7-simplex (gabro) (0,0,0,0,1,2,3,3) 2940 840 21 t0,1,4 Steritruncated 7-simplex (cato) (0,0,0,1,1,1,2,3) 7280 1120 22 t0,2,4 Stericantellated 7-simplex (caro) (0,0,0,1,1,2,2,3) 10080 1680 23 t1,2,4 Biruncitruncated 7-simplex (bipto) (0,0,0,1,1,2,3,3) 8400 1680 24 t0,3,4 Steriruncinated 7-simplex (cepo) (0,0,0,1,2,2,2,3) 5040 1120 25 t1,3,4 Biruncicantellated 7-simplex (bipro) (0,0,0,1,2,2,3,3) 7560 1680 26 t2,3,4 Tricantitruncated 7-simplex (gatroh) (0,0,0,1,2,3,3,3) 3920 1120 27 t0,1,5 Pentitruncated 7-simplex (teto) (0,0,1,1,1,1,2,3) 5460 840 28 t0,2,5 Penticantellated 7-simplex (tero) (0,0,1,1,1,2,2,3) 11760 1680 29 t1,2,5 Bisteritruncated 7-simplex (bacto) (0,0,1,1,1,2,3,3) 9240 1680 30 t0,3,5 Pentiruncinated 7-simplex (tepo) (0,0,1,1,2,2,2,3) 10920 1680 31 t1,3,5 Bistericantellated 7-simplex (bacroh) (0,0,1,1,2,2,3,3) 15120 2520 32 t0,4,5 Pentistericated 7-simplex (teco) (0,0,1,2,2,2,2,3) 4200 840 33 t0,1,6 Hexitruncated 7-simplex (puto) (0,1,1,1,1,1,2,3) 1848 336 34 t0,2,6 Hexicantellated 7-simplex (puro) (0,1,1,1,1,2,2,3) 5880 840 35 t0,3,6 Hexiruncinated 7-simplex (puph) (0,1,1,1,2,2,2,3) 8400 1120 36 t0,1,2,3 Runcicantitruncated 7-simplex (gapo) (0,0,0,0,1,2,3,4) 5880 1680 37 t0,1,2,4 Stericantitruncated 7-simplex (cagro) (0,0,0,1,1,2,3,4) 16800 3360 38 t0,1,3,4 Steriruncitruncated 7-simplex (capto) (0,0,0,1,2,2,3,4) 13440 3360 39 t0,2,3,4 Steriruncicantellated 7-simplex (capro) (0,0,0,1,2,3,3,4) 13440 3360 40 t1,2,3,4 Biruncicantitruncated 7-simplex (gibpo) (0,0,0,1,2,3,4,4) 11760 3360 41 t0,1,2,5 Penticantitruncated 7-simplex (tegro) (0,0,1,1,1,2,3,4) 18480 3360 42 t0,1,3,5 Pentiruncitruncated 7-simplex (tapto) (0,0,1,1,2,2,3,4) 27720 5040 43 t0,2,3,5 Pentiruncicantellated 7-simplex (tapro) (0,0,1,1,2,3,3,4) 25200 5040 44 t1,2,3,5 Bistericantitruncated 7-simplex (bacogro) (0,0,1,1,2,3,4,4) 22680 5040 45 t0,1,4,5 Pentisteritruncated 7-simplex (tecto) (0,0,1,2,2,2,3,4) 15120 3360 46 t0,2,4,5 Pentistericantellated 7-simplex (tecro) (0,0,1,2,2,3,3,4) 25200 5040 47 t1,2,4,5 Bisteriruncitruncated 7-simplex (bicpath) (0,0,1,2,2,3,4,4) 20160 5040 48 t0,3,4,5 Pentisteriruncinated 7-simplex (tacpo) (0,0,1,2,3,3,3,4) 15120 3360 49 t0,1,2,6 Hexicantitruncated 7-simplex (pugro) (0,1,1,1,1,2,3,4) 8400 1680 50 t0,1,3,6 Hexiruncitruncated 7-simplex (pugato) (0,1,1,1,2,2,3,4) 20160 3360 51 t0,2,3,6 Hexiruncicantellated 7-simplex (pugro) (0,1,1,1,2,3,3,4) 16800 3360 52 t0,1,4,6 Hexisteritruncated 7-simplex (pucto) (0,1,1,2,2,2,3,4) 20160 3360 53 t0,2,4,6 Hexistericantellated 7-simplex (pucroh) (0,1,1,2,2,3,3,4) 30240 5040 54 t0,1,5,6 Hexipentitruncated 7-simplex (putath) (0,1,2,2,2,2,3,4) 8400 1680 55 t0,1,2,3,4 Steriruncicantitruncated 7-simplex (gecco) (0,0,0,1,2,3,4,5) 23520 6720 56 t0,1,2,3,5 Pentiruncicantitruncated 7-simplex (tegapo) (0,0,1,1,2,3,4,5) 45360 10080 57 t0,1,2,4,5 Pentistericantitruncated 7-simplex (tecagro) (0,0,1,2,2,3,4,5) 40320 10080 58 t0,1,3,4,5 Pentisteriruncitruncated 7-simplex (tacpeto) (0,0,1,2,3,3,4,5) 40320 10080 59 t0,2,3,4,5 Pentisteriruncicantellated 7-simplex (tacpro) (0,0,1,2,3,4,4,5) 40320 10080 60 t1,2,3,4,5 Bisteriruncicantitruncated 7-simplex (gabach) (0,0,1,2,3,4,5,5) 35280 10080 61 t0,1,2,3,6 Hexiruncicantitruncated 7-simplex (pugopo) (0,1,1,1,2,3,4,5) 30240 6720 62 t0,1,2,4,6 Hexistericantitruncated 7-simplex (pucagro) (0,1,1,2,2,3,4,5) 50400 10080 63 t0,1,3,4,6 Hexisteriruncitruncated 7-simplex (pucpato) (0,1,1,2,3,3,4,5) 45360 10080 64 t0,2,3,4,6 Hexisteriruncicantellated 7-simplex (pucproh) (0,1,1,2,3,4,4,5) 45360 10080 65 t0,1,2,5,6 Hexipenticantitruncated 7-simplex (putagro) (0,1,2,2,2,3,4,5) 30240 6720 66 t0,1,3,5,6 Hexipentiruncitruncated 7-simplex (putpath) (0,1,2,2,3,3,4,5) 50400 10080 67 t0,1,2,3,4,5 Pentisteriruncicantitruncated 7-simplex (geto) (0,0,1,2,3,4,5,6) 70560 20160 68 t0,1,2,3,4,6 Hexisteriruncicantitruncated 7-simplex (pugaco) (0,1,1,2,3,4,5,6) 80640 20160 69 t0,1,2,3,5,6 Hexipentiruncicantitruncated 7-simplex (putgapo) (0,1,2,2,3,4,5,6) 80640 20160 70 t0,1,2,4,5,6 Hexipentistericantitruncated 7-simplex (putcagroh) (0,1,2,3,3,4,5,6) 80640 20160 71 t0,1,2,3,4,5,6 Omnitruncated 7-simplex (guph) (0,1,2,3,4,5,6,7) 141120 40320

