## List of verfs of CRF polyhedra

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### List of verfs of CRF polyhedra

Some of the information here is already given in Marek's and johannes' lists here, here, and here (and perhaps elsewhere). But the first link doesn't give vertex configurations, the second doesn't give dihedral angles, and the third only gives pentavalent vertices. So I'm making my own complete list here.

The vertex configuration (which is topological, with no regard for angles) is given first in bold. Below it are the possible vertex figures. Braces {} are for separation of face sizes and dihedral angles (and may also be interpreted as Schlafli symbols). After the verf is the solid angle, underlined; for comparison, a whole sphere is 720°. After that are the polyhedra which the verf appears in, specified by Johnson index, or Bowers acronym for a uniform polyhedron.

First let's get the odd polygons out of the way so we can focus on 3,4,5,6,8,10 later. We should certainly consider prisms up to at least n=20, because of the augmented 5,20-duoprism. (And johannes said something about n=169, but it wasn't clear.) I'm going up to n=24, and down to n=5 for some kind of completeness. The verc 5.3.3.3 only appears with the angles of an antiprism, but 4.3.3.3 and 3.3.3.3 appear with other angles, so we won't consider them yet. The vercs with n=5,6,8,10 will be listed twice: once in this post and once in the next post.

A prism's angles are obvious: 90° (or π/2 radians), and the angle of the polygon itself, (n - 2)π/n.

24.4.4
{24} 90° {4} 165° {4} 90°; 165°; 24-gonal prism

23.4.4
{23} 90° {4} 164.3478° {4} 90°; 164.3478°; 23-gonal prism

22.4.4
{22} 90° {4} 163.6364° {4} 90°; 163.6364°; 22-gonal prism

21.4.4
{21} 90° {4} 162.8571° {4} 90°; 162.8571°; 21-gonal prism

20.4.4
{20} 90° {4} 162° {4} 90°; 162°; 20-gonal prism

19.4.4
{19} 90° {4} 161.0526° {4} 90°; 161.0526°; 19-gonal prism

18.4.4
{18} 90° {4} 160° {4} 90°; 160°; 18-gonal prism

17.4.4
{17} 90° {4} 158.8235° {4} 90°; 158.8235°; 17-gonal prism

16.4.4
{16} 90° {4} 157.5° {4} 90°; 157.5°; 16-gonal prism

15.4.4
{15} 90° {4} 156° {4} 90°; 156°; 15-gonal prism

14.4.4
{14} 90° {4} 154.2857° {4} 90°; 154.2857°; 14-gonal prism

13.4.4
{13} 90° {4} 152.3077° {4} 90°; 152.3077°; 13-gonal prism

12.4.4
{12} 90° {4} 150° {4} 90°; 150°; 12-gonal prism

11.4.4
{11} 90° {4} 147.2727° {4} 90°; 147.2727°; 11-gonal prism

10.4.4
{10} 90° {4} 144° {4} 90°; 144°; dip, J20-21

9.4.4
{9} 90° {4} 140° {4} 90°; 140°; 9-gonal prism

8.4.4
{8} 90° {4} 135° {4} 90°; 135°; op, J19

7.4.4
{7} 90° {4} 128.5714° {4} 90°; 128.5714°; 7-gonal prism

6.4.4
{6} 90° {4} 120° {4} 90°; 120°; hip, J18, 54-56

5.4.4
{5} 90° {4} 108° {4} 90°; 108°; pip, J9, 52-53

An antiprism's dihedral angles are less obvious; we have the formulas

cos θn3 = (-1/√3) (sin π/n) / (1 + cos π/n),

cos θ33 = (-1/3) (4 cos π/n - 1).

So here are the antiprism verfs:

24.3.3.3
{24} 92.1687° {3} 171.3377° {3} 171.3377° {3} 92.1687°; 167.0127°; 24-gonal antiprism

23.3.3.3
{23} 92.2633° {3} 170.9609° {3} 170.9609° {3} 92.2633°; 166.4483°; 23-gonal antiprism

22.3.3.3
{22} 92.3666° {3} 170.5498° {3} 170.5498° {3} 92.3666°; 165.8327°; 22-gonal antiprism

21.3.3.3
{21} 92.4798° {3} 170.0995° {3} 170.0995° {3} 92.4798°; 165.1585°; 21-gonal antiprism

20.3.3.3
{20} 92.6043° {3} 169.6041° {3} 169.6041° {3} 92.6043°; 164.4169°; 20-gonal antiprism

19.3.3.3
{19} 92.7421° {3} 169.0566° {3} 169.0566° {3} 92.7421°; 163.5973°; 19-gonal antiprism

18.3.3.3
{18} 92.8953° {3} 168.4481° {3} 168.4481° {3} 92.8953°; 162.6868°; 18-gonal antiprism

17.3.3.3
{17} 93.0668° {3} 167.7679° {3} 167.7679° {3} 93.0668°; 161.6694°; 17-gonal antiprism

16.3.3.3
{16} 93.2598° {3} 167.0026° {3} 167.0026° {3} 93.2598°; 160.5249°; 16-gonal antiprism

15.3.3.3
{15} 93.4790° {3} 166.1351° {3} 166.1351° {3} 93.4790°; 159.2281°; 15-gonal antiprism

14.3.3.3
{14} 93.7298° {3} 165.1434° {3} 165.1434° {3} 93.7298°; 157.7464°; 14-gonal antiprism

13.3.3.3
{13} 94.0199° {3} 163.9988° {3} 163.9988° {3} 94.0199°; 156.0373°; 13-gonal antiprism

12.3.3.3
{12} 94.3592° {3} 162.6628° {3} 162.6628° {3} 94.3592°; 154.0442°; 12-gonal antiprism

11.3.3.3
{11} 94.7616° {3} 161.0833° {3} 161.0833° {3} 94.7616°; 151.6898°; 11-gonal antiprism

10.3.3.3
{10} 95.2466° {3} 159.1865° {3} 159.1865° {3} 95.2466°; 148.8663°; dap, J24-25

9.3.3.3
{9} 95.8430° {3} 156.8662° {3} 156.8662° {3} 95.8430°; 145.4184°; 9-gonal antiprism

8.3.3.3
{8} 96.5945° {3} 153.9624° {3} 153.9624° {3} 96.5945°; 141.1138°; oap, J23

7.3.3.3
{7} 97.5723° {3} 150.2223° {3} 150.2223° {3} 97.5723°; 135.5890°; 7-gonal antiprism

6.3.3.3
{6} 98.8994° {3} 145.2219° {3} 145.2219° {3} 98.8994°; 128.2426°; hap, J22

5.3.3.3
{5} 100.8123° {3} 138.1897° {3} 138.1897° {3} 100.8123°; 118.0040°; pap, J11, 62-64, 92
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mr_e_man
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Posts: 531
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### Re: List of verfs of CRF polyhedra

Now we consider only triangles, squares, pentagons, hexagons, octagons, and decagons.

