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 seconds of arc to one decimal place. I believe them to be accurate to +/- 0.1 seconds of arc)
Alternative versions of this page are available showing the dihedral angles in decimal degrees and in radians.
The decimal degree data is also available in comma separated variable format
1. Acrons Found in Uniform Polyhedra
Acron
Dihedral Angles (DD MM' SS")
A-B-C
A-B
B-C
C-A
3-3-3
70 31' 43.6"
70 31' 43.6" 70 31' 43.6"
4-4-3
60
90
90
4-4-4
90
90
90
5-4-4
90
108
90
5-5-5
116 33' 54.2" 116
33' 54.2" 116 33' 54.2"
6-4-4
90
120
90
6-6-3
70 31' 43.6" 109
28' 16.4" 109 28' 16.4"
6-6-4
109 28' 16.4" 125
15' 51.8" 125 15' 51.8"
6-6-5
138 11' 22.9" 142
37' 21.5" 142 37' 21.5"
8-4-4
90
135
90
8-6-4
125 15' 51.8" 144
44' 8.2" 135
8-8-3
90
125 15' 51.8" 125 15' 51.8"
10-4-4
90
144
90
10-6-4 142
37' 21.5" 159 5' 41.4"
148 16' 57.1"
10-10-3 116
33' 54.2" 142 37' 21.5" 142
37' 21.5"
2. Acrons Found In Johnson Solids
Acron
Dihedral Angles (DD MM' SS")
A-B-C
A-B
B-C
C-A
4-3-3
54 44' 8.2"
109 28' 16.4" 54 44' 8.2"
5-3-3
37 22' 38.5" 138
11' 22.9" 37 22' 38.5"
5-5-3
63 26' 5.8"
100 48' 44.3" 100 48' 44.3"
6-4-3
54 44' 8.2"
125 15' 51.8" 70 31' 43.6"
8-4-3
45
144 44' 8.2" 54 44' 8.2"
10-4-3
31 43' 2.9"
159 5' 41.4" 37 22' 38.5"
10-5-3
63 26' 5.8"
142 37' 21.5" 79 11' 15.7"
10-5-4 116
33' 54.2" 148 16' 57.1" 121
43' 2.9"
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 exact correlation between those acrons which have been resolved and those with dihedral angles occurring in the above polyhedra.
3. Polydronic Acrons Not Found In Uniform Polyhedra or Johnson Solids
Acron
Dihedral Angles (DD MM' SS")
A-B-C
A-B
B-C
C-A
5-4-3
58 16' 57.1" 110 54' 18.6"
79 11' 15.7"
5-5-4
96 3' 36.6" 108 57' 38.6"
108 57' 38.6"
6-5-3
65 11' 56.0" 114 48' 4.0"
94 30' 47.6"
6-5-4
100 48' 44.3" 121
43' 2.9" 110
54' 18.6"
6-5-5
124 14' 53.1" 131 10' 29.5"
124 14' 53.1"
8-5-3
65 15' 18.9" 132 8' 18.0"
85 49' 45.6"
8-5-4
108 57' 38.6" 138 1' 48.3"
115 54' 49.1"
8-5-5
141 40' 3.3" 152 32' 23.0"
141 40' 3.3"
8-6-3
76 9' 49.8" 127 33' 5.8"
103 50' 10.2"
10-6-3
79 11' 15.7" 138
11' 22.9" 100 48'
44.3"
10-8-3
99 59' 3.8" 138 3' 12.5"
126 28' 25.0"
Turning finally to apolydronic acrons (with an artificial upper limit of a pentadecagon), we see only the first three have been resolved although the angles do not occur in the polydronic acrons. I believe this to be a feature of the precise solutions found to these acrons where angles are 'cancelled out' by connecting two identical polyhedral sections. This implies that solutions may exist to the outstanding acrons above although I believe such solutions will be 'paired solutions' in similar fashion to the resolved apolydronic acrons.