## The B7 family

The B7 family has symmetry of order 645120 (7 [factorial](/source/Factorial) x 27).

There are 127 forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) with one or more rings. Bowers names and acronym are given for cross-referencing.

See also a [list of B7 polytopes](/source/List_of_B7_polytopes) for symmetric [Coxeter plane](/source/Coxeter_plane) graphs of these polytopes.

B7 uniform polytopes # Coxeter-Dynkin diagram t-notation Name (BSA) Base point Element counts 6 5 4 3 2 1 0 1 t0{3,3,3,3,3,4} 7-orthoplex (zee) (0,0,0,0,0,0,1)√2 128 448 672 560 280 84 14 2 t1{3,3,3,3,3,4} Rectified 7-orthoplex (rez) (0,0,0,0,0,1,1)√2 142 1344 3360 3920 2520 840 84 3 t2{3,3,3,3,3,4} Birectified 7-orthoplex (barz) (0,0,0,0,1,1,1)√2 142 1428 6048 10640 8960 3360 280 4 t3{4,3,3,3,3,3} Trirectified 7-cube (sez) (0,0,0,1,1,1,1)√2 142 1428 6328 14560 15680 6720 560 5 t2{4,3,3,3,3,3} Birectified 7-cube (bersa) (0,0,1,1,1,1,1)√2 142 1428 5656 11760 13440 6720 672 6 t1{4,3,3,3,3,3} Rectified 7-cube (rasa) (0,1,1,1,1,1,1)√2 142 980 2968 5040 5152 2688 448 7 t0{4,3,3,3,3,3} 7-cube (hept) (0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1) 14 84 280 560 672 448 128 8 t0,1{3,3,3,3,3,4} Truncated 7-orthoplex (Taz) (0,0,0,0,0,1,2)√2 142 1344 3360 4760 2520 924 168 9 t0,2{3,3,3,3,3,4} Cantellated 7-orthoplex (Sarz) (0,0,0,0,1,1,2)√2 226 4200 15456 24080 19320 7560 840 10 t1,2{3,3,3,3,3,4} Bitruncated 7-orthoplex (Botaz) (0,0,0,0,1,2,2)√2 4200 840 11 t0,3{3,3,3,3,3,4} Runcinated 7-orthoplex (Spaz) (0,0,0,1,1,1,2)√2 23520 2240 12 t1,3{3,3,3,3,3,4} Bicantellated 7-orthoplex (Sebraz) (0,0,0,1,1,2,2)√2 26880 3360 13 t2,3{3,3,3,3,3,4} Tritruncated 7-orthoplex (Totaz) (0,0,0,1,2,2,2)√2 10080 2240 14 t0,4{3,3,3,3,3,4} Stericated 7-orthoplex (Scaz) (0,0,1,1,1,1,2)√2 33600 3360 15 t1,4{3,3,3,3,3,4} Biruncinated 7-orthoplex (Sibpaz) (0,0,1,1,1,2,2)√2 60480 6720 16 t2,4{4,3,3,3,3,3} Tricantellated 7-cube (Strasaz) (0,0,1,1,2,2,2)√2 47040 6720 17 t2,3{4,3,3,3,3,3} Tritruncated 7-cube (Tatsa) (0,0,1,2,2,2,2)√2 13440 3360 18 t0,5{3,3,3,3,3,4} Pentellated 7-orthoplex (Staz) (0,1,1,1,1,1,2)√2 20160 2688 19 t1,5{4,3,3,3,3,3} Bistericated 7-cube (Sabcosaz) (0,1,1,1,1,2,2)√2 53760 6720 20 t1,4{4,3,3,3,3,3} Biruncinated 7-cube (Sibposa) (0,1,1,1,2,2,2)√2 67200 8960 21 t1,3{4,3,3,3,3,3} Bicantellated 7-cube (Sibrosa) (0,1,1,2,2,2,2)√2 40320 6720 22 t1,2{4,3,3,3,3,3} Bitruncated 7-cube (Betsa) (0,1,2,2,2,2,2)√2 9408 2688 23 t0,6{4,3,3,3,3,3} Hexicated 7-cube (Supposaz) (0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1) 5376 896 24 t0,5{4,3,3,3,3,3} Pentellated 7-cube (Stesa) (0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1) 20160 2688 25 t0,4{4,3,3,3,3,3} Stericated 7-cube (Scosa) (0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1) 35840 4480 26 t0,3{4,3,3,3,3,3} Runcinated 7-cube (Spesa) (0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1) 33600 4480 27 t0,2{4,3,3,3,3,3} Cantellated 7-cube (Sersa) (0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) 16128 2688 28 t0,1{4,3,3,3,3,3} Truncated 7-cube (Tasa) (0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) 142 980 2968 5040 5152 3136 896 29 t0,1,2{3,3,3,3,3,4} Cantitruncated 7-orthoplex (Garz) (0,1,2,3,3,3,3)√2 8400 1680 30 t0,1,3{3,3,3,3,3,4} Runcitruncated 7-orthoplex (Potaz) (0,1,2,2,3,3,3)√2 50400 6720 31 t0,2,3{3,3,3,3,3,4} Runcicantellated 7-orthoplex (Parz) (0,1,1,2,3,3,3)√2 33600 6720 32 t1,2,3{3,3,3,3,3,4} Bicantitruncated 7-orthoplex (Gebraz) (0,0,1,2,3,3,3)√2 30240 6720 33 t0,1,4{3,3,3,3,3,4} Steritruncated 7-orthoplex (Catz) (0,0,1,1,1,2,3)√2 107520 13440 34 t0,2,4{3,3,3,3,3,4} Stericantellated 7-orthoplex (Craze) (0,0,1,1,2,2,3)√2 141120 20160 35 t1,2,4{3,3,3,3,3,4} Biruncitruncated 7-orthoplex (Baptize) (0,0,1,1,2,3,3)√2 120960 20160 36 t0,3,4{3,3,3,3,3,4} Steriruncinated 7-orthoplex (Copaz) (0,1,1,1,2,3,3)√2 67200 13440 37 t1,3,4{3,3,3,3,3,4} Biruncicantellated 7-orthoplex (Boparz) (0,0,1,2,2,3,3)√2 100800 20160 38 t2,3,4{4,3,3,3,3,3} Tricantitruncated 7-cube (Gotrasaz) (0,0,0,1,2,3,3)√2 53760 13440 39 t0,1,5{3,3,3,3,3,4} Pentitruncated 7-orthoplex (Tetaz) (0,1,1,1,1,2,3)√2 87360 13440 40 t0,2,5{3,3,3,3,3,4} Penticantellated 7-orthoplex (Teroz) (0,1,1,1,2,2,3)√2 188160 26880 41 t1,2,5{3,3,3,3,3,4} Bisteritruncated 7-orthoplex (Boctaz) (0,1,1,1,2,3,3)√2 147840 26880 42 t0,3,5{3,3,3,3,3,4} Pentiruncinated 7-orthoplex (Topaz) (0,1,1,2,2,2,3)√2 174720 26880 43 t1,3,5{4,3,3,3,3,3} Bistericantellated 7-cube (Bacresaz) (0,1,1,2,2,3,3)√2 241920 40320 44 t1,3,4{4,3,3,3,3,3} Biruncicantellated 7-cube (Bopresa) (0,1,1,2,3,3,3)√2 120960 26880 45 t0,4,5{3,3,3,3,3,4} Pentistericated 7-orthoplex (Tocaz) (0,1,2,2,2,2,3)√2 67200 13440 46 t1,2,5{4,3,3,3,3,3} Bisteritruncated 7-cube (Bactasa) (0,1,2,2,2,3,3)√2 147840 26880 47 t1,2,4{4,3,3,3,3,3} Biruncitruncated 7-cube (Biptesa) (0,1,2,2,3,3,3)√2 134400 26880 48 t1,2,3{4,3,3,3,3,3} Bicantitruncated 7-cube (Gibrosa) (0,1,2,3,3,3,3)√2 47040 13440 49 t0,1,6{3,3,3,3,3,4} Hexitruncated 7-orthoplex (Putaz) (0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1) 29568 5376 50 t0,2,6{3,3,3,3,3,4} Hexicantellated 7-orthoplex (Puraz) (0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1) 94080 13440 51 t0,4,5{4,3,3,3,3,3} Pentistericated 7-cube (Tacosa) (0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1) 67200 13440 52 t0,3,6{4,3,3,3,3,3} Hexiruncinated 7-cube (Pupsez) (0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1) 134400 17920 53 t0,3,5{4,3,3,3,3,3} Pentiruncinated 7-cube (Tapsa) (0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1) 174720 26880 54 t0,3,4{4,3,3,3,3,3} Steriruncinated 7-cube (Capsa) (0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1) 80640 17920 55 t0,2,6{4,3,3,3,3,3} Hexicantellated 7-cube (Purosa) (0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) 94080 13440 56 t0,2,5{4,3,3,3,3,3} Penticantellated 7-cube (Tersa) (0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) 188160 26880 57 t0,2,4{4,3,3,3,3,3} Stericantellated 7-cube (Carsa) (0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) 161280 26880 58 t0,2,3{4,3,3,3,3,3} Runcicantellated 7-cube (Parsa) (0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) 53760 13440 59 t0,1,6{4,3,3,3,3,3} Hexitruncated 7-cube (Putsa) (0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) 29568 5376 60 t0,1,5{4,3,3,3,3,3} Pentitruncated 7-cube (Tetsa) (0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) 87360 13440 61 t0,1,4{4,3,3,3,3,3} Steritruncated 7-cube (Catsa) (0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) 116480 17920 62 t0,1,3{4,3,3,3,3,3} Runcitruncated 7-cube (Petsa) (0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) 73920 13440 63 t0,1,2{4,3,3,3,3,3} Cantitruncated 7-cube (Gersa) (0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1) 18816 5376 64 t0,1,2,3{3,3,3,3,3,4} Runcicantitruncated 7-orthoplex (Gopaz) (0,1,2,3,4,4,4)√2 60480 13440 65 t0,1,2,4{3,3,3,3,3,4} Stericantitruncated 7-orthoplex (Cogarz) (0,0,1,1,2,3,4)√2 241920 40320 66 t0,1,3,4{3,3,3,3,3,4} Steriruncitruncated 7-orthoplex (Captaz) (0,0,1,2,2,3,4)√2 181440 40320 67 t0,2,3,4{3,3,3,3,3,4} Steriruncicantellated 7-orthoplex (Caparz) (0,0,1,2,3,3,4)√2 181440 40320 68 t1,2,3,4{3,3,3,3,3,4} Biruncicantitruncated 7-orthoplex (Gibpaz) (0,0,1,2,3,4,4)√2 161280 40320 69 t0,1,2,5{3,3,3,3,3,4} Penticantitruncated 7-orthoplex (Tograz) (0,1,1,1,2,3,4)√2 295680 53760 70 t0,1,3,5{3,3,3,3,3,4} Pentiruncitruncated 7-orthoplex (Toptaz) (0,1,1,2,2,3,4)√2 443520 80640 71 t0,2,3,5{3,3,3,3,3,4} Pentiruncicantellated 7-orthoplex (Toparz) (0,1,1,2,3,3,4)√2 403200 80640 72 t1,2,3,5{3,3,3,3,3,4} Bistericantitruncated 7-orthoplex (Becogarz) (0,1,1,2,3,4,4)√2 362880 80640 73 t0,1,4,5{3,3,3,3,3,4} Pentisteritruncated 7-orthoplex (Tacotaz) (0,1,2,2,2,3,4)√2 241920 53760 74 t0,2,4,5{3,3,3,3,3,4} Pentistericantellated 7-orthoplex (Tocarz) (0,1,2,2,3,3,4)√2 403200 80640 75 t1,2,4,5{4,3,3,3,3,3} Bisteriruncitruncated 7-cube (Bocaptosaz) (0,1,2,2,3,4,4)√2 322560 80640 76 t0,3,4,5{3,3,3,3,3,4} Pentisteriruncinated 7-orthoplex (Tecpaz) (0,1,2,3,3,3,4)√2 241920 53760 77 t1,2,3,5{4,3,3,3,3,3} Bistericantitruncated 7-cube (Becgresa) (0,1,2,3,3,4,4)√2 362880 80640 78 t1,2,3,4{4,3,3,3,3,3} Biruncicantitruncated 7-cube (Gibposa) (0,1,2,3,4,4,4)√2 188160 53760 79 t0,1,2,6{3,3,3,3,3,4} Hexicantitruncated 7-orthoplex (Pugarez) (0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1) 134400 26880 80 t0,1,3,6{3,3,3,3,3,4} Hexiruncitruncated 7-orthoplex (Papataz) (0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1) 322560 53760 81 t0,2,3,6{3,3,3,3,3,4} Hexiruncicantellated 7-orthoplex (Puparez) (0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1) 268800 53760 82 t0,3,4,5{4,3,3,3,3,3} Pentisteriruncinated 7-cube (Tecpasa) (0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1) 241920 53760 83 t0,1,4,6{3,3,3,3,3,4} Hexisteritruncated 7-orthoplex (Pucotaz) (0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) 322560 53760 84 t0,2,4,6{4,3,3,3,3,3} Hexistericantellated 7-cube (Pucrosaz) (0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) 483840 80640 85 t0,2,4,5{4,3,3,3,3,3} Pentistericantellated 7-cube (Tecresa) (0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) 403200 80640 86 t0,2,3,6{4,3,3,3,3,3} Hexiruncicantellated 7-cube (Pupresa) (0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) 268800 53760 87 t0,2,3,5{4,3,3,3,3,3} Pentiruncicantellated 7-cube (Topresa) (0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) 403200 80640 88 t0,2,3,4{4,3,3,3,3,3} Steriruncicantellated 7-cube (Copresa) (0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) 215040 53760 89 t0,1,5,6{4,3,3,3,3,3} Hexipentitruncated 7-cube (Putatosez) (0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) 134400 26880 90 t0,1,4,6{4,3,3,3,3,3} Hexisteritruncated 7-cube (Pacutsa) (0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) 322560 53760 91 t0,1,4,5{4,3,3,3,3,3} Pentisteritruncated 7-cube (Tecatsa) (0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) 