For a vertex with face angles a,b,c, the dihedral angle between faces a and b is given by

cos θab = (cos c - cos a cos b) / (sin a sin b),

a form of the spherical law of cosines. We use this to calculate the angles at trivalent vertices:

10.10.3
{10} 116.5651° {10} 142.6226° {3} 142.6226°; 221.8103°; tid, J68-71

10.6.4
{10} 142.6226° {6} 159.0948° {4} 148.2825°; 270°; grid

10.5.4
{10} 116.5651° {5} 148.2825° {4} 121.7175°; 206.5651°; J76-83

10.5.3
{10} 63.4349° {5} 142.6226° {3} 79.1877°; 105.2453°; J6

10.4.4
{10} 90° {4} 144° {4} 90°; 144°; dip, J20-21

10.4.3
{10} 31.7175° {4} 159.0948° {3} 37.3774°; 48.1897°; J5

8.8.3
{8} 90° {8} 125.2644° {3} 125.2644°; 160.5288°; tic, J66-67

8.6.4
{8} 125.2644° {6} 144.7356° {4} 135°; 225°; girco

8.4.4
{8} 90° {4} 135° {4} 90°; 135°; op, J19

8.4.3
{8} 45° {4} 144.7356° {3} 54.7356°; 64.4712°; J4

6.6.5
{6} 138.1897° {6} 142.6226° {5} 142.6226°; 243.4349°; ti

6.6.4
{6} 109.4712° {6} 125.2644° {4} 125.2644°; 180°; toe

6.6.3
{6} 70.5288° {6} 109.4712° {3} 109.4712°; 109.4712°; tut, J65

6.4.4
{6} 90° {4} 120° {4} 90°; 120°; hip, J18, 54-56

6.4.3
{6} 54.7356° {4} 125.2644° {3} 70.5288°; 70.5288°; J3

5.5.5
{5} 116.5651° {5} 116.5651° {5} 116.5651°; 169.6952°; doe, J58-61

5.5.3
{5} 63.4349° {5} 100.8123° {3} 100.8123°; 85.0596°; J62-64, 91

5.4.4
{5} 90° {4} 108° {4} 90°; 108°; pip, J9, 52-53

5.3.3
{5} 37.3774° {3} 138.1897° {3} 37.3774°; 32.9444°; J2

4.4.4
{4} 90° {4} 90° {4} 90°; 90°; cube, J8

4.4.3
{4} 60° {4} 90° {3} 90°; 60°; trip, J7, 26, 49

4.3.3
{4} 54.7356° {3} 109.4712° {3} 54.7356°; 38.9424°; J1

3.3.3
{3} 70.5288° {3} 70.5288° {3} 70.5288°; 31.5863°; tet, J7, 12, 14, 64

Next are the tetravalent vertices. Most of these are combinations of trivalent vertices; this will be indicated after the verf.

10.3.4.3
{10} 142.6226° {3} 174.3401° {4} 159.0948° {3} 153.9424° (10.3.10 + 10.4.3); 270°; J68-71

10.3.3.3
{10} 95.2466° {3} 159.1865° {3} 159.1865° {3} 95.2466°; 148.8663°; dap, J24-25

8.3.4.3
{8} 125.2644° {3} 170.2644° {4} 144.7356° {3} 144.7356° (8.3.8 + 8.4.3); 225°; J66-67

8.3.3.3
{8} 96.5945° {3} 153.9624° {3} 153.9624° {3} 96.5945°; 141.1138°; oap, J23

6.4.3.3
A: {6} 90° {4} 174.7356° {3} 109.4712° {3} 144.7356° (6.4.4 + 4.3.3); 158.9424°; J54-57
B: {6} 110.9052° {4} 159.0948° {3} 138.1897° {3} 138.1897° (6.4.5 + 5.3.3) (4.3.10 + 10.3.6); 186.3794°; J92

6.3.4.3
{6} 109.4712° {3} 164.2068° {4} 125.2644° {3} 141.0576° (6.3.6 + 6.4.3); 180°; J65

6.3.3.3
{6} 98.8994° {3} 145.2219° {3} 145.2219° {3} 98.8994°; 128.2426°; hap, J22

5.5.3.3
A: {5} 126.8699° {5} 142.6226° {3} 158.3754° {3} 142.6226° (5.3.10 + 10.3.5); 210.4905°; J34
B: {5} 116.5651° {5} 153.9424° {3} 138.1897° {3} 153.9424° (5.5.5 + 5.3.3); 202.6396°; J58-61
C: {5} 63.4349° {5} 171.3411° {3} 70.5288° {3} 171.3411° (5.5.3 + 3.3.3); 116.6459°; J64

5.3.5.3
{5} 142.6226° {3} 142.6226° {5} 142.6226° {3} 142.6226° (5.3.10 + 10.5.3); 210.4905°; id, J6, 21, 25, 32-34, 40-43, 47-48, 91-92

5.4.4.3
A: {5} 153.4349° {4} 144° {4} 169.1877° {3} 142.6226° (5.3.10 + 10.4.4); 249.2453°; J21, 40-43
B: {5} 148.2825° {4} 153.4349° {4} 159.0948° {3} 153.9424° (5.4.10 + 10.4.3); 254.7547°; J72-75, 77-79, 82

5.4.3.4
{5} 148.2825° {4} 159.0948° {3} 159.0948° {4} 148.2825° (5.4.10 + 10.3.4); 254.7547°; srid, J5, 20, 24, 30-33, 38-41, 46-47, 68-83

5.4.3.3
A: {5} 95.1524° {4} 159.0948° {3} 116.5651° {3} 142.6226° (5.3.10 + 10.3.4); 153.4349°; J33
B: {5} 90° {4} 162.7356° {3} 109.4712° {3} 144.7356° (5.4.4 + 4.3.3); 146.9424°; J52-53

