Dihedral Angles of Polydronic Trihedral Acrons (decimal degrees)

It is fascinating to look at the dihedral angles in trihedral acrons and compare those acrons which have been resolved with those that have not.

We begin by looking at Uniform Polyhedra and Johnson Solids:

(note: all dihedral angles are presented in degrees to five decimal places.  I believe them to be accurate to +/- 0.00001 degrees)

Alternative versions of this page are available showing the dihedral angles in dd-mm-ss form and in radians.

The decimal degree data is also available in comma separated variable format

1. Acrons Found in Uniform Polyhedra

Acron         Dihedral Angles (Degrees)
A-B-C              A-B        B-C        C-A
3-3-3          70.52878   70.52878   70.52878
4-4-3                60         90         90
4-4-4                90         90         90
5-4-4                90        108         90
5-5-5         116.56505  116.56505  116.56505
6-4-4                90        120         90
6-6-3          70.52878  109.47122  109.47122
6-6-4         109.47122  125.26439  125.26439
6-6-5         138.18969  142.62263  142.62263
8-4-4                90        135         90
8-6-4         125.26439  144.73561        135
8-8-3                90  125.26439  125.26439
10-4-4               90        144         90
10-6-4        142.62263  159.09484  148.28253
10-10-3       116.56505  142.62263  142.62263

2. Acrons Found In Johnson Solids

Acron         Dihedral Angles (Degrees)
A-B-C              A-B        B-C        C-A
4-3-3          54.73561  109.47122   54.73561
5-3-3          37.37737  138.18969   37.37737
5-5-3          63.43495  100.81232  100.81232
6-4-3          54.73561  125.26439   70.52878
8-4-3                45  144.73561   54.73561
10-4-3         31.71747  159.09484   37.37737
10-5-3         63.43495  142.62263   79.18768
10-5-4        116.56505  148.28253  121.71747

Looking now at the remaining polydronic acrons, we can colour code the acrons themselves: green if resolved, black otherwise.  We can also colour code the dihedral angles: red if they occur in a Uniform Polyhedron, blue if they occur in a Johnson Solid and black otherwise.  Note the precise correlation between the solved acrons and those with a dihedral angle occurring in the above polyhedra.

Where a dihedral angle is shared between either two unsolved acrons, then this is noted.

Acron         Dihedral Angles (Degrees)
A-B-C              A-B        B-C        C-A      Shared angles with
5-4-3          58.28253  110.90516   79.18768
5-5-4          96.06017  108.96071  108.96071     5-5-4 = 8-5-4
6-5-3          65.19890  114.80110   94.51323     6-5-3 = 15-5-3
6-5-4         100.81232  121.71747  110.90516
6-5-5         124.24809  131.17486  124.24809     6-5-5 = 12-5-4
8-5-3          65.25526  132.13833   85.82933
8-5-4         108.96071  138.03008  115.91365     8-5-4 = 5-5-4
8-5-5         141.66758  152.53973  141.66758     8-5-5 = 16-5-4
8-6-3          76.16384  127.55160  103.83616
10-6-3         79.18768  138.18969  100.81232
10-8-3          99.9844  138.05347  126.47361

4. Apolydronic Acrons

Turning finally to apolydronic acrons (with an artificial upper limit of a pentadecagon), we see only a few have been resolved.  None of the angles involved in these occur in the polydronic acrons.

Acron         Dihedral Angles (Degrees)
A-B-C              A-B        B-C        C-A      Shared angles with
7-4-3          50.24367  136.04986   62.58569     7-4-3 = 7-6-4
9-4-3          38.93477  152.19643   46.52332     9-4-3 = 9-6-4
11-4-3         22.35750  166.26298   26.05477     11-4-3 = 11-6-4