241920 53760 92 t0,1,3,6{4,3,3,3,3,3} Hexiruncitruncated 7-cube (Pupetsa) (0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) 322560 53760 93 t0,1,3,5{4,3,3,3,3,3} Pentiruncitruncated 7-cube (Toptosa) (0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) 443520 80640 94 t0,1,3,4{4,3,3,3,3,3} Steriruncitruncated 7-cube (Captesa) (0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) 215040 53760 95 t0,1,2,6{4,3,3,3,3,3} Hexicantitruncated 7-cube (Pugrosa) (0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1) 134400 26880 96 t0,1,2,5{4,3,3,3,3,3} Penticantitruncated 7-cube (Togresa) (0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1) 295680 53760 97 t0,1,2,4{4,3,3,3,3,3} Stericantitruncated 7-cube (Cogarsa) (0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1) 268800 53760 98 t0,1,2,3{4,3,3,3,3,3} Runcicantitruncated 7-cube (Gapsa) (0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1) 94080 26880 99 t0,1,2,3,4{3,3,3,3,3,4} Steriruncicantitruncated 7-orthoplex (Gocaz) (0,0,1,2,3,4,5)√2 322560 80640 100 t0,1,2,3,5{3,3,3,3,3,4} Pentiruncicantitruncated 7-orthoplex (Tegopaz) (0,1,1,2,3,4,5)√2 725760 161280 101 t0,1,2,4,5{3,3,3,3,3,4} Pentistericantitruncated 7-orthoplex (Tecagraz) (0,1,2,2,3,4,5)√2 645120 161280 102 t0,1,3,4,5{3,3,3,3,3,4} Pentisteriruncitruncated 7-orthoplex (Tecpotaz) (0,1,2,3,3,4,5)√2 645120 161280 103 t0,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantellated 7-orthoplex (Tacparez) (0,1,2,3,4,4,5)√2 645120 161280 104 t1,2,3,4,5{4,3,3,3,3,3} Bisteriruncicantitruncated 7-cube (Gabcosaz) (0,1,2,3,4,5,5)√2 564480 161280 105 t0,1,2,3,6{3,3,3,3,3,4} Hexiruncicantitruncated 7-orthoplex (Pugopaz) (0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1) 483840 107520 106 t0,1,2,4,6{3,3,3,3,3,4} Hexistericantitruncated 7-orthoplex (Pucagraz) (0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) 806400 161280 107 t0,1,3,4,6{3,3,3,3,3,4} Hexisteriruncitruncated 7-orthoplex (Pucpotaz) (0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) 725760 161280 108 t0,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantellated 7-cube (Pucprosaz) (0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) 725760 161280 109 t0,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantellated 7-cube (Tocpresa) (0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) 645120 161280 110 t0,1,2,5,6{3,3,3,3,3,4} Hexipenticantitruncated 7-orthoplex (Putegraz) (0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) 483840 107520 111 t0,1,3,5,6{4,3,3,3,3,3} Hexipentiruncitruncated 7-cube (Putpetsaz) (0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) 806400 161280 112 t0,1,3,4,6{4,3,3,3,3,3} Hexisteriruncitruncated 7-cube (Pucpetsa) (0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) 725760 161280 113 t0,1,3,4,5{4,3,3,3,3,3} Pentisteriruncitruncated 7-cube (Tecpetsa) (0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) 645120 161280 114 t0,1,2,5,6{4,3,3,3,3,3} Hexipenticantitruncated 7-cube (Putgresa) (0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1) 483840 107520 115 t0,1,2,4,6{4,3,3,3,3,3} Hexistericantitruncated 7-cube (Pucagrosa) (0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1) 806400 161280 116 t0,1,2,4,5{4,3,3,3,3,3} Pentistericantitruncated 7-cube (Tecgresa) (0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1) 645120 161280 117 t0,1,2,3,6{4,3,3,3,3,3} Hexiruncicantitruncated 7-cube (Pugopsa) (0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1) 483840 107520 118 t0,1,2,3,5{4,3,3,3,3,3} Pentiruncicantitruncated 7-cube (Togapsa) (0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1) 725760 161280 119 t0,1,2,3,4{4,3,3,3,3,3} Steriruncicantitruncated 7-cube (Gacosa) (0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1) 376320 107520 120 t0,1,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantitruncated 7-orthoplex (Gotaz) (0,1,2,3,4,5,6)√2 1128960 322560 121 t0,1,2,3,4,6{3,3,3,3,3,4} Hexisteriruncicantitruncated 7-orthoplex (Pugacaz) (0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) 1290240 322560 122 t0,1,2,3,5,6{3,3,3,3,3,4} Hexipentiruncicantitruncated 7-orthoplex (Putgapaz) (0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) 1290240 322560 123 t0,1,2,4,5,6{4,3,3,3,3,3} Hexipentistericantitruncated 7-cube (Putcagrasaz) (0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1) 1290240 322560 124 t0,1,2,3,5,6{4,3,3,3,3,3} Hexipentiruncicantitruncated 7-cube (Putgapsa) (0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1) 1290240 322560 125 t0,1,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantitruncated 7-cube (Pugacasa) (0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1) 1290240 322560 126 t0,1,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantitruncated 7-cube (Gotesa) (0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1) 1128960 322560 127 t0,1,2,3,4,5,6{4,3,3,3,3,3} Omnitruncated 7-cube (Guposaz) (0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1) 2257920 645120

