## Dichoral angles of CRF's ["split" from IncMats Update]

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

### Dichoral angles of CRF's ["split" from IncMats Update]

I thought it to be inappropriate to spam the Incmats Website update with all the diheddral angle stuff, so I decided to make a threat for it.

Klitzing wrote:Sadly by mere short sight I cannot envision how you derived that ...
(Your a,b,c seem to be sides of some planar triangle with circumradius 1, b' and c' then would be respective inradii.)

I don't find that surprising, as my deriviation was long and complex. In addition, your formula is much neater. (my deriviation was based on the 3-cut perpendicular to an edge, cut of at a length 1, yielding a pyramid T ABC where TA,TB and TC are 1. then you have an awful lot of algebra, after which everything lets itself be simplified to what I had).
Klitzings formula: arccos[ (C - A B) / sqrt[(1 - A²)(1 - B²)] ]

As I wanted to calculate the dichoral angles of all the CRF's, I needed some more tools.
case 1: 4 things meeting at an edge, 1 dichoral angle is known.
you have a "pyramid" T ABCD, and the dichoral angle is known at TA. now you can calculate the length of BD in T ABD, and then calculate the dichoral angle at TC in T BCD. This is just done by reversing the formula of Klitzing, and you'd get "C"=cos(alpha)*sqrt( (1-A²)(1-B²) ) + AB
And thus "beta"=arccos( (cos(alpha)*sqrt( (1-A²)(1-B²) ) + AB - CD)/sqrt( (1-C²)(1-D²) )
case 2: 4 things meet at an edge, the bottom vertices of the pyramid form a trapezohedron ADBD
derivation is possible, but I just programmed my calculator to use the "angle"-program.

I have already calculated all dichoral angles for all the segmentochora, but they're in dutch, not exact, and cryptically denotated. i'll give them anyway:
the cryptix works as follows: [n] denotates the face at which the angle is measured, {0,1} denotates the nodes that are ringed in the CD-diagram. furthermore, (2+) etc. means some deriviation of the polytope listed "2" on a list that I lost. kpl=cupola; pr=prism; pir=pyr=pyramid; ico=icosahedron; kub=cube;

for the {5,3} CD-graph:
{0}||{2}: ([5] {0} 5pir) 72˚; ([3] {2} Tet) 142,2387561˚; ([3] Tet tet) 164,4775122˚; ([3] 5pir tet) 142,2387561˚

{0}||{2(4+)}: “{0}||{2}” ({2}→{2(4+)}) + ([5] {0} 5ap) 36˚; ([5] {2(4+)} 5ap) 142,2387561˚; ([3] 5ap 5pir) 120˚

{0}||{0,2}: ([5] {0} 5pr) 162˚; ([5] {0,2} 5pr) 18˚; ([4] {0,2} 3pr) 20,90515745˚; ([3] {0,2} tet) 22,23875609˚; ([4] 5pr 3pr) 169,1876830˚; ([3] tet 3pr) 172,2387561˚

{0}||{0,2(1+)}: “{0}||{0,2}” ({0,2}→{0,2(1+)} ) + ([5] {0} 5kpl) 144˚; ([10] {0,2(1+)} 5kpl) 36˚; ([4] 5kpl 5pr) 148,2825256˚; ([3] 5kpl 3pr) 150˚

{0,1}||{1,2}: ([10] {0,1} 5kpl) 72˚; ([3] {0,1} 3kpl) 37,76124391˚; ([6] {1,2} 3kpl) 142,2387561˚; ([5] {1,2} 5kpl) 108˚; ([3] 3kpl tet) 164,4775122˚; ([3] 5kpl tet)142,2387561˚; ([4] 3kpl 5kpl) 135˚

{0,1}||{0,2}: ([10] {0,1} 5kpl) 36˚; ([3] {0,1} oct) 22,23875609˚; ([5] {0,2} 5kpl) 144˚; ([3] {0,2} oct) 157,7612439˚;([4] {0,2} 3pr) 159,0948426˚; ([4] 5kpl 3pr) 159,0948426˚; ([3] oct 5kpl) 157,7612439˚; ([3] oct 3pr) 172,2387561˚

{0,1}||{0,2(1+)}: “{0,1}||{0,2}” ({0,2}→{0,2(1+)}; oct→4pir) + ([10] {0,1} 10pr) 18˚; ([10] {0,2(1+)} 10pr) 162˚; ([4] 10pr 5kpl) 148,2825256˚; ([4] 10pr 4pir) 166,7174744˚

{0,1}||{0,1,2}: ([10] {0,1} 10pr) 162˚; ([3] {0,1} 3kpl) 157,7612439˚; ([10] {0,1,2} 10pr) 18˚; ([6] {0,1,2} 3kpl) 22,23875609˚; ([4] {0,1,2} 3pr) 20,90515745˚; ([4] 10pr 3pr) 169,1876830˚; ([4] 10pr 3kpl) 166,7174744˚; ([3] 3pr 3kpl) 172,2387561˚