7-5-3          65.58640   124.7085   89.76916
9-5-3          64.49003  137.94314   82.36033
11-5-3         62.17081  146.49043   76.20928
12-5-3         60.74583   149.7539   73.36009
13-5-3         59.18983   152.5559   70.59566
14-5-3         57.52145  154.99833   67.88362
15-5-3         55.75191  157.15596   65.19890     15-5-3 = 6-5-3
16-5-3         53.88726  159.08513   62.52085
17-5-3         51.92953   160.8293   59.83133
18-5-3         49.87739  162.42279   57.11335
19-5-3         47.72632  163.89334   54.34991
20-5-3         45.46847  165.26399   51.52297
21-5-3         43.09215  166.55447   48.61217
22-5-3         40.58078  167.78241   45.59331
23-5-3         37.91107  168.96442   42.43602
24-5-3         35.04990  170.11732   39.10005
25-5-3         31.94850  171.25983   35.52881
26-5-3         28.53095  172.41550   31.63661
27-5-3         24.66831  173.61891   27.28028
28-5-3         20.10772  174.93222   22.18145
29-5-3         14.19374  176.51051   15.62123

7-5-4         105.01754  130.96334  113.28142
9-5-4         112.78189  143.65536  118.73408
11-5-4        120.37019  152.19574  124.86016     11-5-4 = 8-5-5
12-5-4        124.24809  155.58743  128.17271     12-5-4 = 6-5-5
13-5-4        128.24918  158.59494  131.67838
14-5-4        132.43107  161.32225  135.41514     14-5-4 = 7-5-5
15-5-4        136.86809  163.85448  139.44191
16-5-4        141.66758  166.26987  143.85242     16-5-4 = 8-5-5
17-5-4        147.00461  168.65467  148.80728
18-5-4        153.21572  171.13392  154.62271     18-5-4 = 9-5-5
19-5-4        161.16616  173.98314  162.11999

7-5-5         132.43107  142.64451  132.43107     7-5-5 = 14-5-4
9-5-5         153.21572  162.26785  153.21572     9-5-5 = 18-5-4

7-6-3          73.85690  119.86687  106.14310
9-6-3          77.86954  133.47668  102.13046     9-6-3 = 9-6-4 = 9-9-3 , 9-6-3 = 9-9-3
11-6-3         80.23978  142.03047   99.76022
12-6-3         81.10057  145.22189   98.89943
13-6-3         81.81881  147.91648   98.18119
14-6-3         82.42774  150.22226   97.57226
15-6-3         82.95092  152.21798   97.04908
16-6-3         83.40548  153.96238   96.59452
17-6-3         83.80426  155.50024   96.19574
18-6-3         84.15703  156.86624   95.84297
19-6-3         84.47139  158.08772   95.52861
20-6-3         84.75336  159.18651   95.24664
21-6-3         85.00772  160.18021   94.99228
22-6-3         85.23837  161.08325   94.76163
23-6-3         85.44851  161.90750   94.55149
24-6-3         85.64077  162.66285   94.35923
25-6-3         85.81734  163.35760   94.18266
26-6-3         85.98010  163.99877   94.01990
27-6-3         86.13059  164.59233   93.86941
28-6-3         86.27018  165.14339   93.72982
29-6-3         86.39999  165.65638   93.60001
30-6-3         86.52104  166.13510   93.47896
31-6-3         86.63418  166.58288   93.36582
32-6-3         86.74017  167.00263   93.25983
33-6-3         86.83967  167.39689   93.16033
34-6-3         86.93325  167.76793   93.06675
35-6-3         87.02143  168.11774   92.97857
36-6-3         87.10467  168.44809   92.89533
37-6-3         87.18337  168.76056   92.81663
38-6-3         87.25789  169.05656   92.74211
39-6-3         87.32856  169.33737   92.67144
40-6-3         87.39567  169.60412   92.60433
41-6-3         87.45949  169.85785   92.54051
42-6-3         87.52024  170.09948   92.47976
43-6-3         87.57815  170.32986   92.42185
44-6-3         87.63342  170.54977   92.36658
45-6-3         87.68621  170.75989   92.31379
46-6-3         87.73670  170.96087   92.26330
47-6-3         87.78502  171.15328   92.21498
48-6-3         87.83132  171.33768   92.16868
49-6-3         87.87572  171.51454   92.12428
50-6-3         87.91834  171.68433   92.08166
... etc

