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In seven-dimensional geometry , a 7-POLYTOPE is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets .

A UNIFORM 7-POLYTOPE is one which is vertex-transitive , and constructed from uniform 6-polytope facets.

CONTENTS

* 1 Regular 7-polytopes * 2 Characteristics * 3 Uniform 7-polytopes by fundamental Coxeter groups * 4 The A7 family * 5 The B7 family * 6 The D7 family * 7 The E7 family

* 8 Regular and uniform honeycombs

* 8.1 Regular and uniform hyperbolic honeycombs

* 9 Notes on the Wythoff construction
Wythoff construction
for the uniform 7-polytopes * 10 References * 11 External links

REGULAR 7-POLYTOPES

Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6-polytopes facets around each 4-face.

There are exactly three such convex regular 7-polytopes :

* {3,3,3,3,3,3} - 7-simplex * {4,3,3,3,3,3} - 7-cube
7-cube
* {3,3,3,3,3,4} - 7-orthoplex

There are no nonconvex regular 7-polytopes.

CHARACTERISTICS

The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients .

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

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.

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 :

# COXETER GROUP REGULAR AND SEMIREGULAR FORMS UNIFORM COUNT

1 A7

* 7-simplex - {36},

71

2 B7

* 7-cube
7-cube
- {4,35}, * 7-orthoplex - {35,4}, * 7-demicube - h{4,35},

127 + 32

3 D7

* 7-demicube , {3,34,1}, * 7-orthoplex , {34,31,1},

95 (0 unique)

4 E7

* 321 - * 132 - * 231 -

127

PRISMATIC FINITE COXETER GROUPS

# COXETER GROUP COXETER DIAGRAM

6+1

1 A6A1 ×

2 BC6A1 ×

3 D6A1 ×

4 E6A1 ×

5+2

1 A5I2(p) ×

2 BC5I2(p) ×

3 D5I2(p) ×

5+1+1

1 A5A12 ×2

2 BC5A12 ×2

3 D5A12 ×2

4+3

1 A4A3 ×

2 A4B3 ×

3 A4H3 ×

4 BC4A3 ×

5 BC4B3 ×

6 BC4H3 ×

7 H4A3 ×

8 H4B3 ×

9 H4H3 ×

10 F4A3 ×

11 F4B3 ×

12 F4H3 ×

13 D4A3 ×

14 D4B3 ×

15 D4H3 ×

4+2+1

1 A4I2(p)A1 ××

2 BC4I2(p)A1 ××

3 F4I2(p)A1 ××

4 H4I2(p)A1 ××

5 D4I2(p)A1 ××

4+1+1+1

1 A4A13 ×3

2 BC4A13 ×3

3 F4A13 ×3

4 H4A13 ×3

5 D4A13 ×3

3+3+1

1 A3A3A1 ××

2 A3B3A1 ××

3 A3H3A1 ××

4 BC3B3A1 ××

5 BC3H3A1 ××

6 H3A3A1 ××

3+2+2

1 A3I2(p)I2(q) ××

2 BC3I2(p)I2(q) ××

3 H3I2(p)I2(q) ××

3+2+1+1

1 A3I2(p)A12 ××2

2 BC3I2(p)A12 ××2

3 H3I2(p)A12 ××2

3+1+1+1+1

1 A3A14 ×4

2 BC3A14 ×4

3 H3A14 ×4

2+2+2+1

1 I2(p)I2(q)I2(r)A1 ×××

2+2+1+1+1

1 I2(p)I2(q)A13 ××3

2+1+1+1+1+1

1 I2(p)A15 ×5

1+1+1+1+1+1+1

1 A17 7

THE A7 FAMILY

The A7 family has symmetry of order 40320 (8 factorial ).

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

See also a list of A7 polytopes for symmetric 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 x 27).

There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Johnson and Bowers names.

See also a list of B7 polytopes for symmetric 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 x 26).