{1}||{0,2}: ([5] {1} 5ap) 144˚; ([3] {1} oct) 142,2387561˚; ([5] {0,2} 5ap) 36˚; ([3] {0,2} oct) 37,76124391˚; ([4] {0,2} 4pir) 45˚; ([3] 5ap 4pir) 157,7612439˚; ([3] oct 4pir) 164,4775122˚; ([3] oct 5ap)157,7612439˚

{1}||{2}: ([3] {1} oct) 22,23875609˚; ([5] {1} 5pir) 36˚; ([3] {2} oct) 157,7612439˚; ([3] 5pir oct) 157,7612439˚; ([3] oct oct) 164,4775122˚

{1}||{2(1+)}: “{1}||{2}” ({2}→{2(1+)}; oct→4pir) ([5] {1} 5pr) 18˚; ([5] {2(1+)} 5pr) 162˚; ([4] 5pr 4pir) 166,71747441˚

{1}||{2(3+)} “++” (not sure what i ment by this)

{1(1+)}||{2(4+)}: “{1}||{2(1+)}”({1}→{1(1+)}; {2(1+)}→{2(4+)}; oct→4pir) + ([10] {1(1+)} 5kpl) 144˚; ([5] {2(4+)} 5kpl) 36˚; ([4] 5kpl 4pir) 45˚; ([3] 5kpl oct) 37,76124391˚; ([4] 5kpl 5pr) 31,71747441˚

{1,2}||{0,2}: ([5] {1,2} 5ap) 72˚; ([6] {1,2} 3kpl) 82,23875609˚; ([5] {0,2} 5ap) 108˚; ([3] {0,2} 3kpl) 97,76124391˚; ([4] {0,2} 3pr) 110,9051574˚; ([3] 3pr 5ap) 150˚; ([4] 3kpl 3pr) 155,9051574˚; ([3] 5ap 3kpl) 142,2387561˚

for the {4,3} CD-graph:

{0}||{2}: ([4] {0} 4pir) 107,0312485˚; ([3] {2} Tet) 124,1014653˚; ([3] Tet tet) 144,1604822˚; ([3] 4pir tet) 124,1014653˚

{0}||{1}: ([4] {0} 4ap) 104,2582505˚; ([4] {1} 4ap) 75,74174952˚; ([3] {1} tet) 93,47770685˚; ([3] tet 4ap) 122,9286782˚; ([3] 4ap 4ap) 93,47770685˚

{0}||{0,2}: ([4] {0} kub) 135˚; ([4] {0,2} kub) 45˚; ([4] {0,2} 3pr) 54,73561032˚; ([3] {0,2} tet) 60˚; ([4] kub 3pr) 144,7356103˚; ([3] tet 3pr) 150˚

{0}||Ico: ([4] {0} 3pr) 110,9051574˚; ([3] Ico 4pir) 82,23875609˚; ([3] Ico tet) 97,76124391˚; ([4] 3pr 4pir) 155,9051574˚; ([3] tet 4pir) 135,5224878˚; ([3] 3pr 4pir) 122,2387561˚;

{0,1}||{1,2}: ([8] {0,1} 4kpl) 107,0312485˚; ([3] {0,1} 3kpl) 55,89853468˚; ([6] {1,2} 3kpl) 124,1014653˚; ([4] {1,2} 4kpl) 72,96875154˚; ([4] 4kpl 3kpl) 107,0312485˚; ([3] tet 4kpl) 124,1014653˚; ([3] tet 3kpl) 79,08574016˚ 144,16904822175˚
{0,1}||{0,2}: ([8] {0,1} 4kpl) 90˚; ([3] {0,1} oct) 60˚; ([4] {0,2} 4kpl) 90˚; ([3] {0,2} oct) 120˚; ([4] {0,2} 3pr) 125,2643897˚; ([3] oct 4kpl) 120˚; ([3] oct 3pr) 150˚; ([4] 3pr 4kpl) 125,2643897˚

{0,1}||{0,2(4+)}: “{0,1}||{0,2}” ({0,2}→{0,2(4+)}; oct→4pir) + ([8] {0,1} 8pr) 45˚; ([8] {0,2(4+)} 8pr) 135˚; ([4] 4pir 8pr) 135˚; ([4] 4kpl 8pr) 90˚

{0,1}||{0,1,2}: ([8] {0,1} 8pr) 135˚; ([3] {0,1} 3kpl) 120˚; ([8] {0,1,2} 8pr) 45˚; ([6] {0,1,2} 3kpl) 60˚; ([4] {0,1,2} 3pr) 54,73561032˚; ([4] 3pr 8pr) 144,9356103˚; ([3] 3pr 3kpl) 150˚; ([4] 3kpl 8pr) 135˚

{1}||{0,2}: ([3] {1} oct) 124,1014653˚; ([4] {1} 4ap) 126,4843738˚; ([4] {0,2} 4ap)53,51562423˚; ([4] {0,2} 4pir) 72,96875154˚; ([3] {0,2} oct) 55,89853462˚; ([3] oct 4pir) 144,1604822˚; ([3] oct 4ap) 135,8690262˚; ([3] 4pir 4ap) 135,8690262˚