## The D7 family

The D7 family has symmetry of order 322560 (7 [factorial](/source/Factorial) × 26).

This family has 3 × 32 − 1 = 95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 [Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram). Of these, 63 (2 × 32 − 1) are repeated from the B7 family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.

See also [list of D7 polytopes](/source/List_of_D7_polytopes) for Coxeter plane graphs of these polytopes.

D7 uniform polytopes # Coxeter diagram Names Base point (Alternately signed) Element counts 6 5 4 3 2 1 0 1 = 7-cube demihepteract (hesa) (1,1,1,1,1,1,1) 78 532 1624 2800 2240 672 64 2 = cantic 7-cube truncated demihepteract (thesa) (1,1,3,3,3,3,3) 142 1428 5656 11760 13440 7392 1344 3 = runcic 7-cube small rhombated demihepteract (sirhesa) (1,1,1,3,3,3,3) 16800 2240 4 = steric 7-cube small prismated demihepteract (sphosa) (1,1,1,1,3,3,3) 20160 2240 5 = pentic 7-cube small cellated demihepteract (sochesa) (1,1,1,1,1,3,3) 13440 1344 6 = hexic 7-cube small terated demihepteract (suthesa) (1,1,1,1,1,1,3) 4704 448 7 = runcicantic 7-cube great rhombated demihepteract (girhesa) (1,1,3,5,5,5,5) 23520 6720 8 = stericantic 7-cube prismatotruncated demihepteract (pothesa) (1,1,3,3,5,5,5) 73920 13440 9 = steriruncic 7-cube prismatorhomated demihepteract (prohesa) (1,1,1,3,5,5,5) 40320 8960 10 = penticantic 7-cube cellitruncated demihepteract (cothesa) (1,1,3,3,3,5,5) 87360 13440 11 = pentiruncic 7-cube cellirhombated demihepteract (crohesa) (1,1,1,3,3,5,5) 87360 13440 12 = pentisteric 7-cube celliprismated demihepteract (caphesa) (1,1,1,1,3,5,5) 40320 6720 13 = hexicantic 7-cube tericantic demihepteract (tuthesa) (1,1,3,3,3,3,5) 43680 6720 14 = hexiruncic 7-cube terirhombated demihepteract (turhesa) (1,1,1,3,3,3,5) 67200 8960 15 = hexisteric 7-cube teriprismated demihepteract (tuphesa) (1,1,1,1,3,3,5) 53760 6720 16 = hexipentic 7-cube tericellated demihepteract (tuchesa) (1,1,1,1,1,3,5) 21504 2688 17 = steriruncicantic 7-cube great prismated demihepteract (gephosa) (1,1,3,5,7,7,7) 94080 26880 18 = pentiruncicantic 7-cube celligreatorhombated demihepteract (cagrohesa) (1,1,3,5,5,7,7) 181440 40320 19 = pentistericantic 7-cube celliprismatotruncated demihepteract (capthesa) (1,1,3,3,5,7,7) 181440 40320 20 = pentisteriruncic 7-cube celliprismatorhombated demihepteract (coprahesa) (1,1,1,3,5,7,7) 120960 26880 21 = hexiruncicantic 7-cube terigreatorhombated demihepteract (tugrohesa) (1,1,3,5,5,5,7) 120960 26880 22 = hexistericantic 7-cube teriprismatotruncated demihepteract (tupthesa) (1,1,3,3,5,5,7) 221760 40320 23 = hexisteriruncic 7-cube teriprismatorhombated demihepteract (tuprohesa) (1,1,1,3,5,5,7) 134400 26880 24 = hexipenticantic 7-cube tericellitruncated demihepteract (tucothesa) (1,1,3,3,3,5,7) 147840 26880 25 = hexipentiruncic 7-cube tericellirhombated demihepteract (tucrohesa) (1,1,1,3,3,5,7) 161280 26880 26 = hexipentisteric 7-cube tericelliprismated demihepteract (tucophesa) (1,1,1,1,3,5,7) 80640 13440 27 = pentisteriruncicantic 7-cube great cellated demihepteract (gochesa) (1,1,3,5,7,9,9) 282240 80640 28 = hexisteriruncicantic 7-cube terigreatoprismated demihepteract (tugphesa) (1,1,3,5,7,7,9) 322560 80640 29 = hexipentiruncicantic 7-cube tericelligreatorhombated demihepteract (tucagrohesa) (1,1,3,5,5,7,9) 322560 80640 30 = hexipentistericantic 7-cube tericelliprismatotruncated demihepteract (tucpathesa) (1,1,3,3,5,7,9) 362880 80640 31 = hexipentisteriruncic 7-cube tericelliprismatorhombated demihepteract (tucprohesa) (1,1,1,3,5,7,9) 241920 53760 32 = hexipentisteriruncicantic 7-cube great terated demihepteract (guthesa) (1,1,3,5,7,9,11) 564480 161280

