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

Postby mr_e_man » Wed Nov 11, 2020 7:50 pm

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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Re: List of verfs of CRF polyhedra

Postby mr_e_man » Wed Nov 11, 2020 7:54 pm

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); 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); 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); 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°; 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°; 198.7714°; J22, 44
c: {4} 144.7356° {3} 151.3301° {3} 153.9624° {3} 153.9624° {3} 141.5945°; 205.5850°; J23, 45
d: {4} 159.0948° {3} 132.6240° {3} 159.1865° {3} 159.1865° {3} 126.9641°; 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°; 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°; 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°; 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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