{1}||{0,2(4+)}: “{1}||{0,2}” ({0,2}→{0,2(4+)}; oct→4pir) + ([4] {1} 4kpl) 72,96875154˚; ([8] {0,2(4+)} 4kpl) 107,0312485˚; ([4] 4kpl 4pir) 107,0312485˚; ([3] 4kpl 4ap) 100,0295084˚

{1}||{1,2}: ([4] {1} kub) 135˚; ([3] {1} 3kpl) 120˚; ([6] {1,2} 3kpl) 60˚; ([4] {1,2} kub) 45˚; ([4] kub 3kpl) 135˚; ([3] 3kpl 3kpl) 120˚

{1}||{2}: ([3] {1} oct) 60˚; ([4] {1} 4pir) 90˚; ([3] {2} oct) 120˚; ([3] 4pir oct) 120˚; ([3] oct oct) 120˚

{1,2}||{0,2}: ([6] {1,2} 3kpl) 93,47770685˚; ([4] {1,2} 4ap) 75,74174952˚; ([3] {0,2} 3kpl) 86,52229315˚; ([4] {0,2} 4ap) 104,2582505˚; ([4] {0,2} 3pr) 109,7674872˚; ([4] 3kpl 3pr) 138,2839881˚; ([3] 4ap 3kpl) 136,7388534˚; ([3] 4ap 3pr) 136,7388534˚
{1,2}||{0,1,2}: ([6] {1,2} 6pr) 150˚; ([4] {1,2} 4kpl) 135˚; ([6] {0,1,2} 6pr) 30˚; ([8] {0,1,2} 4kpl) 45˚; ([4] {0,1,2} 3pr) 35,26438968˚; ([3] 3pr 4kpl) 150˚; ([4] 4kpl 6pr) 144,7356103˚; ([4] pr 6pr) 160,5287794˚

{2}||{0,2}: ([3] {2} 3Pr) 150˚; ([3] {0,2} 3Pr) 30˚; ([4] {0,2} 4pir) 45˚; ([4] {0,2} 3pr) 35,26438968˚; ([4] 3pr 3Pr) 160,5287794˚; ([3] 3pr 4pir) 150˚;

{2(1+)}||{0,2(2+)}: “{2}||{0,2}”({2}→{2(1+)}; {0,2}→{0,2(2+)}) + ([8] {0,2(2+)} 4kpl) 45˚; ([4] {2(1+)} 4kpl) 135˚; ([4] 3pr 4kpl) 125,264389˚; ([3] 4pir 4kpl) 120˚

{2(1+)}||{0,2(3+)}: ([4] {2(1+)} {0,2(3+)}) 45˚; ([8] {0,2(3+)} {0,2(3+)}) 135˚; ([4] {2(1+)} 3pr) 150˚; ([4] {0,2(3+)} 3pr) 35,264389268˚; ([3] {0,2(3+)} 3pr) 30˚; ([4] 3pr 3pr) 160,5287794˚

for the {3,3} CD-graph:
{0}||{2}: ([3] tet tet) 120˚

{0}||{1}: ([3] tet oct) 104,4775122˚; ([3] oct oct) 75,52248781˚

{0}||{0,2}: ([3] {0} 3pr)55,77113367˚; ([4] {0,2} 3pr) 65,90515745˚; ([3] {0,2} tet) 75,52248781

{0}||{0,2 (1+)}: “0||0,2”({0,2}→{0,2(1+)}) + ([6] {0,2(1+)} 3kpl) 104,4775122˚

{0}||{0,1}: ([3] {0} 3kpl) 90˚; ([3] {0,1} tet) 60˚; ([6] {0,1} 3kpl) 60˚; ([3] 3kpl tet) 120˚; ([4] 3kpl 3kpl) 120˚

{0,1}||{0,2}: ([3] {0,1} oct) 75,52248781˚; ([6] {0,1} 3kpl) 104,4775122˚; ([3] {0,2} 3kpl) 75,52248781˚; ([3] {0,2} oct) 104,4775122˚; ([4] {0,2} 3pr) 114,0948426˚; ([3] 3kpl oct) 104,4775122˚; ([4] 3kpl 3pr) 114,0948426˚; ([3] oct 3pr) 127,7612439˚

{0,1}||{0,2(1+)}: “0,1||0,2”({0,2}→{0,2(1+)} ) + ([6] {0,1} 6pr) 52,23875609˚; ([6] {0,2(1+)} 6pr) 127,7612439˚; ([4] 4pir 6pr) 114,0948426˚; ([4] 3kpl 6pr) 65,90515745˚

{0,1}||{0,1,2}: ([3] {0,1} 3kpl) 75,52248781˚; ([6] {0,1} 6pr) 52,23875609˚; ([4] {0,1,2} 3pr) 65,90515745˚; ([6] {0,1,2} 6pr) 52,23875609˚; ([6] {0,1,2} 3kpl) 75,52248781˚; ([4] 6pr 3pr) 131,8103149˚; ([4] 6pr 3kpl) 114,0948426˚; ([3] 3kpl 3pr) 127,7612439˚

{0,1}||{1}: ([3] {0,1} 3pr) 52,23875609˚; ([6] {0,1} 3kpl) 75,52248781˚; ([3] {1} 3pr) 127,7612439˚; ([3] {1} 3kpl) 104,4775122˚; ([4] 3kpl pr) 114,0948426˚; ([3] 3kpl 3kpl) 75,52248781˚