4. Apolydronic Acrons
Acron
Dihedral Angles (Degrees)
A-B-C
A-B B-C
C-A
7-4-3
50 14' 37.2" 136 2' 59.5"
62 35' 8.5"
9-4-3
38 56' 5.2" 152 11' 47.2"
46 31' 24.0"
11-4-3
22 21' 27.0" 166 15' 46.7"
26 3' 17.2"
7-5-3
65 35' 11.1" 124 42' 30.6"
89 46' 9.0"
9-5-3
64 29' 24.1" 137 56' 35.3"
82 21' 37.2"
11-5-3
62 10' 14.9" 146 29' 25.6"
76 12' 33.4"
12-5-3
60 44' 45" 149 45'
14" 73 21' 36.3"
13-5-3
59 11' 23.4" 152 33' 21.3"
70 35' 44.4"
14-5-3
57 31' 17.2" 154 59' 54"
67 53' 1.0"
15-5-3
55 45' 6.9" 157 9' 21.4"
65 11' 56"
16-5-3
53 53' 14.1" 159 5' 6.5"
62 31' 15.1"
17-5-3
51 55' 46.3" 160 49' 45.5"
59 49' 52.8"
18-5-3
49 52' 38.6" 162 25' 22.1"
57 6' 48.1"
19-5-3
47 43' 34.8" 163 53' 36.0"
54 20' 59.7"
20-5-3
45 28' 6.5" 165 15' 50.4"
51 31' 22.7"
21-5-3
43 5' 31.8" 166 33' 16.1"
48 36' 43.8"
22-5-3
40 34' 50.8" 167 46' 56.7"
45 35' 35.9"
23-5-3
37 54' 39.8" 168 57' 51.9"
42 26' 9.7"
24-5-3
35 2' 59.6" 170 7' 2.3"
39 6' 0.2"
25-5-3
31 56' 54.6" 171 15' 35.4"
35 31' 43.7"
26-5-3
28 31' 51.4" 172 24' 55.8"
31 38' 11.8"
27-5-3
24 40' 5.9" 173 37' 8.1"
27 16' 49.0"
28-5-3
20 6' 27.8" 174 55' 56.0"
22 10' 53.2"
29-5-3
14 11' 37.4" 176 30' 37.8"
15 37' 16.4"
7-5-4
105 1' 3.2" 130 57' 48.0"
113 16' 53.1"
9-5-4
112 46' 54.8" 143 39' 19.3"
118 44' 2.7"
11-5-4 120
22' 12.7" 152 11' 44.7" 124
51' 36.6"
12-5-4 124
14' 53.1" 155 35' 14.7" 128
10' 21.7"
13-5-4 128
14' 57.1" 158 35' 41.8" 131
40' 42.2"
14-5-4 132
25' 51.8" 161 19' 20.1" 135
24' 54.5"
15-5-4 136
52' 5.1" 163 51' 16.1"
139 26' 30.9"
16-5-4 141
40' 3.3" 166 16' 11.5"
143 51' 8.7"
17-5-4 147
0' 16.6" 168 39' 16.8" 148
48' 26.2"
18-5-4 153
12' 56.6" 171 8' 2.1"
154 37' 21.8"
19-5-4 161
9' 58.2" 173 58' 59.3" 162
7' 12.0"
7-5-5
132 25' 51.8" 142 38' 40.2"
132 25' 51.8"
9-5-5
153 12' 56.6" 162 16' 4.2"
153 12' 56.6"
7-6-3
73 51' 24.9" 119 52' 0.7"
106 8' 35.1"
9-6-3
77 52' 10.4" 133 28' 36.0"
102 7' 49.6"
11-6-3
80 14' 23.2" 142 1' 49.7"
99 45' 36.8"
12-6-3
81 6' 2.1" 145 13' 18.8"
98 53' 57.9"
13-6-3
81 49' 7.7" 147 54' 59.3"
98 10' 52.3"
14-6-3
82 25' 39.9" 150 13' 20.1"
97 34' 20.1"
15-6-3
82 57' 3.3" 152 13' 4.7"
97 2' 56.7"
16-6-3
83 24' 19.7" 153 57' 44.6"
96 35' 40.3"
17-6-3
83 48' 15.3" 155 30' 0.8"
96 11' 44.7"
18-6-3
84 9' 25.3" 156 51' 58.5"
95 50' 34.7"
19-6-3
84 28' 17" 158 5' 15.8"
95 31' 43"
20-6-3
84 45' 12.1" 159 11' 11.4"
95 14' 47.9"
21-6-3
85 0' 27.8" 160 10' 48.8"