This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 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 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
7-cube
demihepteract (hesa) (1,1,1,1,1,1,1) 78 532 1624 2800 2240 672 64

2 = cantic 7-cube
7-cube
truncated demihepteract (thesa) (1,1,3,3,3,3,3) 142 1428 5656 11760 13440 7392 1344

3 = runcic 7-cube
7-cube
small rhombated demihepteract (sirhesa) (1,1,1,3,3,3,3)

16800 2240

4 = steric 7-cube
7-cube
small prismated demihepteract (sphosa) (1,1,1,1,3,3,3)

20160 2240

5 = pentic 7-cube
7-cube
small cellated demihepteract (sochesa) (1,1,1,1,1,3,3)

13440 1344

6 = hexic 7-cube
7-cube
small terated demihepteract (suthesa) (1,1,1,1,1,1,3)

4704 448

7 = runcicantic 7-cube
7-cube
great rhombated demihepteract (Girhesa) (1,1,3,5,5,5,5)

23520 6720

8 = stericantic 7-cube
7-cube
prismatotruncated demihepteract (pothesa) (1,1,3,3,5,5,5)

73920 13440

9 = steriruncic 7-cube
7-cube
prismatorhomated demihepteract (prohesa) (1,1,1,3,5,5,5)

40320 8960

10 = penticantic 7-cube
7-cube
cellitruncated demihepteract (cothesa) (1,1,3,3,3,5,5)

87360 13440

11 = pentiruncic 7-cube
7-cube
cellirhombated demihepteract (crohesa) (1,1,1,3,3,5,5)

87360 13440

12 = pentisteric 7-cube
7-cube
celliprismated demihepteract (caphesa) (1,1,1,1,3,5,5)

40320 6720

13 = hexicantic 7-cube
7-cube
tericantic demihepteract (tuthesa) (1,1,3,3,3,3,5)

43680 6720

14 = hexiruncic 7-cube
7-cube
terirhombated demihepteract (turhesa) (1,1,1,3,3,3,5)

67200 8960

15 = hexisteric 7-cube
7-cube
teriprismated demihepteract (tuphesa) (1,1,1,1,3,3,5)

53760 6720

16 = hexipentic 7-cube
7-cube
tericellated demihepteract (tuchesa) (1,1,1,1,1,3,5)

21504 2688

17 = steriruncicantic 7-cube
7-cube
great prismated demihepteract (Gephosa) (1,1,3,5,7,7,7)

94080 26880

18 = pentiruncicantic 7-cube
7-cube
celligreatorhombated demihepteract (cagrohesa) (1,1,3,5,5,7,7)

181440 40320

19 = pentistericantic 7-cube
7-cube
celliprismatotruncated demihepteract (capthesa) (1,1,3,3,5,7,7)

181440 40320

20 = pentisteriruncic 7-cube
7-cube
celliprismatorhombated demihepteract (coprahesa) (1,1,1,3,5,7,7)

120960 26880

21 = hexiruncicantic 7-cube
7-cube
terigreatorhombated demihepteract (tugrohesa) (1,1,3,5,5,5,7)

120960 26880

22 = hexistericantic 7-cube
7-cube
teriprismatotruncated demihepteract (tupthesa) (1,1,3,3,5,5,7)

221760 40320

23 = hexisteriruncic 7-cube
7-cube
teriprismatorhombated demihepteract (tuprohesa) (1,1,1,3,5,5,7)

134400 26880

24 = hexipenticantic 7-cube
7-cube
teriCellitruncated demihepteract (tucothesa) (1,1,3,3,3,5,7)

147840 26880

25 = hexipentiruncic 7-cube
7-cube
tericellirhombated demihepteract (tucrohesa) (1,1,1,3,3,5,7)

161280 26880

26 = hexipentisteric 7-cube
7-cube
tericelliprismated demihepteract (tucophesa) (1,1,1,1,3,5,7)