{0,1}||{1,2}: {0,1}≡{1,2} ([6] {0,1} 3kpl) 120˚; ([3] {0,1} 3kpl) 60˚; ([4] 3kpl 3kpl) 90˚; ([3] tet 3kpl) 120˚

pyramids:

Tetraëder: ([3]tet tet) 75,52248781˚
Octaëder: ([3]tet oct) 60˚; ([3]tet tet) 120˚
Kubus: ([4]kub 4pir) 45˚; ([3]4pir 4pir)120˚
Icosaëder: ([3]ico tet) 22,23875609˚; ([3]tet tet) 164,4775122˚
tridico: ([3] tridico tet) 22,23875609˚; ([5] tridico 5pir) 36˚; ([3] tet tet) 164,4775122˚; ([3]5pir tet) 142,2387561˚; ([3] 5pir 5pir) 120˚

4pir(=4pyr): ([4]4pir 4pir) 90˚; ([3]4pir tet) 60˚; ([3]tet tet) 120˚
5pir(=5pyr): ([5]5pir 5pir) 144˚; ([3]5pir tet) 22,23875609˚; ([3]tet tet) 164,4775122˚

3pr: ([3]3pr tet) 52,23875609˚; ([4]3pr 4pir) 65,90515745˚; ([3]tet 4pir) 104,4775122˚; ([3]4pir 4pir)75,52248781˚
5pr: ([5]5pr 5pir) 18˚; ([4]5pr 4pir) 13,28252559˚; ([3]5pir 4pir) 157,7612439˚; ([3]4pir 4pir)164,4775122˚

4ap: ([3]4ap tet) 44,13097377˚; ([4]4ap 4pir) 53,51562423˚; ([3]4pir tet) 124,1014653˚; ([3]tet tet) 144,1604822˚
5ap: ([3]5ap tet)22,23875609˚; ([5]5ap 5pir) 36˚; ([3]5pir tet) 142,2387561˚; ([3]tet tet) 164,4775122˚

gyrobicupolic rings:

n=2: ([3] tet 4pir) 104,4775122˚ ([3] 3pr 4pir) 114,0948462˚ ([4] 3pr 4pir) 52,23875609˚
([4] 3pr 3pr) 48,1896851˚ ([3] 4pir 4pir) 75,52248781˚
n=3: ([3] oct 3kpl) 60˚ ([3] oct 4pir) 120˚ ([3] 3kpl 4pir) 60˚
([4]3kpl 4pir) 90˚ ([6] 3kpl 3kpl) 60˚ ([3] 4pir 4pir) 120˚
n=4: ([4] 4ap 4kpl) 53,51562423˚ ([3] 4ap 4pir)135,86902624˚ ([3] 4kpl 4pir) 55,89853468˚
([4] 4kpl 4pir) 72,96875154˚ ([8] 4kpl 4kpl) 72,96875154˚ ([3] 4pir 4pir) 144,1604822˚
n=5: ([5] 5ap 5kpl) 36˚ ([3] 5ap 4pir) 157,7612439˚ ([3] 5kpl 4pir) 37,76124391˚
([4] 5kpl 4pir) 45˚ ([10] 5kpl 5kpl) 108˚ ([3] 4pir 4pir) 164,4775122˚

orthobicupolic rings:

n=3: ([4] 3pr pr) 131,8103149˚ ([3] 3pr 3kpl) 52,23875609˚ ([4] 3kpl pr) 65,90515745˚
([3] 3kpl tet) 75,52248781˚ ([6] 3kpl 3kpl) 75,522487681˚ ([3]tet pr) 127,7612439˚
n=4: ([4] kub pr) 144,7356103˚ ([4] kub 4kpl) 45˚ ([4] 4kpl pr) 54,73561032˚
([3] 4kpl tet) 60˚ ([8] 4kpl 4kpl) 90˚ ([3] tet pr) 150˚
n=5: ([4] 5pr pr) 169,187683˚ ([5] 5pr 5kpl) 18˚ ([4] 5kpl pr)20,90515745˚
([3] 5kpl tet) 22,23875609˚ ([10] 5kpl 5kpl) 144˚ ([3] tet pr) 172,2387561˚

magnabicupolic rings:

n=3: ([6] 6pr 3kpl) 52,23875609˚ ([4] 6pr pr) 48,1896851˚ ([4] 6pr 4pir) 65,90515745˚
([4] 3kpl pr) 114,0948426˚ ([3] 3kpl 4pir) 104,4775122˚ ([3] 3kpl 3kpl) 65,52248781˚
([3] pr 4pir) 127,7612439˚
n=4: ([8] 8pr 4kpl) 45˚ ([4] 8pr pr) 35,26438968˚ ([4] 8pr 4pir) 45˚
([4] 4kpl pr) 125,2643897˚ ([3] 4kpl 4pir) 120˚ ([4] 4kpl 4kpl) 90˚
([3] pr 4pir) 150˚
n=5: ([10] 10pr 5kpl) 18˚ ([4] 10pr pr) 10,81231696˚ ([4] 10pr 4pir) 13,28252559˚
([4] 5kpl pr) 159,0948426˚ ([3] 5kpl 4pir) 157,7612439˚ ([5] 5kpl 5kpl) 144˚
([3] pr 4pir) 172,2387561˚