5.3.4.3
{5} 142.6226° {3} 110.9052° {4} 159.0948° {3} 100.8123° (5.3.10 + 10.4.3) (3.4.5 + 5.3.5); 153.4349°; J32, 91-92

5.3.3.3
{5} 100.8123° {3} 138.1897° {3} 138.1897° {3} 100.8123° (5.3.5 + 5.3.3); 118.0040°; pap, J11, 62-64, 92

4.4.4.3
A: {4} 135° {4} 135° {4} 144.7356° {3} 144.7356° (4.4.8 + 8.4.3); 199.4712°; sirco, J4, 19, 23, 28-29, 37, 45, 66-67
B: {4} 120° {4} 144.7356° {4} 125.2644° {3} 160.5288° (4.4.6 + 6.4.3); 190.5288°; J18, 35-36
C: {4} 144° {4} 121.7175° {4} 159.0948° {3} 127.3774° (4.4.10 + 10.4.3); 192.1897°; J20, 38-41

4.4.3.3
A: {4} 60° {4} 160.5288° {3} 70.5288° {3} 160.5288° (4.4.3 + 3.3.3); 91.5863°; J7, 14
B: {4} 90° {4} 144.7356° {3} 109.4712° {3} 144.7356° (4.4.4 + 4.3.3) (4.3.8 + 8.3.4); 128.9424°; J8, 15, 28
C: {4} 108° {4} 127.3774° {3} 138.1897° {3} 127.3774° (4.4.5 + 5.3.3); 140.9444°; J9, 16
D: {4} 109.4712° {4} 125.2644° {3} 141.0576° {3} 125.2644° (4.3.6 + 6.3.4); 141.0576°; J27
E: {4} 63.4349° {4} 159.0948° {3} 74.7547° {3} 159.0948° (4.3.10 + 10.3.4); 96.3794°; J30
F: {4} 117.0190° {4} 109.5240° {3} 159.8924° {3} 109.5240°; 135.9595°; J86
G: {4} 72.9730° {4} 154.7223° {3} 86.7268° {3} 154.7223°; 109.1444°; J88
H: {4} 102.5238° {4} 133.9728° {3} 128.4960° {3} 133.9728°; 138.9654°; J89
I: {4} 100.1939° {4} 136.3359° {3} 124.7019° {3} 136.3359°; 137.5677°; J90

4.3.4.3
A: {4} 125.2644° {3} 125.2644° {4} 125.2644° {3} 125.2644° (4.3.6 + 6.4.3); 141.0576°; co, J3, 18, 22, 27, 35-36, 44, 65
B: {4} 90° {3} 150° {4} 90° {3} 150° (4.3.4 + 4.4.3); 120°; J26
C: {4} 144.7356° {3} 99.7356° {4} 144.7356° {3} 99.7356° (4.3.8 + 8.4.3); 128.9424°; J29
D: {4} 159.0948° {3} 69.0948° {4} 159.0948° {3} 69.0948° (4.3.10 + 10.4.3); 96.3794°; J31

4.3.3.3
A: {4} 103.8362° {3} 127.5516° {3} 127.5516° {3} 103.8362°; 102.7755°; squap, J10
B: {4} 90° {3} 144.7356° {3} 109.4712° {3} 114.7356° (4.3.4 + 4.3.3); 98.9424°; J49-50
C: {4} 97.4555° {3} 135.9915° {3} 118.8922° {3} 109.5240°; 101.8633°; J86-87

3.3.3.3
A: {3} 109.4712° {3} 109.4712° {3} 109.4712° {3} 109.4712° (3.3.4 + 4.3.3); 77.8849°; oct, J1, 8, 10, 15, 17, 49-57, 87
B: {3} 70.5288° {3} 141.0576° {3} 70.5288° {3} 141.0576° (3.3.3 + 3.3.3); 63.1727°; J12
C: {3} 138.1897° {3} 74.7547° {3} 138.1897° {3} 74.7547° (3.3.5 + 5.3.3); 65.8888°; J13
D: {3} 96.1983° {3} 121.7432° {3} 96.1983° {3} 121.7432°; 75.8830°; J84
E: {3} 86.7268° {3} 129.4446° {3} 86.7268° {3} 129.4446°; 72.3428°; J88

(See here for the locations of these angles in J84-90.)

And finally the pentavalent vertices:

5.3.3.3.3
A: {5} 152.9299° {3} 164.1754° {3} 164.1754° {3} 164.1754° {3} 152.9299°; 258.3859°; snid
B: {5} 142.6226° {3} 174.4343° {3} 159.1865° {3} 159.1865° {3} 158.6816° (5.3.10 + 10.3.3.3); 254.1116°; J25, 47-48

4.3.3.3.3
A: {4} 142.9834° {3} 153.2346° {3} 153.2346° {3} 153.2346° {3} 142.9834°; 205.6706°; snic
B: {4} 125.2644° {3} 169.4282° {3} 145.2219° {3} 145.2219° {3} 153.6350° (4.3.6 + 6.3.3.3); 198.7714°; J22, 44
C: {4} 144.7356° {3} 151.3301° {3} 153.9624° {3} 153.9624° {3} 141.5945° (4.3.8 + 8.3.3.3); 205.5850°; J23, 45
D: {4} 159.0948° {3} 132.6240° {3} 159.1865° {3} 159.1865° {3} 126.9641° (4.3.10 + 10.3.3.3); 197.0560°; J24, 46-47
E: {4} 145.4406° {3} 144.1436° {3} 164.2574° {3} 144.1436° {3} 145.4406°; 203.4259°; J85
F: {4} 171.7546° {3} 109.4712° {3} 164.2596° {3} 159.8924° {3} 109.5240° (3.3.4 + 4.3.3.4 F); 174.9019°; J87
G: {4} 137.2401° {3} 143.7383° {3} 171.6457° {3} 129.4446° {3} 154.7223°; 196.7910°; J88
H: {4} 152.9756° {3} 141.3411° {3} 157.1481° {3} 157.1481° {3} 133.9728°; 202.5858°; J89
I: {4} 154.4188° {3} 133.5912° {3} 166.8114° {3} 148.4340° {3} 136.3359°; 199.5913°; J90