## The E7 family

The E7 [Coxeter group](/source/Coxeter_group) has order 2,903,040.

There are 127 forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) with one or more rings. Bowers names and acronym are given for cross-referencing.

See also a [list of E7 polytopes](/source/List_of_E7_polytopes) for symmetric Coxeter plane graphs of these polytopes.

E7 uniform polytopes # Coxeter-Dynkin diagram Names Element counts 6 5 4 3 2 1 0 1 231 (laq) 632 4788 16128 20160 10080 2016 126 2 Rectified 231 (rolaq) 758 10332 47880 100800 90720 30240 2016 3 Rectified 132 (rolin) 758 12348 72072 191520 241920 120960 10080 4 132 (lin) 182 4284 23688 50400 40320 10080 576 5 Birectified 321 (branq) 758 12348 68040 161280 161280 60480 4032 6 Rectified 321 (ranq) 758 44352 70560 48384 11592 12096 756 7 321 (naq) 702 6048 12096 10080 4032 756 56 8 Truncated 231 (talq) 758 10332 47880 100800 90720 32256 4032 9 Cantellated 231 (sirlaq) 131040 20160 10 Bitruncated 231 (botlaq) 30240 11 small demified 231 (shilq) 2774 22428 78120 151200 131040 42336 4032 12 demirectified 231 (hirlaq) 12096 13 truncated 132 (tolin) 20160 14 small demiprismated 231 (shiplaq) 20160 15 birectified 132 (berlin) 758 22428 142632 403200 544320 302400 40320 16 tritruncated 321 (totanq) 40320 17 demibirectified 321 (hobranq) 20160 18 small cellated 231 (scalq) 7560 19 small biprismated 231 (sobpalq) 30240 20 small birhombated 321 (sabranq) 60480 21 demirectified 321 (harnaq) 12096 22 bitruncated 321 (botnaq) 12096 23 small terated 321 (stanq) 1512 24 small demicellated 321 (shocanq) 12096 25 small prismated 321 (spanq) 40320 26 small demified 321 (shanq) 4032 27 small rhombated 321 (sranq) 12096 28 Truncated 321 (tanq) 758 11592 48384 70560 44352 12852 1512 29 great rhombated 231 (girlaq) 60480 30 demitruncated 231 (hotlaq) 24192 31 small demirhombated 231 (sherlaq) 60480 32 demibitruncated 231 (hobtalq) 60480 33 demiprismated 231 (hiptalq) 80640 34 demiprismatorhombated 231 (hiprolaq) 120960 35 bitruncated 132 (batlin) 120960 36 small prismated 231 (spalq) 80640 37 small rhombated 132 (sirlin) 120960 38 tritruncated 231 (tatilq) 80640 39 cellitruncated 231 (catalaq) 60480 40 cellirhombated 231 (crilq) 362880 41 biprismatotruncated 231 (biptalq) 181440 42 small prismated 132 (seplin) 60480 43 small biprismated 321 (sabipnaq) 120960 44 small demibirhombated 321 (shobranq) 120960 45 cellidemiprismated 231 (chaplaq) 60480 46 demibiprismatotruncated 321 (hobpotanq) 120960 47 great birhombated 321 (gobranq) 120960 48 demibitruncated 321 (hobtanq) 60480 49 teritruncated 231 (totalq) 24192 50 terirhombated 231 (trilq) 120960 51 demicelliprismated 321 (hicpanq) 120960 52 small teridemified 231 (sethalq) 24192 53 small cellated 321 (scanq) 60480 54 demiprismated 321 (hipnaq) 80640 55 terirhombated 321 (tranq) 60480 56 demicellirhombated 321 (hocranq) 120960 57 prismatorhombated 321 (pranq) 120960 58 small demirhombated 321 (sharnaq) 60480 59 teritruncated 321 (tetanq) 15120 60 demicellitruncated 321 (hictanq) 60480 61 prismatotruncated 321 (potanq) 120960 62 demitruncated 321 (hotnaq) 24192 63 great rhombated 321 (granq) 24192 64 great demified 231 (gahlaq) 120960 65 great demiprismated 231 (gahplaq) 241920 66 prismatotruncated 231 (potlaq) 241920 67 prismatorhombated 231 (prolaq) 241920 68 great rhombated 132 (girlin) 241920 69 celligreatorhombated 231 (cagrilq) 362880 70 cellidemitruncated 231 (chotalq) 241920 71 prismatotruncated 132 (patlin) 362880 72 biprismatorhombated 321 (bipirnaq) 362880 73 tritruncated 132 (tatlin) 241920 74 cellidemiprismatorhombated 231 (chopralq) 362880 75 great demibiprismated 321 (ghobipnaq) 362880 76 celliprismated 231 (caplaq) 241920 77 biprismatotruncated 321 (boptanq) 362880 78 great trirhombated 231 (gatralaq) 241920 79 terigreatorhombated 231 (togrilq) 241920 80 teridemitruncated 231 (thotalq) 120960 81 teridemirhombated 231 (thorlaq) 241920 82 celliprismated 321 (capnaq) 241920 83 teridemiprismatotruncated 231 (thoptalq) 241920 84 teriprismatorhombated 321 (tapronaq) 362880 85 demicelliprismatorhombated 321 (hacpranq) 362880 86 teriprismated 231 (toplaq) 241920 87 cellirhombated 321 (cranq) 362880 88 demiprismatorhombated 321 (hapranq) 241920 89 tericellitruncated 231 (tectalq) 120960 90 teriprismatotruncated 321 (toptanq) 362880 91 demicelliprismatotruncated 321 (hecpotanq) 362880 92 teridemitruncated 321 (thotanq) 120960 93 cellitruncated 321 (catnaq) 241920 94 demiprismatotruncated 321 (hiptanq) 241920 95 terigreatorhombated 321 (tagranq) 120960 96 demicelligreatorhombated 321 (hicgarnq) 241920 97 great prismated 321 (gopanq) 241920 98 great demirhombated 321 (gahranq) 120960 99 great prismated 231 (gopalq) 483840 100 great cellidemified 231 (gechalq) 725760 101 great birhombated 132 (gebrolin) 725760 102 prismatorhombated 132 (prolin) 725760 103 celliprismatorhombated 231 (caprolaq) 725760 104 great biprismated 231 (gobpalq) 725760 105 tericelliprismated 321 (ticpanq) 483840 106 teridemigreatoprismated 231 (thegpalq) 725760 107 teriprismatotruncated 231 (teptalq) 725760 108 teriprismatorhombated 231 (topralq) 725760 109 cellipriemsatorhombated 321 (copranq) 725760 110 tericelligreatorhombated 231 (tecgrolaq) 725760 111 tericellitruncated 321 (tectanq) 483840 112 teridemiprismatotruncated 321 (thoptanq) 725760 113 celliprismatotruncated 321 (coptanq) 725760 114 teridemicelligreatorhombated 321 (thocgranq) 483840 115 terigreatoprismated 321 (tagpanq) 725760 116 great demicellated 321 (gahcnaq) 725760 117 tericelliprismated laq (tecpalq) 483840 118 celligreatorhombated 321 (cogranq) 725760 119 great demified 321 (gahnq) 483840 120 great cellated 231 (gocalq) 1451520 121 terigreatoprismated 231 (tegpalq) 1451520 122 tericelliprismatotruncated 321 (tecpotniq) 1451520 123 tericellidemigreatoprismated 231 (techogaplaq) 1451520 124 tericelligreatorhombated 321 (tacgarnq) 1451520 125 tericelliprismatorhombated 231 (tecprolaq) 1451520 126 great cellated 321 (gocanq) 1451520 127 great terated 321 (gotanq) 2903040