biantiprismatic rings:

n=3: ([4] 3pr 4pir) 114,0948426˚ ([3] 3pr oct) 52,23875609˚ ([3] oct 4pir) 75,52248781˚
([3] tet oct) 104,4775122˚ ([3] oct oct)75,52248781˚ ([3] tet 4pir) 104,4775122˚
n=4: ([4] kub 4pir) 107,0312485˚ ([4] kub 4ap) 53,51562423˚ ([3] 4ap 4pir) 79,97049155˚
([3] tet 4ap) 100,0295084˚ ([4] 4ap 4ap) 72,96875154˚ ([3] tet 4pir) 124,1014653˚
n=5: ([4] 5pr 4pir) 103,2825256˚ ([5] 5pr 5ap) 54˚ ([3] 5ap 4pir) 82,23875609˚
([3] tet 5ap) 97,76124391˚ ([5] 5ap 5ap) 72˚ ([3] tet 4pir) 135,5224878˚
n=6: ([4] 6pr 4pir) 100,9217792˚ ([6] 6pr 6ap) 54,24041347˚ ([3] 6ap 4pir) 83,64301964˚
([3] tet 6ap) 96,35698036˚ ([6] 6ap 6ap) 71,51917307˚ ([3] tet 4pir) 143,0383461˚
n=8: ([4] 8pr 4pir) 98,08559285˚ ([8] 8pr 8ap) 54,46532121˚ ([3] 8ap 4pir) 85,31103898˚
([3] tet 8ap) 94,68896102˚ ([8] 8ap 8ap) 71,06935758˚ ([3] tet 4pir) 152,3525137˚
n=10: ([4] 10pr 4pir) 96,43031477˚ ([10] 10pr 10ap) 54,56497312˚ ([3] 10ap 4pir) 86,27703833˚
([3] tet 10ap) 93,72296167˚ ([10] 10ap 10ap) 70,87005377˚ ([3] tet 4pir) 157,9085518˚

K4.109: ...coming soon

K4.33: ([3] oct 3pr) 112,23875609297˚;([5]teddi 5pir) 108˚; ([3]teddi oct) 37,76124390703˚;([3] teddi 4pir) 37,76124390703˚; ([3]teddi 3pr) 30˚; ([3]oct 5pir) 82,23875609296˚; ([3] oct tet) 135,5224878141˚; ([3] tet 5pir) 97,76124390702˚; ([3] 5pir 4pir) 82,23875609297˚; ([3] tet 4pir) 135,5224878141˚; ([4] 3pr 4pir) 155,9051574479˚
Last edited by student91 on Thu Feb 13, 2014 12:14 am, edited 3 times in total.
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student91
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

student91 wrote:for the {4,3} CD-graph:

{0}||{2}: ([4] {0} 4pir) 107,0312485˚; ([3] {2} Tet) 124,1014653˚; ([3] Tet tet) 144,1604822˚; ([3] 4pir tet) 124,1014653˚

...

{0,1}||{1,2}: ([8] {0,1} 4kpl) 107,0312485˚; ([3] {0,1} 3kpl) 55,89853468˚; ([6] {1,2} 3kpl) 124,1014653˚; ([4] {1,2} 4kpl) 72,96875154˚; ([4] 4kpl 3kpl) 107,0312485˚; ([3] tet 4kpl) 124,1014653˚; ([3] tet 3kpl) 79,08574016˚

Just wanted to check my formula and calculated the tic||toe, which so far was missing. Further by Stott reduction of the central nodes, i.e. x3x4o -> x3o4o and o3x4x -> o3o4x, it is then clear that cube||oct should have the same dihedral angles.

Thus I appreciated your Independent calculation and checked just those 2 cases.

Spotted that your last value of tic||toe is wrong: 79,0857° should rather be 144,1605°, which in cube||oct you've got right.

Just for your record. No further checks so far.

--- rk
Klitzing
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

Recently I've been mulling over an automated tool that will calculate all dihedral/diacral/dichoral/etc angles of any polytope, and store them in a queryable database.

In my understanding, this should not be too difficult (please correct me if I'm wrong): given two q-dimensional surtopes A, B of an n-dimensional polytope P, where q ≤ n, if they are adjacent (i.e., there is a common p-dimensional surtope C where both A and B are incident, for p<q), then calculating the angle between A and B is simply a matter of computing the angle of the triangle formed by taking the centroids of A, B, and C. If all vertices are known, calculating the centroids should be trivial, so a program can automatically compute this for every pair of adjacent elements on P.

The only drawback of this approach is that the resulting angles will be in floating-point format rather than algebraic, so we won't be able to extract digits from it to arbitrary accuracy (or know how it relates to other values algebraically). Unless the computation was performed symbolically, of course.
quickfur
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

quickfur wrote:Recently I've been mulling over an automated tool that will calculate all dihedral/diacral/dichoral/etc angles of any polytope, and store them in a queryable database.