3.3.3.3.3
A: {3} 138.1897° {3} 138.1897° {3} 138.1897° {3} 138.1897° {3} 138.1897° (3.3.5 + 5.3.5 + 5.3.3); 150.9484°; ike, J2, 9, 11, 13, 16, 58-62
B: {3} 109.4712° {3} 158.5718° {3} 127.5516° {3} 127.5516° {3} 158.5718° (3.3.4 + 4.3.3.3 A); 141.7180°; J10, 17
C: {3} 109.4712° {3} 144.7356° {3} 144.7356° {3} 109.4712° {3} 169.4712° (3.3.4 + 4.3.4 + 4.3.3); 137.8849°; J50-51
D: {3} 96.1983° {3} 166.4406° {3} 121.7432° {3} 121.7432° {3} 166.4406°; 132.5657°; J84
E: {3} 164.2574° {3} 114.6452° {3} 144.1436° {3} 144.1436° {3} 114.6452°; 141.8351°; J85
F: {3} 159.8924° {3} 118.8922° {3} 143.4787° {3} 143.4787° {3} 118.8922°; 144.6343°; J86-87
G: {3} 131.4416° {3} 143.4787° {3} 135.9915° {3} 135.9915° {3} 143.4787°; 150.3820°; J86-87
H: {3} 152.1911° {3} 109.4712° {3} 164.2596° {3} 118.8922° {3} 135.9915° (3.3.4 + 4.3.3.3 C); 140.8057°; J87
I: {3} 86.7268° {3} 171.6457° {3} 117.3556° {3} 117.3556° {3} 171.6457°; 124.7294°; J88
J: {3} 161.4828° {3} 117.3556° {3} 143.7383° {3} 143.7383° {3} 117.3556°; 143.6706°; J88
K: {3} 111.7348° {3} 157.1481° {3} 128.4960° {3} 128.4960° {3} 157.1481°; 143.0230°; J89
L: {3} 149.5648° {3} 128.4960° {3} 141.3411° {3} 141.3411° {3} 128.4960°; 149.2390°; J89
M: {3} 124.7019° {3} 148.4340° {3} 133.5912° {3} 133.5912° {3} 148.4340°; 148.7523°; J90
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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mr_e_man
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Posts: 531
Joined: Tue Sep 18, 2018 4:10 am

### Re: List of verfs of CRF polyhedra

Some of these verfs can be broken into smaller verfs in interesting ways.

Here I'm just considering polygons on a sphere. A verf decomposing doesn't imply that the polyhedron decomposes. For example, the square orthobicupola's vertex 4.4.3.3 decomposes as 4.4.4 + 4.3.3, even though squobcu itself doesn't decompose into anything with cubes and pyramids (without introducing non-CRF polyhedra).

The decompositions in the list above, in parentheses, only involve cutting a line straight through the polygon. Here are some decompositions involving cutting partway through, to a point inside the polygon:

10.10.3 = 4.10.3 + 3.10.5 + 5.3.4

10.5.4 = 5.10.3 + 3.5.3 + 3.4.5

8.8.3 = 3.8.4 + 4.8.3 + 3.3.3

8.6.4 = 4.8.3 + 3.6.4 + 4.4.4

6.6.4 = 3.6.4 + 4.6.3 + 3.4.3

5.5.5 = 3.5.4 + 4.5.3 + 3.5.3

4.3.4.3 A = 3.4.3 + 3.3.3 + 3.4.3 + 3.3.3

(Of course some of these were already known. See viewtopic.php?p=27629#p27629 .)

Those keep the original edges intact. Here are some decompositions involving splitting a 120° edge (for a hexagon) into two 60° edges (for triangles):

10.6.4 = 10.3.10 + 10.3.4

10.6.3 = 10.3.5 + 5.3.3
(relevant to thawro; see 6.4.3.3 B)

8.6.4 = 8.3.8 + 8.3.4

6.6.5 = 6.3.10 + 10.3.5 = 3.3.5 + 5.3.5.3

6.6.4 = 6.3.6 + 6.3.4 = 3.3.4 + 4.3.4.3 A

6.6.3 = 6.3.4 + 4.3.3 = 3.3.3 + 3.3.3.3 A

6.5.4 = 3.5.5 + 5.4.3
(relevant to thawro)

6.4.4 = 3.4.4 + 4.4.3

6.4.3 = 3.4.3 + 3.3.3

In fact anything with a 120° edge can be decomposed in this way, except 6.3.3.3 (for an antiprism).
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mr_e_man
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### Re: List of verfs of CRF polyhedra

So here's a list of verfs which can't be decomposed, except by splitting 120° edges or using false verfs like 5.4.3 . The polyhedra will not be listed; it's not clear whether 4.4.4 appears in squobcu, for example.

10.10.3
{10} 116.5651° {10} 142.6226° {3} 142.6226° (4.10.3 + 3.10.5 + 5.3.4); 221.8103°

10.6.4
{10} 142.6226° {6} 159.0948° {4} 148.2825° (10.3.10 + 10.3.4) (3.10.4 + 4.3.5 + 5.10.3 + 10.3.4); 270°

10.5.4
{10} 116.5651° {5} 148.2825° {4} 121.7175° (5.10.3 + 3.5.3 + 3.4.5); 206.5651°

10.5.3
{10} 63.4349° {5} 142.6226° {3} 79.1877°; 105.2453°

10.4.4
{10} 90° {4} 144° {4} 90°; 144°

10.4.3
{10} 31.7175° {4} 159.0948° {3} 37.3774°; 48.1897°

8.4.4
{8} 90° {4} 135° {4} 90°; 135°

8.4.3
{8} 45° {4} 144.7356° {3} 54.7356°; 64.4712°

6.6.5
{6} 138.1897° {6} 142.6226° {5} 142.6226° (6.3.10 + 10.3.5) (3.3.5 + 5.10.3 + 10.3.5); 243.4349°

6.6.3
{6} 70.5288° {6} 109.4712° {3} 109.4712° (3.3.3 + 3.4.3 + 4.3.3); 109.4712°

6.4.4
{6} 90° {4} 120° {4} 90° (3.4.4 + 4.4.3); 120°

6.4.3
{6} 54.7356° {4} 125.2644° {3} 70.5288° (3.4.3 + 3.3.3); 70.5288°

5.5.5
{5} 116.5651° {5} 116.5651° {5} 116.5651° (3.5.4 + 4.5.3 + 3.5.3); 169.6952°