7-6-4         117.41431  136.04986  129.75633     7-6-4 = 7-7-3 = 14-7-3 , 7-6-4 = 7-4-3
9-6-4         133.47668  152.19643  141.06523     9-6-4 = 9-6-3 = 9-9-3 , 9-6-4 = 9-4-3
11-6-4        153.94523  166.26298  157.64250     11-6-4 = 11-11-3 , 11-6-4 = 11-4-3

7-6-5         156.46468  160.83663  158.67776

7-7-3          79.51267  117.41431  117.41431     7-7-3 = 7-6-4 = 14-7-3
8-7-3          83.86047  125.72651  116.15521
9-7-3          87.44772  132.14143  115.59295
10-7-3         90.55032  137.25906  115.47946
11-7-3         93.32461  141.44779  115.67557
12-7-3         95.86682  144.94754  116.09674
13-7-3         98.23994  147.92179  116.68852
14-7-3        100.48733  150.48598  117.41431     14-7-3 = 7-7-3 = 7-6-4
15-7-3        102.64004  152.72406  118.24870

16-7-3        104.72111  154.69863  119.17361
17-7-3        106.74823  156.45735  120.17599
18-7-3        108.73533  158.03721  121.24634
19-7-3        110.69377  159.46735  122.37774
20-7-3        112.63304  160.77112  123.56525
21-7-3        114.56130  161.96742  124.80545
22-7-3        116.48583  163.07180  126.09615
23-7-3        118.41329  164.09718  127.43621
24-7-3        120.34995  165.05442  128.82542
25-7-3        122.30198  165.95275  130.26442
26-7-3        124.27556  166.80013  131.75468
27-7-3        126.27711  167.60351  133.29853
28-7-3        128.31354  168.36907  134.89924
29-7-3        130.39240  169.10234  136.56114
30-7-3        132.52227  169.80846  138.28982
31-7-3        134.71311  170.49228  140.09242
32-7-3        136.97681  171.15853  141.97811
33-7-3        139.32793  171.81202  143.95868
34-7-3        141.78491  172.45787  146.04964
35-7-3        144.37185  173.10188  148.27180
36-7-3        147.12158  173.75101  150.65402
37-7-3        150.08109  174.41437  153.23812
38-7-3        153.32231  175.10498  156.08865
39-7-3        156.96616  175.84386  159.31466
40-7-3        161.24847  176.67109  163.12933
41-7-3        166.77900  177.68961  168.08426

7-7-4         129.49150   142.8895  142.88950
9-7-4         161.87715   168.4664  165.92493

9-8-3          95.26088  132.34514  125.60430
11-8-3        104.36661  142.78873  127.72440
12-8-3        108.53218  146.81036  129.27130
13-8-3        112.56773  150.29559  131.06415
14-8-3        116.53898  153.37078  133.07479
15-8-3        120.50069  156.12961  135.29052
16-8-3        124.50336  158.64446  137.71111
17-8-3        128.59881  160.97396  140.34827
18-8-3        132.84642  163.16868  143.22756
19-8-3        137.32188  165.27633  146.39351
20-8-3        142.13302  167.34804  149.92121
21-8-3        147.45373  169.44940  153.94337
22-8-3        153.61603  171.68787  158.72538
23-8-3        161.47047  174.31820  164.96096

9-9-3         102.13046  133.47668  133.47668      9-9-3 = 9-6-3 = 9-6-4 , 9-9-3 = 9-6-3
10-9-3        108.47739  139.92968  135.25405
11-9-3        114.55882  145.40284  137.53951
12-9-3        120.56090  150.18837  140.27348
13-9-3        126.64670  154.49792  143.45085
14-9-3        132.99331  158.50301  147.11943
15-9-3        139.84265  162.36946  151.40246
16-9-3        147.61804  166.31073  156.57752
17-9-3        157.36421  170.76175  163.40155

11-10-3       124.63037  149.09086  146.04985
12-10-3       133.05200  155.04605  150.26685
13-10-3       142.37566  160.87717  155.52201
14-10-3       153.83611  167.23730  162.58604

11-11-3       135.28500  153.94523  153.94523     11-11-3 = 11-6-4
12-11-3       147.72198  162.04191  160.52582
13-11-3       167.08561  173.11188  171.97975