94 59' 32.2"
22-6-3
85 14' 18.1" 161 4' 59.7"
94 45' 41.9"
23-6-3
85 26' 54.6" 161 54' 27.0"
94 33' 5.4"
24-6-3
85 38' 26.8" 162 39' 46.2"
94 21' 33.2"
25-6-3
85 49' 2.4" 163 21' 27.3"
94 10' 57.6"
26-6-3
85 58' 48.3" 163 59' 55.6"
94 1' 11.7"
27-6-3
86 7' 50.1" 164 35' 32.4"
93 52' 9.9"
28-6-3
86 16' 12.6" 165 8' 36.2"
93 43' 47.4"
29-6-3
86 24' 0.0" 165 39' 23.0"
93 36' 0.0"
30-6-3
86 31' 15.8" 166 8' 6.4"
93 28' 44.2"
31-6-3
86 38' 3.1" 166 34' 58.4"
93 21' 56.9"
32-6-3
86 44' 24.6" 167 0' 9.5"
93 15' 35.4"
33-6-3
86 50' 22.8" 167 23' 48.8"
93 9' 37.2"
34-6-3
86 55' 59.7" 167 46' 4.6"
93 4' 0.3"
35-6-3
87 1' 17.1" 168 7' 3.9"
92 58' 42.9"
36-6-3
87 6' 16.8" 168 26' 53.1"
92 53' 43.2"
37-6-3
87 11' 0.1" 168 45' 38.0"
92 48' 59.9"
38-6-3
87 15' 28.4" 169 3' 23.6"
92 44' 31.6"
39-6-3
87 19' 42.8" 169 20' 14.5"
92 40' 17.2"
40-6-3
87 23' 44.4" 169 36' 14.8"
92 36' 15.6"
41-6-3
87 27' 34.2" 169 51' 28.3"
92 32' 25.8"
42-6-3
87 31' 12.9" 170 5' 58.1"
92 28' 47.1"
43-6-3
87 34' 41.4" 170 19' 47.5"
92 25' 18.6"
44-6-3
87 38' 0.3" 170 32' 59.2"
92 21' 59.7"
45-6-3
87 41' 10.4" 170 45' 35.6"
92 18' 49.6"
46-6-3
87 44' 12.1" 170 57' 39.1"
92 15' 47.9"
47-6-3
87 47' 6.1" 171 9' 11.8"
92 12' 53.9"
48-6-3
87 49' 52.8" 171 20' 15.6"
92 10' 7.2"
49-6-3
87 52' 32.6" 171 30' 52.4"
92 7' 27.4"
50-6-3
87 55' 6.0" 171 41' 3.6"
92 4' 54.0"
etc ...
7-6-4
117 24' 51.5" 136 2' 59.5"
129 45' 22.8"
9-6-4
133 28' 36.0" 152 11' 47.2"
141 3' 54.8"
11-6-4 153
56' 42.8" 166 15' 46.7" 157
38' 33.0"
7-6-5 156 27' 52.8" 160 50' 11.9" 158 40' 39.9"
7-7-3
79 30' 45.6" 117 24' 51.5"
117 24' 51.5"
8-7-3
83 51' 37.7" 125 43' 35.4"
116 9' 18.8"
9-7-3
87 26' 51.8" 132 8' 29.1"
115 35' 34.6"
10-7-3
90 33' 1.1" 137 15' 32.6"
115 28' 46.0"
11-7-3
93 19' 28.6" 141 26' 52.0"
115 40' 32.0"
12-7-3
95 52' 0.5" 144 56' 51.1"
116 5' 48.3"
13-7-3
98 14' 23.8" 147 55' 18.4"
116 41' 18.7"
14-7-3 100
29' 14.4" 150 29' 9.5"
117 24' 51.5"
15-7-3 102
38' 24.1" 152 43' 26.6" 118
14' 55.3"
16-7-3 104
43' 16.0" 154 41' 55.1" 119
10' 25.0"
17-7-3 106
44' 53.6" 156 27' 26.5" 120
10' 33.6"
18-7-3 108
44' 7.2" 158 2' 13.9"
121 14' 46.8"
19-7-3 110
41' 37.6" 159 28' 2.5"
122 22' 39.9"
20-7-3 112
37' 58.9" 160 46' 16.0" 123
33' 54.9"
21-7-3 114
33' 40.7" 161 58' 2.7"
124 48' 19.6"
22-7-3 116
29' 9.0" 163 4' 18.5"
126 5' 46.1"
23-7-3 118
24' 47.8" 164 5' 49.9"
127 26' 10.3"
24-7-3 120
20' 59.8" 165 3' 15.9"