5.5.3
{5} 63.4349° {5} 100.8123° {3} 100.8123°; 85.0596°

5.4.4
{5} 90° {4} 108° {4} 90°; 108°

5.3.3
{5} 37.3774° {3} 138.1897° {3} 37.3774°; 32.9444°

4.4.4
{4} 90° {4} 90° {4} 90°; 90°

4.4.3
{4} 60° {4} 90° {3} 90°; 60°

4.3.3
{4} 54.7356° {3} 109.4712° {3} 54.7356°; 38.9424°

3.3.3
{3} 70.5288° {3} 70.5288° {3} 70.5288°; 31.5863°

10.3.3.3
{10} 95.2466° {3} 159.1865° {3} 159.1865° {3} 95.2466°; 148.8663°

8.3.3.3
{8} 96.5945° {3} 153.9624° {3} 153.9624° {3} 96.5945°; 141.1138°

6.4.3.3
B: {6} 110.9052° {4} 159.0948° {3} 138.1897° {3} 138.1897° (6.4.5 + 5.3.3) (3.4.5 + 5.5.3 + 5.3.3) (4.3.10 + 10.3.6) (4.3.10 + 10.5.3 + 5.3.3); 186.3794°

6.3.3.3
{6} 98.8994° {3} 145.2219° {3} 145.2219° {3} 98.8994°; 128.2426°

4.4.3.3
F: {4} 117.0190° {4} 109.5240° {3} 159.8924° {3} 109.5240°; 135.9595°
G: {4} 72.9730° {4} 154.7223° {3} 86.7268° {3} 154.7223°; 109.1444°
H: {4} 102.5238° {4} 133.9728° {3} 128.4960° {3} 133.9728°; 138.9654°
I: {4} 100.1939° {4} 136.3359° {3} 124.7019° {3} 136.3359°; 137.5677°

4.3.3.3
A: {4} 103.8362° {3} 127.5516° {3} 127.5516° {3} 103.8362°; 102.7755°
C: {4} 97.4555° {3} 135.9915° {3} 118.8922° {3} 109.5240°; 101.8633°

3.3.3.3
D: {3} 96.1983° {3} 121.7432° {3} 96.1983° {3} 121.7432°; 75.8830°
E: {3} 86.7268° {3} 129.4446° {3} 86.7268° {3} 129.4446°; 72.3428°

5.3.3.3.3
A: {5} 152.9299° {3} 164.1754° {3} 164.1754° {3} 164.1754° {3} 152.9299°; 258.3859°

4.3.3.3.3
A: {4} 142.9834° {3} 153.2346° {3} 153.2346° {3} 153.2346° {3} 142.9834°; 205.6706°
E: {4} 145.4406° {3} 144.1436° {3} 164.2574° {3} 144.1436° {3} 145.4406°; 203.4259°
G: {4} 137.2401° {3} 143.7383° {3} 171.6457° {3} 129.4446° {3} 154.7223°; 196.7910°
H: {4} 152.9756° {3} 141.3411° {3} 157.1481° {3} 157.1481° {3} 133.9728°; 202.5858°
I: {4} 154.4188° {3} 133.5912° {3} 166.8114° {3} 148.4340° {3} 136.3359°; 199.5913°

3.3.3.3.3
D: {3} 96.1983° {3} 166.4406° {3} 121.7432° {3} 121.7432° {3} 166.4406°; 132.5657°
E: {3} 164.2574° {3} 114.6452° {3} 144.1436° {3} 144.1436° {3} 114.6452°; 141.8351°
F: {3} 159.8924° {3} 118.8922° {3} 143.4787° {3} 143.4787° {3} 118.8922°; 144.6343°
G: {3} 131.4416° {3} 143.4787° {3} 135.9915° {3} 135.9915° {3} 143.4787°; 150.3820°
I: {3} 86.7268° {3} 171.6457° {3} 117.3556° {3} 117.3556° {3} 171.6457°; 124.7294°
J: {3} 161.4828° {3} 117.3556° {3} 143.7383° {3} 143.7383° {3} 117.3556°; 143.6706°
K: {3} 111.7348° {3} 157.1481° {3} 128.4960° {3} 128.4960° {3} 157.1481°; 143.0230°
L: {3} 149.5648° {3} 128.4960° {3} 141.3411° {3} 141.3411° {3} 128.4960°; 149.2390°
M: {3} 124.7019° {3} 148.4340° {3} 133.5912° {3} 133.5912° {3} 148.4340°; 148.7523°

It should be somewhat easier to build spherical polyhedra (for polychoron verfs) with this smaller set of polygons. If some dihedral angles end up being exactly 180°, then we can combine those faces into a single face (provided that it exists in the longer list above; otherwise we discard the polyhedron). Often we can't tell if it's slightly greater or less than 180°; this step (combining coplanar faces) can be delayed, or it may not be necessary at all, if that polyhedron doesn't work as a verf for some other reason.
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mr_e_man
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Posts: 531
Joined: Tue Sep 18, 2018 4:10 am

### Re: List of verfs of CRF polyhedra

Here's my complete list of dihedral angles, in polyhedra with 3,4,5,6,8,10-gon faces. This shows which verfs the angle appears in, along with the relevant pairs of faces (in bold).