In my understanding, this should not be too difficult (please correct me if I'm wrong): given two q-dimensional surtopes A, B of an n-dimensional polytope P, where q ≤ n, if they are adjacent (i.e., there is a common p-dimensional surtope C where both A and B are incident, for p<q), then calculating the angle between A and B is simply a matter of computing the angle of the triangle formed by taking the centroids of A, B, and C. If all vertices are known, calculating the centroids should be trivial, so a program can automatically compute this for every pair of adjacent elements on P.

The only drawback of this approach is that the resulting angles will be in floating-point format rather than algebraic, so we won't be able to extract digits from it to arbitrary accuracy (or know how it relates to other values algebraically). Unless the computation was performed symbolically, of course.

I'm not sure, but as far as I know ditopal angles are always measured between (n-1)-dimensional surtopes. This means that q should be exactly n-1 instead of ≤n. If you'd take q to be ≤n, you'd be calculating ditopal angles of it's facets instead of P itself.
The aproach itself seems correct to me. In fact, it was this approach I used to prove to myself there weren't any regulars for q>4.
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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student91
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### Re: Margin Angles of CRH's ["split" from IncMats Update]

The stott matrix should handle most of this. for the wme figures, the angle is simply the cross product of the pair of bevels, ie aij aji / aii ajj = sin ma.

for the lace prisms of two bases, a similar thing ought hold, but it is tricker.
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the dream we dream together is reality.

wendy
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

I just noticed that K4.109 doesn't have a page on Klitzings website.
I think K4.109 should really get a page, as it is a special segmentochoron. because it's the only 4D ring with length 4 that is not a prism (as far as I know).

to help you with the page (and help myself), I derived its incidence matrix.
Code: Select all
`oxxo4xxoo&#xr, all heights=sqrt(1/2) o...4o...     |  4  *  *  *  |  2  0  0  0  1  1  0  0  |  1  0  0  1  0  2  0  0  2  2  |  1  0  0  1  1  2  |.o..4.o..     |  *  8  *  *  |  0  2  2  0  1  0  1  0  |  0  1  0  2  2  2  2  0  0  2  |  1  1  0  0  1  1  |..o.4..o.     |  *  *  4  *  |  0  0  0  2  0  0  1  1  |  0  0  1  0  1  0  2  2  0  2  |  0  1  1  0  2  1  |...o4...o     |  *  *  *  1  |  0  0  0  0  0  1  0  1  |  0  0  0  0  0  0  0  4  4  8  |  1  0  0  1  4  4  |--------------+--------------+--------------------------+--------------------------------+--------------------+.... x...     |  2  0  0  0  |  4  *  *  *  *  *  *  *  |  1  0  0  0  0  1  0  0  1  0  |  1  0  0  1  0  1  |.... .x..     |  0  2  0  0  |  *  4  *  *  *  *  *  *  |  0  1  0  0  0  1  0  1  0  0  |  1  1  0  0  0  1  |.x.. ....     |  0  2  0  0  |  *  *  4  *  *  *  *  *  |  0  1  0  1  1  0  0  0  0  0  |  0  1  1  0  1  0  |..x. ....     |  0  0  2  0  |  *  *  *  4  *  *  *  *  |  0  0  1  0  1  0  0  1  0  0  |  0  0  1  1  1  0  |oo..4oo..&#xr |  1  1  0  0  |  *  *  *  *  8  *  *  *  |  0  0  0  1  0  1  0  0  0  1  |  1  0  0  0  1  1  |o..o4o..o&#xr |  1  0  0  1  |  *  *  *  *  *  4  *  *  |  0  0  0  0  0  0  0  0  2  1  |  0  0  0  1  1  1  |.oo.4.oo.&#xr |  0  1  1  0  |  *  *  *  *  *  *  8  *  |  0  0  0  0  1  0  1  0  0  1  |  0  1  0  0  1  1  |..oo4..oo&#xr |  0  0  1  1  |  *  *  *  *  *  *  *  4  |  0  0  0  0  0  0  0  2  0  1  |  0  0  1  0  1  1  |--------------+--------------+--------------------------+--------------------------------+--------------------+o...4x...     |  4  0  0  0  |  4  0  0  0  0  0  0  0  |  1  *  *  *  *  *  *  *  *  *  |  1  0  0  1  0  0  |.x..4.x..     |  0  8  0  0  |  0  4  4  0  0  0  0  0  |  *  1  *  *  *  *  *  *  *  *  |  1  1  0  0  0  0  |..x.4..o.     |  0  0  4  0  |  0  0  0  4  0  0  0  0  |  *  *  1  *  *  *  *  *  *  *  |  0  1  1  0  0  0  |ox.. ....&#xr |  1  2  0  0  |  0  0  1  0  2  0  0  0  |  *  *  *  4  *  *  *  *  *  *  |  1  0  0  0  1  0  |.xx. ....&#xr |  0  2  2  0  |  0  0  1  1  0  2  0  0  |  *  *  *  *  4  *  *  *  *  *  |  0  1  0  0  1  0  |.... xx..&#xr |  2  2  0  0  |  1  1  0  0  2  0  0  0  |  *  *  *  *  *  4  *  *  *  *  |  1  0  0  0  0  1  |.... .xo.&#xr |  0  2  1  0  |  0  1  0  0  0  0  2  0  |  *  *  *  *  *  *  4  *  *  *  |  0  1  0  0  0  1  |..xo ....&#xr |  0  0  2  1  |  0  0  0  1  0  0  0  2  |  *  *  *  *  *  *  *  4  *  *  |  0  0  1  0  2  1  |.... x..o&#xr |  2  0  0  1  |  1  0  0  0  0  2  0  0  |  *  *  *  *  *  *  *  *  4  *  |  0  0  0  1  1  2  |oooo4oooo&#xr |  1  1  1  1  |  0  0  0  0  1  1  1  1  |  *  *  *  *  *  *  *  *  *  8  |  0  0  0  0  1  1  |--------------+--------------+--------------------------+--------------------------------+--------------------+ox..4xx..&#xr |  4  8  0  0  |  4  4  4  0  8  0  0  0  |  1  1  0  4  0  4  0  0  0  0  |  1  *  *  *  *  *  |  <- 4cup.xx.4.xo.&#xr |  0  8  4  0  |  0  4  4  4  0  0  8  0  |  0  1  1  0  4  0  4  0  0  0  |  *  1  *  *  *  *  |  <- 4cup..xo4..oo&#xr |  0  0  4  1  |  0  0  0  4  0  0  0  4  |  0  0  1  0  0  0  0  4  0  0  |  *  *  1  *  *  *  |  <- 4pyro..o4x..o&#xr |  4  0  0  1  |  4  0  0  0  0  4  0  0  |  1  0  0  0  0  0  0  0  4  0  |  *  *  *  1  *  *  |  <- 4pyroxxo ....&#xr |  1  2  2  1  |  0  0  1  1  2  1  2  2  |  0  0  0  1  1  0  0  1  0  2  |  *  *  *  *  4  *  |  <-  3pr.... xxoo&#xr |  2  2  1  1  |  1  1  0  0  2  2  2  1  |  0  0  0  0  0  1  1  0  1  2  |  *  *  *  *  *  4  |  <-  3pr`
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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student91
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

