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)
{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˚