31.7175°: 10.4.3
37.3774°: 10.4.3, 5.3.3
45.0000°: 8.4.3
54.7356°: 8.4.3, 6.4.3, 4.3.3
60.0000°: 4.4.3, 4.4.3.3 A
63.4349°: 10.5.3, 5.5.3, 5.5.3.3 C, 4.4.3.3 E
69.0948°: 4.3.4.3 D
70.5288°: 6.6.3, 6.4.3, 3.3.3, 5.5.3.3 C, 4.4.3.3 A, 3.3.3.3 B
72.9730°: 4.4.3.3 G
74.7547°: 4.4.3.3 E, 3.3.3.3 C
79.1877°: 10.5.3
86.7268°: 4.4.3.3 G, 3.3.3.3 E, 3.3.3.3.3 I
90.0000°: 10.4.4, 8.8.3, 8.4.4, 6.4.4, 5.4.4, 4.4.4, 4.4.3, 6.4.3.3 A, 5.4.3.3 B, 4.4.3.3 B, 4.3.4.3 B, 4.3.3.3 B
95.1524°: 5.4.3.3 A
95.2466°: 10.3.3.3
96.1983°: 3.3.3.3 D, 3.3.3.3.3 D
96.5945°: 8.3.3.3
97.4555°: 4.3.3.3 C
98.8994°: 6.3.3.3
99.7356°: 4.3.4.3 C
100.1939°: 4.4.3.3 I
100.8123°: 5.5.3, 5.3.4.3, 5.3.3.3
102.5238°: 4.4.3.3 H
103.8362°: 4.3.3.3 A
108.0000°: 5.4.4, 4.4.3.3 C
109.4712°: 6.6.4, 6.6.3, 4.3.3, 6.4.3.3 A, 6.3.4.3, 5.4.3.3 B, 4.4.3.3 B, 4.4.3.3 D, 4.3.3.3 B, 3.3.3.3 A, 4.3.3.3.3 F, 3.3.3.3.3 B, 3.3.3.3.3 C, 3.3.3.3.3 H
109.5240°: 4.4.3.3 F, 4.3.3.3 C, 4.3.3.3.3 F
110.9052°: 6.4.3.3 B, 5.3.4.3
111.7348°: 3.3.3.3.3 K
114.6452°: 3.3.3.3.3 E
114.7356°: 4.3.3.3 B
116.5651°: 10.10.3, 10.5.4, 5.5.5, 5.5.3.3 B, 5.4.3.3 A
117.0190°: 4.4.3.3 F
117.3556°: 3.3.3.3.3 I, 3.3.3.3.3 J
118.8922°: 4.3.3.3 C, 3.3.3.3.3 F, 3.3.3.3.3 H
120.0000°: 6.4.4, 4.4.4.3 B
121.7175°: 10.5.4, 4.4.4.3 C
121.7432°: 3.3.3.3 D, 3.3.3.3.3 D
124.7019°: 4.4.3.3 I, 3.3.3.3.3 M
125.2644°: 8.8.3, 8.6.4, 6.6.4, 6.4.3, 8.3.4.3, 6.3.4.3, 4.4.4.3 B, 4.4.3.3 D, 4.3.4.3 A, 4.3.3.3.3 B
126.8699°: 5.5.3.3 A
126.9641°: 4.3.3.3.3 D
127.3774°: 4.4.4.3 C, 4.4.3.3 C
127.5516°: 4.3.3.3 A, 3.3.3.3.3 B
128.4960°: 4.4.3.3 H, 3.3.3.3.3 K, 3.3.3.3.3 L
129.4446°: 3.3.3.3 E, 4.3.3.3.3 G
131.4416°: 3.3.3.3.3 G
132.6240°: 4.3.3.3.3 D
133.5912°: 4.3.3.3.3 I, 3.3.3.3.3 M
133.9728°: 4.4.3.3 H, 4.3.3.3.3 H
135.0000°: 8.6.4, 8.4.4, 4.4.4.3 A
135.9915°: 4.3.3.3 C, 3.3.3.3.3 G, 3.3.3.3.3 H
136.3359°: 4.4.3.3 I, 4.3.3.3.3 I
137.2401°: 4.3.3.3.3 G
138.1897°: 6.6.5, 5.3.3, 6.4.3.3 B, 6.4.3.3 B, 5.5.3.3 B, 5.3.3.3, 4.4.3.3 C, 3.3.3.3 C, 3.3.3.3.3 A
141.0576°: 6.3.4.3, 4.4.3.3 D, 3.3.3.3 B
141.3411°: 4.3.3.3.3 H, 3.3.3.3.3 L
141.5945°: 4.3.3.3.3 C
142.6226°: 10.10.3, 10.6.4, 10.5.3, 6.6.5, 10.3.4.3, 5.5.3.3 A, 5.3.5.3, 5.4.4.3 A, 5.4.3.3 A, 5.3.4.3, 5.3.3.3.3 B
142.9834°: 4.3.3.3.3 A
143.4787°: 3.3.3.3.3 F, 3.3.3.3.3 G
143.7383°: 4.3.3.3.3 G, 3.3.3.3.3 J
144.0000°: 10.4.4, 5.4.4.3 A, 4.4.4.3 C
144.1436°: 4.3.3.3.3 E, 3.3.3.3.3 E
144.7356°: 8.6.4, 8.4.3, 8.3.4.3, 8.3.4.3, 6.4.3.3 A, 5.4.3.3 B, 4.4.4.3 A, 4.4.4.3 B, 4.4.3.3 B, 4.3.4.3 C, 4.3.3.3 B, 4.3.3.3.3 C, 3.3.3.3.3 C
145.2219°: 6.3.3.3, 4.3.3.3.3 B, 4.3.3.3.3 B
145.4406°: 4.3.3.3.3 E
148.2825°: 10.6.4, 10.5.4, 5.4.4.3 B, 5.4.3.4
148.4340°: 4.3.3.3.3 I, 3.3.3.3.3 M
149.5648°: 3.3.3.3.3 L
150.0000°: 4.3.4.3 B
151.3301°: 4.3.3.3.3 C
152.1911°: 3.3.3.3.3 H
152.9299°: 5.3.3.3.3 A
152.9756°: 4.3.3.3.3 H
153.2346°: 4.3.3.3.3 A, 4.3.3.3.3 A
153.4349°: 5.4.4.3 A, 5.4.4.3 B
153.6350°: 4.3.3.3.3 B
153.9424°: 10.3.4.3, 5.5.3.3 B, 5.4.4.3 B
153.9624°: 8.3.3.3, 4.3.3.3.3 C, 4.3.3.3.3 C
154.4188°: 4.3.3.3.3 I
154.7223°: 4.4.3.3 G, 4.3.3.3.3 G
157.1481°: 4.3.3.3.3 H, 4.3.3.3.3 H, 3.3.3.3.3 K
158.3754°: 5.5.3.3 A
158.5718°: 3.3.3.3.3 B
158.6816°: 5.3.3.3.3 B
159.0948°: 10.6.4, 10.4.3, 10.3.4.3, 6.4.3.3 B, 5.4.4.3 B, 5.4.3.4, 5.4.3.3 A, 5.3.4.3, 4.4.4.3 C, 4.4.3.3 E, 4.3.4.3 D, 4.3.3.3.3 D
159.1865°: 10.3.3.3, 5.3.3.3.3 B, 5.3.3.3.3 B, 4.3.3.3.3 D, 4.3.3.3.3 D
159.8924°: 4.4.3.3 F, 4.3.3.3.3 F, 3.3.3.3.3 F
160.5288°: 4.4.4.3 B, 4.4.3.3 A
161.4828°: 3.3.3.3.3 J
162.7356°: 5.4.3.3 B
164.1754°: 5.3.3.3.3 A, 5.3.3.3.3 A
164.2068°: 6.3.4.3
164.2574°: 4.3.3.3.3 E, 3.3.3.3.3 E
164.2596°: 4.3.3.3.3 F, 3.3.3.3.3 H
166.4406°: 3.3.3.3.3 D
166.8114°: 4.3.3.3.3 I
169.1877°: 5.4.4.3 A
169.4282°: 4.3.3.3.3 B
169.4712°: 3.3.3.3.3 C
170.2644°: 8.3.4.3
171.3411°: 5.5.3.3 C
171.6457°: 4.3.3.3.3 G, 3.3.3.3.3 I
171.7546°: 4.3.3.3.3 F
174.3401°: 10.3.4.3
174.4343°: 5.3.3.3.3 B
174.7356°: 6.4.3.3 A
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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mr_e_man
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Posts: 531
Joined: Tue Sep 18, 2018 4:10 am

### Re: List of verfs of CRF polyhedra

mr_e_man wrote:An antiprism's dihedral angles are less obvious; we have the formulas

cos θn3 = (-1/√3) (sin π/n) / (1 + cos π/n),

cos θ33 = (-1/3) (4 cos π/n - 1).