Here now is the corrected matrix:
Code: Select all
`oxxo4xxoo&#xr -> heights = 1/sqrt(2) = 0.707107o...4o...     |  4  *  *  *  |  2  0  0  0  2  1  0  0  |  1  0  0  1  0  2  0  0  2  2  |  1  0  0  1  1  2.o..4.o..     |  *  8  *  *  |  0  1  1  0  1  0  1  0  |  0  1  0  1  1  1  1  0  0  1  |  1  1  0  0  1  1..o.4..o.     |  *  *  4  *  |  0  0  0  2  0  0  2  1  |  0  0  1  0  1  0  2  2  0  2  |  0  1  1  0  2  1...o4...o     |  *  *  *  1  |  0  0  0  0  0  4  0  4  |  0  0  0  0  0  0  0  4  4  8  |  0  0  1  1  4  4--------------+--------------+--------------------------+--------------------------------+------------------.... x...     |  2  0  0  0  |  4  *  *  *  *  *  *  *  |  1  0  0  0  0  1  0  0  1  0  |  1  0  0  1  0  1.... .x..     |  0  2  0  0  |  *  4  *  *  *  *  *  *  |  0  1  0  0  0  1  1  0  0  0  |  1  1  0  0  0  1.x.. ....     |  0  2  0  0  |  *  *  4  *  *  *  *  *  |  0  1  0  1  1  0  0  0  0  0  |  1  1  0  0  1  0..x. ....     |  0  0  2  0  |  *  *  *  4  *  *  *  *  |  0  0  1  0  1  0  0  1  0  0  |  0  1  1  0  1  0oo..4oo..&#x  |  1  1  0  0  |  *  *  *  *  8  *  *  *  |  0  0  0  1  0  1  0  0  0  1  |  1  0  0  0  1  1o..o4o..o&#x  |  1  0  0  1  |  *  *  *  *  *  4  *  *  |  0  0  0  0  0  0  0  0  2  2  |  0  0  0  1  1  2.oo.4.oo.&#x  |  0  1  1  0  |  *  *  *  *  *  *  8  *  |  0  0  0  0  1  0  1  0  0  1  |  0  1  0  0  1  1..oo4..oo&#x  |  0  0  1  1  |  *  *  *  *  *  *  *  4  |  0  0  0  0  0  0  0  2  0  2  |  0  0  1  0  2  1--------------+--------------+--------------------------+--------------------------------+------------------o...4x...     |  4  0  0  0  |  4  0  0  0  0  0  0  0  |  1  *  *  *  *  *  *  *  *  *  |  1  0  0  1  0  0.x..4.x..     |  0  8  0  0  |  0  4  4  0  0  0  0  0  |  *  1  *  *  *  *  *  *  *  *  |  1  1  0  0  0  0..x.4..o.     |  0  0  4  0  |  0  0  0  4  0  0  0  0  |  *  *  1  *  *  *  *  *  *  *  |  0  1  1  0  0  0ox.. ....&#x  |  1  2  0  0  |  0  0  1  0  2  0  0  0  |  *  *  *  4  *  *  *  *  *  *  |  1  0  0  0  1  0.xx. ....&#x  |  0  2  2  0  |  0  0  1  1  0  0  2  0  |  *  *  *  *  4  *  *  *  *  *  |  0  1  0  0  1  0.... xx..&#x  |  2  2  0  0  |  1  1  0  0  2  0  0  0  |  *  *  *  *  *  4  *  *  *  *  |  1  0  0  0  0  1.... .xo.&#x  |  0  2  1  0  |  0  1  0  0  0  0  2  0  |  *  *  *  *  *  *  4  *  *  *  |  0  1  0  0  0  1..xo ....&#x  |  0  0  2  1  |  0  0  0  1  0  0  0  2  |  *  *  *  *  *  *  *  4  *  *  |  0  0  1  0  1  0.... x..o&#x  |  2  0  0  1  |  1  0  0  0  0  2  0  0  |  *  *  *  *  *  *  *  *  4  *  |  0  0  0  1  0  1oooo4oooo&#xr |  1  1  1  1  |  0  0  0  0  1  1  1  1  |  *  *  *  *  *  *  *  *  *  8  |  0  0  0  0  1  1--------------+--------------+--------------------------+--------------------------------+------------------ox..4xx..&#x  |  4  8  0  0  |  4  4  4  0  8  0  0  0  |  1  1  0  4  0  4  0  0  0  0  |  1  *  *  *  *  *  <- 4cup.xx.4.xo.&#x  |  0  8  4  0  |  0  4  4  4  0  0  8  0  |  0  1  1  0  4  0  4  0  0  0  |  *  1  *  *  *  *  <- 4cup..xo4..oo&#x  |  0  0  4  1  |  0  0  0  4  0  0  0  4  |  0  0  1  0  0  0  0  4  0  0  |  *  *  1  *  *  *  <- 4pyro..o4x..o&#x  |  4  0  0  1  |  4  0  0  0  0  4  0  0  |  1  0  0  0  0  0  0  0  4  0  |  *  *  *  1  *  *  <- 4pyroxxo ....&#xr |  1  2  2  1  |  0  0  1  1  2  1  2  2  |  0  0  0  1  1  0  0  1  0  2  |  *  *  *  *  4  *  <-  3pr.... xxoo&#xr |  2  2  1  1  |  1  1  0  0  2  2  2  1  |  0  0  0  0  0  1  1  0  1  2  |  *  *  *  *  *  4  <-  3pr`