## Regular and uniform honeycombs

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](/source/Coxeter_groups) and sixteen prismatic groups that generate regular and uniform tessellations in 6-space:

# Coxeter group Coxeter diagram Forms 1 A ~ 6 {\displaystyle {\tilde {A}}_{6}} [3[7]] 17 2 C ~ 6 {\displaystyle {\tilde {C}}_{6}} [4,34,4] 71 3 B ~ 6 {\displaystyle {\tilde {B}}_{6}} h[4,34,4] [4,33,31,1] 95 (32 new) 4 D ~ 6 {\displaystyle {\tilde {D}}_{6}} q[4,34,4] [31,1,32,31,1] 41 (6 new) 5 E ~ 6 {\displaystyle {\tilde {E}}_{6}} [32,2,2] 39

Regular and uniform tessellations include:

- A ~ 6 {\displaystyle {\tilde {A}}_{6}} , 17 forms - Uniform [6-simplex honeycomb](/source/6-simplex_honeycomb): {3[7]} - Uniform [Cyclotruncated 6-simplex honeycomb](https://en.wikipedia.org/w/index.php?title=Cyclotruncated_6-simplex_honeycomb&action=edit&redlink=1): t0,1{3[7]} - Uniform [Omnitruncated 6-simplex honeycomb](/source/Omnitruncated_6-simplex_honeycomb): t0,1,2,3,4,5,6,7{3[7]}

- C ~ 6 {\displaystyle {\tilde {C}}_{6}} , [4,34,4], 71 forms - Regular [6-cube honeycomb](/source/6-cube_honeycomb), represented by symbols {4,34,4},

- B ~ 6 {\displaystyle {\tilde {B}}_{6}} , [31,1,33,4], 95 forms, 64 shared with C ~ 6 {\displaystyle {\tilde {C}}_{6}} , 32 new - Uniform [6-demicube honeycomb](/source/6-demicube_honeycomb), represented by symbols h{4,34,4} = {31,1,33,4}, =

- D ~ 6 {\displaystyle {\tilde {D}}_{6}} , [31,1,32,31,1], 41 unique ringed permutations, most shared with B ~ 6 {\displaystyle {\tilde {B}}_{6}} and C ~ 6 {\displaystyle {\tilde {C}}_{6}} , and 6 are new. Coxeter calls the first one a [quarter 6-cubic honeycomb](/source/Quarter_6-cubic_honeycomb). - = - = - = - = - = - =