80640 13440

27 = pentisteriruncicantic 7-cube
7-cube
great cellated demihepteract (gochesa) (1,1,3,5,7,9,9)

282240 80640

28 = hexisteriruncicantic 7-cube
7-cube
terigreatoprimated demihepteract (tugphesa) (1,1,3,5,7,7,9)

322560 80640

29 = hexipentiruncicantic 7-cube
7-cube
tericelligreatorhombated demihepteract (tucagrohesa) (1,1,3,5,5,7,9)

322560 80640

30 = hexipentistericantic 7-cube
7-cube
tericelliprismatotruncated demihepteract (tucpathesa) (1,1,3,3,5,7,9)

362880 80640

31 = hexipentisteriruncic 7-cube
7-cube
tericellprismatorhombated demihepteract (tucprohesa) (1,1,1,3,5,7,9)

241920 53760

32 = hexipentisteriruncicantic 7-cube
7-cube
great terated demihepteract (guthesa) (1,1,3,5,7,9,11)

564480 161280

THE E7 FAMILY

The E7 Coxeter group has order 2,903,040.

There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.

E7 UNIFORM POLYTOPES

# Coxeter-Dynkin diagram Schläfli symbol 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 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}} ]

17

2 C 6 {displaystyle {tilde {C}}_{6}}

71

3 B 6 {displaystyle {tilde {B}}_{6}} h

95 (32 new)

4 D 6 {displaystyle {tilde {D}}_{6}} q

41 (6 new)

5 E 6 {displaystyle {tilde {E}}_{6}}

39

Regular and uniform tessellations include:

* A 6 {displaystyle {tilde {A}}_{6}} , 17 forms

* Uniform 6-simplex honeycomb : {3} * Uniform Cyclotruncated 6-simplex honeycomb : t0,1{3} * Uniform Omnitruncated 6-simplex honeycomb : t0,1,2,3,4,5,6,7{3}

* C 6 {displaystyle {tilde {C}}_{6}} , , 71 forms

* Regular 6-cube honeycomb , represented by symbols {4,34,4},

* B 6 {displaystyle {tilde {B}}_{6}} , , 95 forms, 64 shared with C 6 {displaystyle {tilde {C}}_{6}} , 32 new

* Uniform 6-demicube honeycomb , represented by symbols h{4,34,4} = {31,1,33,4}, =

* D 6 {displaystyle {tilde {D}}_{6}} , , 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 .

* = * = * = * = * = * =

* E 6 {displaystyle {tilde {E}}_{6}} : , 39 forms

* Uniform 222 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}} ,2,∞]

2 B 5 {displaystyle {tilde {B}}_{5}} x I 1 {displaystyle {tilde {I}}_{1}}

3 C 5 {displaystyle {tilde {C}}_{5}} x I 1 {displaystyle {tilde {I}}_{1}}

4 D 5 {displaystyle {tilde {D}}_{5}} x I 1 {displaystyle {tilde {I}}_{1}}

5 A 4 {displaystyle {tilde {A}}_{4}} x I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}} ,2,∞,2,∞,2,∞]

6 B 4 {displaystyle {tilde {B}}_{4}} x I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}}

7 C 4 {displaystyle {tilde {C}}_{4}} x I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}}

8 D 4 {displaystyle {tilde {D}}_{4}} x I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}}

9 F 4 {displaystyle {tilde {F}}_{4}} x I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}}

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}}

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}}

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}} ,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}}

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}}

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}} ,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}}

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 . However, there are 3 noncompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

P 6 {displaystyle {bar {P}}_{6}} = ]: Q 6 {displaystyle {bar {Q}}_{6}} = : S 6 {displaystyle {bar {S}}_{6}} = :

NOTES ON THE WYTHOFF CONSTRUCTION FOR THE UNIFORM 7-POLYTOPES

The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction
Wythoff construction
process, and represented by a 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 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 half way 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