I derived those using the standard Cartesian coordinates, but here's a neater derivation. Draw the n.3.3.3 tetragon on the sphere, and extend the 'n' edge and the opposite '3' edge in both directions until they meet, forming a lune. By symmetry, the extensions have the same length in both directions; we can use this to determine what those lengths must be. (Note that the total length of a lune edge is π.) Then analyze one of the trigons (formed by extending the edges in one direction) to figure out the angles.

antiprismVerf.png (17.38 KiB) Viewed 23000 times

Thus the n.3.3.3 tetragon has the same angles as 2n.6.3 .

cos θn,3 = (cos(2π/3) - cos(π/3) cos(π - π/n)) / (sin(π/3) sin(π - π/n))
= (-1/2 + 1/2 cos(π/n)) / (√3/2 sin(π/n))
= (-1/√3) (1 - cos(π/n)) / sin(π/n)
= (-1/√3) sin(π/n) / (1 + cos(π/n))
= (-1/√3) tan(π/(2n))

cos θ3,3 = (cos(π - π/n) - cos(π/3) cos(2π/3)) / (sin(π/3) sin(2π/3))
= (-cos(π/n) + 1/4) / (3/4)
= (-1/3) (4 cos(π/n) - 1)

By the same reasoning, any symmetric n.k.m.k tetragon has the same angles as 2n.k.2m .

Take a look at one of the thin spheric trigons. The edge lengths are π/3, π/3, π/n, so the trigon is isosceles, and can be bisected. Perpendicular to the bisecting line is another line that bisects the supplementary angle. (In 3D, this corresponds to a plane that bisects the 3.3 dihedral angle, and passes through the centre of the antiprism.) This gives another way to calculate the angles.

antiprismVerf2.png (22.72 KiB) Viewed 21133 times

cos θn,3 = (cos(π/2) - cos(π/3) cos(π/2 - π/(2n))) / (sin(π/3) sin(π/2 - π/(2n)))
= (0 - 1/2 sin(π/(2n))) / (√3/2 cos(π/(2n)))
= (-1/√3) tan(π/(2n))

cos ½θ3,3 = (cos(π/2 - π/(2n)) - cos(π/2) cos(π/3)) / (sin(π/2) sin(π/3))
= (sin(π/(2n)) - 0) / (√3/2)
= (2/√3) sin(π/(2n))

I also have a formula for the solid angle Ω at a vertex of the antiprism; that is the area of the spheric tetragon:

cos ½Ω = (√3/9) tan(π/(2n)) (5 + 4 cos(π/n))
Last edited by mr_e_man on Thu Aug 17, 2023 11:05 pm, edited 1 time in total.
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mr_e_man
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### Re: List of verfs of CRF polyhedra

mr_e_man wrote:It should be somewhat easier to build spherical polyhedra (for polychoron verfs) with this smaller set of polygons. If some dihedral angles end up being exactly 180°, then we can combine those faces into a single face (provided that it exists in the longer list above; otherwise we discard the polyhedron). Often we can't tell if it's slightly greater or less than 180°; this step (combining coplanar faces) can be delayed, or it may not be necessary at all, if that polyhedron doesn't work as a verf for some other reason.

Marek14 wrote:EDIT: This also means that we might actually not need to search for several skew verfs (like triangular dipyramid) directly -- instead they would be a special case whenever verf would have two tetrahedra and they would turn to have dihedral angle exactly 180 degrees. If we implement this, we'll only have to search for "primitive" verfs which can't be split, i.e. verfs where no diagonal is a valid chord, i.e. where the polyhedron can't be cut along any diagonal of that particular vertex. This means that the number of polygons we have to "try" will be reduced substantially and we can generate new verfs from "invalid" cases where some dihedral angles are exactly 180! So number of quadrangles to try is reduced from 45 to mere 14 and number of pentagons from 24 to 15. And majority of those is from the "weird" Johnson solids at the end of the line (most of the corona line's verfs are primitive except for two that occur in augmented sphenocorona).

The disadvantage is that some symmetrical verfs like octahedral one have more than one diagonal as valid chord, meaning that verfs containing them might be generated more than once. Also, some uniforms would have to be abandoned as primitive cells: octahedral verf would have to be built from two square pyramids, icosahedral from gyroelongated pentagonal pyramid and pentagonal pyramid, or even from metabidiminished icosahedron and two pentagonal pyramids, cuboctahedron from two square cupolas, rhombicuboctahedron from square cupola and octagonal prism, icosidodecahedron from two pentagonal rotundas and rhombicosidodecahedron from pentagonal cupola and diminished rhombicosidodecahedron.
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mr_e_man
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### Re: List of verfs of CRF polyhedra

Many of the 3,4,5,6,8,10-gon dihedral angles decompose in terms of these seven smaller angles:

31.7175° = arccos √(φ/√5)
37.3774° = arccos (φ/√3) √(φ/√5)
45.0000° = arccos 1/√2
54.7356° = arccos 1/√3
60.0000° = arccos 1/2
70.5288° = arccos 1/3
79.1877° = arccos (φ⁻¹/√3) √(φ⁻¹/√5)