128 49' 31.5"
25-7-3 122
18' 7.1" 165 57' 9.9"
130 15' 51.9"
26-7-3 124
16' 32.0" 166 48' 0.5"
131 45' 16.8"
27-7-3 126
16' 37.6" 167 36' 12.7" 133
17' 54.7"
28-7-3 128
18' 48.7" 168 22' 8.6"
134 53' 57.3"
29-7-3 130
23' 32.6" 169 6' 8.4"
136 33' 40.1"
30-7-3 132
31' 20.2" 169 48' 30.5" 138
17' 23.4"
31-7-3 134
42' 47.2" 170 29' 32.2" 140
5' 32.7"
32-7-3 136
58' 36.5" 171 9' 30.7"
141 58' 41.2"
33-7-3 139
19' 40.5" 171 48' 43.3" 143
57' 31.3"
34-7-3 141
47' 5.7" 172 27' 28.3"
146 2' 58.7"
35-7-3 144
22' 18.7" 173 6' 6.8"
148 16' 18.5"
36-7-3 147
7' 17.7" 173 45' 3.6" 150
39' 14.5"
37-7-3 150
4' 51.9" 174 24' 51.7" 153
14' 17.2"
38-7-3 153
19' 20.3" 175 6' 17.9"
156 5' 19.2"
39-7-3 156
57' 58.2" 175 50' 37.9" 159
18' 52.8"
40-7-3 161
14' 54.5" 176 40' 15.9" 163
7' 45.6"
41-7-3 166
46' 44.4" 177 41' 22.6" 168
5' 3.3"
7-7-4
129 29' 29.4" 142 53' 22.2"
142 53' 22.2"
9-7-4
161 52' 37.8" 168 27' 59.0"
165 55' 29.8"
9-8-3
95 15' 39.2" 132 20' 42.5"
125 36' 15.5"
11-8-3 104
21' 59.8" 142 47' 19.4" 127
43' 27.8"
12-8-3 108
31' 55.9" 146 48' 37.3" 129
16' 16.7"
13-8-3 112
34' 3.8" 150 17' 44.1"
131 3' 50.9"
14-8-3 116
32' 20.3" 153 22' 14.8" 133
4' 29.2"
15-8-3 120
30' 2.5" 156 7' 46.6"
135 17' 25.9"
16-8-3 124
30' 12.1" 158 38' 40.1" 137
42' 40.0"
17-8-3 128
35' 55.7" 160 58' 26.3" 140
20' 53.8"
18-8-3 132
50' 47.1" 163 10' 7.2"
143 13' 39.2"
19-8-3 137
19' 18.8" 165 16' 34.8" 146
23' 36.6"
20-8-3 142
7' 58.9" 167 20' 52.9" 149
55' 16.4"
21-8-3 147
27' 13.4" 169 26' 57.8" 153
56' 36.1"
22-8-3 153
36' 57.7" 171 41' 16.3" 158
43' 31.4"
23-8-3 161
28' 13.7" 174 19' 5.5"
164 57' 39.4"
9-9-3
102 7' 49.6" 133 28' 36"
133 28' 36.0"
10-9-3 108
28' 38.6" 139 55' 46.8" 135
15' 14.6"
11-9-3 114
33' 31.7" 145 24' 10.2" 137
32' 22.2"
12-9-3 120
33' 39.2" 150 11' 18.1" 140
16' 24.5"
13-9-3 126
38' 48.1" 154 29' 52.5" 143
27' 3.1"
14-9-3 132
59' 35.9" 158 30' 10.8" 147
7' 10.0"
15-9-3 139
50' 33.5" 162 22' 10.1" 151
24' 8.9"
16-9-3 147
37' 4.9" 166 18' 38.6"
156 34' 39.1"
17-9-3 157
21' 51.2" 170 45' 42.3" 163
24' 5.6"
11-10-3 124
37' 49.3" 149 5' 27.1"
146 2' 59.5"
12-10-3 133
3' 7.2" 155 2' 45.8"
150 16' 0.7"
13-10-3 142
22' 32.4" 160 52' 37.8" 155
31' 19.2"
14-10-3 153
50' 10.0" 167 14' 14.3" 162
35' 9.8"
11-11-3 135
17' 6.0" 153 56' 42.8"
153 56' 42.8"
12-11-3 147
43' 19.1" 162 2' 30.9"
160 31' 32.9"
13-11-3 167
5' 8.2" 173 6' 42.8"
171 58' 47.1"