--- rk
Klitzing
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

Klitzing wrote:Sadly esp. in the superdiagonal part there had be several errors.
Here now is the corrected matrix:
[...]
--- rk

Thank you
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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student91
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

Btw., as segmetochoron squippy || squacu has height 1/2.
The vertices belong to a circumradius of sqrt[2+sqrt(2)] = 1.847759.
In fact it is nothing but a tiny section from spic = x3o4o3x.

How can this be seen?
The octahedron first cap of spic is oct || sirco.
If you would chopp off similarily 2 such octahedra, those segments would intersect.
The bottom sirco of the second would intersect the top octahedron of the first in its diagonal square and the bottom sirco in its upper octagon.
Similarily the other way round.

The lace city of oct || sirco is
Code: Select all
`        o4o         o4x         o4o                                                                                                                                         x4o         x4x             x4x         x4o`

That of squippy || squacu then is
Code: Select all
`        o4o         o4x                                                                     x4o         x4x        `

--- rk
Klitzing
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

And here is the corresponding full lace city of spic = x3o4o3x
Code: Select all
`                o4o         o4x         o4o                                                                                                                                                                                                         x4o         x4x             x4x         x4o                                                                                                                                                                                         o4o                         o4w                         o4o                                                                   x4x         w4o             w4o         x4x                                                                                                                                                                                         o4x             o4w         q4q         o4w             o4x                                                                                                                                                                                         x4x         w4o             w4o         x4x                                                                   o4o                         o4w                         o4o                                                                                                                                                                                         x4o         x4x             x4x         x4o                                                                                                                                                                                                         o4o         o4x         o4o                `

--- rk
Klitzing
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### Re: Dichoral angles of CRF's ["split" from IncMats Update]

student91: some time ago, I suggested a combinatorical approach to CRF polychora consisting of first enumerating all possible types of vertices, which would naturally include possible dichoral angles.

Basically:

If you're trying to build a Johnson solid, then you can imagine each vertex surrounded by a unit sphere, with adjacent vertices on that sphere. Then you connect those vertices by "struts" corresponding to chords of polygons.
But it won't work perfectly, because if you have more than 3 adjacent vertices, the polygon won't be rigid, as the vertices can move with some freedom on the sphere -- this is probably the reason why there's so many anomalous Johnson solids.

However, if we move one dimension higher, it might work. We'll get a skelet of polyhedron inscribed in hypersphere, but it has more edges, so the freedom must be lower. If those polyhedra are completely rigid, then we can combinatorically build all possible "vertices" for CRF polychora, and then combine them together like pieces of puzzle until they close or until we get an impossible configuration.

I tried enumerating tetrahedral CRF vertices, it's probably somewhere in the thread -- even with just those it might be possible to find something new...
Marek14
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