31.7175°
37.3774°
45.0000°
54.7356°
60.0000°
63.4349° = 31.7175° + 31.7175°
69.0948° = 31.7175° + 37.3774°
70.5288°
72.9730°
74.7547° = 37.3774° + 37.3774°
79.1877°
86.7268°
90.0000° = 45° + 45°
95.1524° = 31.7175° + 31.7175° + 31.7175°
95.2466°
96.1983°
96.5945°
97.4555°
98.8994°
99.7356° = 45° + 54.7356°
100.1939°
100.8123° = 31.7175° + 31.7175° + 37.3774°
102.5238°
103.8362°
108.0000°
109.4712° = 54.7356° + 54.7356°
109.5240°
110.9052° = 31.7175° + 79.1877°
111.7348°
114.6452°
114.7356° = 54.7356° + 60°
116.5651° = 37.3774° + 79.1877°
117.0190°
117.3556°
118.8922°
120.0000° = 60° + 60°
121.7175° = 31.7175° + 45° + 45°
121.7432°
124.7019°
125.2644° = 54.7356° + 70.5288°
126.8699° = 31.7175° + 31.7175° + 31.7175° + 31.7175°
126.9641° = 31.7175° + 95.2466°
127.3774° = 37.3774° + 45° + 45°
127.5516°
128.4960°
129.4446°
131.4416°
132.6240° = 37.3774° + 95.2466°
133.5912°
133.9728°
135.0000° = 45° + 45° + 45°
135.9915°
136.3359°
137.2401°
138.1897° = 31.7175° + 31.7175° + 37.3774° + 37.3774°
141.0576° = 70.5288° + 70.5288°
141.3411°
141.5945° = 45° + 96.5945°
142.6226° = 31.7175° + 31.7175° + 79.1877°
142.9834°
143.4787°
143.7383°
144.0000°
144.1436°
144.7356° = 45° + 45° + 54.7356°
145.2219°
145.4406°
148.2825° = 31.7175° + 37.3774° + 79.1877°
148.4340°
149.5648°
150.0000° = 45° + 45° + 60°
151.3301° = 54.7356° + 96.5945°
152.1911° = 54.7356° + 97.4555°
152.9299°
152.9756°
153.2346°
153.4349° = 31.7175° + 31.7175° + 45° + 45°
153.6350° = 54.7356° + 98.8994°
153.9424° = 37.3774° + 37.3774° + 79.1877°
153.9624°
154.4188°
154.7223°
157.1481°
158.3754° = 79.1877° + 79.1877°
158.5718° = 54.7356° + 103.8362°
158.6816° = 31.7175° + 31.7175° + 95.2466°
159.0948° = 31.7175° + 37.3774° + 45° + 45°
159.1865°
159.8924°
160.5288° = 45° + 45° + 70.5288°
161.4828°
162.7356° = 54.7356° + 108°
164.1754°
164.2068° = 54.7356° + 54.7356° + 54.7356°
164.2574°
164.2596° = 54.7356° + 109.5240°
166.4406°
166.8114°
169.1877° = 45° + 45° + 79.1877°
169.4282° = 70.5288° + 98.8994°
169.4712° = 54.7356° + 54.7356° + 60°
170.2644° = 45° + 54.7356° + 70.5288°
171.3411° = 31.7175° + 31.7175° + 37.3774° + 70.5288°
171.6457°
171.7546° = 54.7356° + 117.0190°
174.3401° = 31.7175° + 31.7175° + 31.7175° + 79.1877°
174.4343° = 79.1877° + 95.2466°
174.7356° = 54.7356° + 60° + 60°

Also, any 360° sum of those angles, except 108°+108°+144°, is a result of these four simple 180° sums:

60° + 60° + 60° = 180°
45° + 45° + 45° + 45° = 180°
54.7356° + 54.7356° + 70.5288° = 180°
31.7175° + 31.7175° + 37.3774° + 79.1877° = 180°

For example, in the cube-doe-bilbiro honeycomb, a cube edge has dihedral angle sum
90° + 110.9052° + 159.0948°
= (2*45°) + (31.7175° + 79.1877°) + (31.7175° + 37.3774° + 2*45°)
= (4*45°) + (2*31.7175° + 37.3774° + 79.1877°)
= (180°) + (180°)
= 360°.
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mr_e_man
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### Re: List of verfs of CRF polyhedra

If large prisms/antiprisms are included, then there are a few more decompositions:

165° = 45° + 60° + 60°
168° = 60° + 108°
173.5714° = 45° + 128.5714°

(These are the angles in the 24-gon, 30-gon, and 56-gon prisms.) There are no other decompositions of CRF dihedral angles; I did an exhaustive search.

However, there may be other 360° sums. For example, I can't rule out the possibility that two antiprism angles near 90°, and one prism or antiprism angle near 180°, combine exactly to 360°. But it's unlikely. I don't expect anything other than combinations of prism angles like 60°+140°+160°, possibly including 45°.
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mr_e_man
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### Re: List of verfs of CRF polyhedra

Previously I considered combinations of angles with positive integer coefficients. Here I allow negative or fractional coefficients.

I was able to prove (using algebraic number theory on their complex exponentials (cosθ + i sinθ)) that these six angles,

70.5288° (tetrahedron),
116.5651° (dodecahedron),
138.1897° (icosahedron),
103.8362° (square antiprism),
127.5516° (square antiprism),
360°,

are linearly independent over the rational numbers. That means there are no rational numbers a,b,c,d,e,f such that

a*70.5288° + b*116.5651° + c*138.1897° + d*103.8362° + e*127.5516° = f*360°,

other than the trivial solution a=b=c=d=e=f=0. Equivalently, the coefficients in such a sum are unique;

a*70.5288° + b*116.5651° + c*138.1897° + d*103.8362° + e*127.5516° + f*360°
= A*70.5288° + B*116.5651° + C*138.1897° + D*103.8362° + E*127.5516° + F*360°

only if a=A, b=B, c=C, d=D, e=E, f=F.

We could use smaller angles instead; e.g. these six are also linearly independent:

54.7356° = 1/4*360° - 1/2*70.5288° (square pyramid),
31.7175° = 1/4*360° - 1/2*116.5651° (pentagonal cupola),
37.3774° = 1/2*138.1897° + 1/2*116.5651° - 1/4*360° (pentagonal pyramid),
103.8362°,
127.5516°,
1° = 1/360*360°.

It seems a bit more natural to use 3° instead of 1°, because the 3,4,5,6,8,10-gon angles are multiples of 3°, and conversely 3° = 108° - 60° - 45° = 108° + 135° - 4*60° (the coefficients are integers).

Of course the square antiprism angles could be omitted, or more than six angles could be included. But, as of now, I haven't checked other specific angles, to see that they are in fact independent.

In any case, I don't expect this (linear independence) to be useful in the search for CRH polytopes....
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