- E ~ 6 {\displaystyle {\tilde {E}}_{6}} : [32,2,2], 39 forms - Uniform [222 honeycomb](/source/Gosset_2_22_honeycomb): represented by symbols {3,3,32,2}, - Uniform t4(222) honeycomb: 4r{3,3,32,2}, - Uniform 0222 honeycomb: {32,2,2}, - Uniform t2(0222) honeycomb: 2r{32,2,2},

Prismatic groups # Coxeter group Coxeter-Dynkin diagram 1 A ~ 5 {\displaystyle {\tilde {A}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[6],2,∞] 2 B ~ 5 {\displaystyle {\tilde {B}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,3,31,1,2,∞] 3 C ~ 5 {\displaystyle {\tilde {C}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,33,4,2,∞] 4 D ~ 5 {\displaystyle {\tilde {D}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [31,1,3,31,1,2,∞] 5 A ~ 4 {\displaystyle {\tilde {A}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[5],2,∞,2,∞,2,∞] 6 B ~ 4 {\displaystyle {\tilde {B}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,3,31,1,2,∞,2,∞] 7 C ~ 4 {\displaystyle {\tilde {C}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,3,3,4,2,∞,2,∞] 8 D ~ 4 {\displaystyle {\tilde {D}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [31,1,1,1,2,∞,2,∞] 9 F ~ 4 {\displaystyle {\tilde {F}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3,4,3,3,2,∞,2,∞] 10 C ~ 3 {\displaystyle {\tilde {C}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,3,4,2,∞,2,∞,2,∞] 11 B ~ 3 {\displaystyle {\tilde {B}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,31,1,2,∞,2,∞,2,∞] 12 A ~ 3 {\displaystyle {\tilde {A}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[4],2,∞,2,∞,2,∞] 13 C ~ 2 {\displaystyle {\tilde {C}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [4,4,2,∞,2,∞,2,∞,2,∞] 14 H ~ 2 {\displaystyle {\tilde {H}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [6,3,2,∞,2,∞,2,∞,2,∞] 15 A ~ 2 {\displaystyle {\tilde {A}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [3[3],2,∞,2,∞,2,∞,2,∞] 16 I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} [∞,2,∞,2,∞,2,∞,2,∞]

### Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 7, groups that can generate honeycombs with all finite facets, and a finite [vertex figure](/source/Vertex_figure). However, there are [3 paracompact hyperbolic Coxeter groups](/source/Coxeter-Dynkin_diagram#Rank_4_to_10) of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

P ¯ 6 {\displaystyle {\bar {P}}_{6}} = [3,3[6]]: Q ¯ 6 {\displaystyle {\bar {Q}}_{6}} = [31,1,3,32,1]: S ¯ 6 {\displaystyle {\bar {S}}_{6}} = [4,3,3,32,1]:

## Notes on the Wythoff construction for the uniform 7-polytopes

The reflective 7-dimensional [uniform polytopes](/source/Uniform_polytope) are constructed through a [Wythoff construction](/source/Wythoff_construction) process, and represented by a [Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram), where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the [regular polytopes](/source/Regular_polytope) in each family. Some families have two regular constructors and thus may be named in two equally valid ways.

Here are the primary operators available for constructing and naming the uniform 7-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 Coxeter- Dynkin diagram Description Parent t0{p,q,r,s,t,u} Any regular 7-polytope Rectified t1{p,q,r,s,t,u} The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual. Birectified t2{p,q,r,s,t,u} Birectification reduces cells to their duals. Truncated t0,1{p,q,r,s,t,u} 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 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual. Bitruncated t1,2{p,q,r,s,t,u} Bitrunction transforms cells to their dual truncation. Tritruncated t2,3{p,q,r,s,t,u} Tritruncation transforms 4-faces to their dual truncation. Cantellated t0,2{p,q,r,s,t,u} 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 t1,3{p,q,r,s,t,u} 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 t0,3{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges. Biruncinated t1,4{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges. Stericated t0,4{p,q,r,s,t,u} Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps. Pentellated t0,5{p,q,r,s,t,u} Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. Hexicated t0,6{p,q,r,s,t,u} Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes) Omnitruncated t0,1,2,3,4,5,6{p,q,r,s,t,u} All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

## References

1. ^ [***a***](#cite_ref-richeson_1-0) [***b***](#cite_ref-richeson_1-1) [***c***](#cite_ref-richeson_1-2) Richeson, D.; *Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy*, Princeton, 2008.

- [T. Gosset](/source/Thorold_Gosset): *On the Regular and Semi-Regular Figures in Space of n Dimensions*, [Messenger of Mathematics](/source/Messenger_of_Mathematics), Macmillan, 1900

- [A. Boole Stott](/source/Alicia_Boole_Stott) (1910). ["Geometrical deduction of semiregular from regular polytopes and space fillings"](https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF). *Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam*. **XI** (1). Amsterdam: Johannes Müller. Archived from [the original](https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF) on 29 April 2025.

- [H.S.M. Coxeter](/source/Harold_Scott_MacDonald_Coxeter): - H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: *Uniform Polyhedra*, Philosophical Transactions of the Royal Society of London, 1954 - H.S.M. Coxeter, *Regular Polytopes*, 3rd edition, Dover, New York, 1973

- **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, [wiley.com](https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036), [ISBN](/source/ISBN_(identifier)) [978-0-471-01003-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-01003-6) - (Paper 22) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes I*, [Math. Zeit. 46 (1940) 380–407, MR 2,10] - (Paper 23) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559–591] - (Paper 24) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3–45]

- [N.W. Johnson](/source/Norman_Johnson_(mathematician)): *The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966

- Klitzing, Richard. ["7D uniform polytopes (polyexa) with acronyms"](https://bendwavy.org/klitzing/dimensions/polyexa.htm).

## External links

- [Polytope names](http://www.steelpillow.com/polyhedra/ditela.html)

- [Polytopes of Various Dimensions](http://www.polytope.net/hedrondude/topes.htm)

- [Multi-dimensional Glossary](http://tetraspace.alkaline.org/glossary.htm)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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