mr_e_man wrote:Two tids, sharing a decagon, don't fit with anything else. So one tid must be surrounded by 12 peros. Each pero has two possible orientations; we might call them ortho and gyro. If two triangles in two adjacent peros meet along an edge (with dihedral angles 79.19°+116.57°+79.19°), then nothing else fits at that edge; so we can't have two gyro peros next to each other. An ortho pero has a triangle meeting a triangle of tid (with dihedral angles 79.19°+142.62° = 360° - 138.19°); this only fits with the blunt end of a teddi. And we already know that this allows only one configuration; all peros must be ortho. The result is symmetrical:
tid + 12 pero + 20 teddi + 30 doe
This leaves 12 pero pentagons exposed, which can't be covered by id or pero or bilbiro (142.62°+142.62°) and must be covered by 12 doe's. Now we have a triangular depression between three doe's, which must be filled by a teddi. So any one of the 20 original teddis shares an edge with a new teddi, pointing in the opposite direction. But two of these new teddis, sharing an edge, point in the same direction; the required pattern cannot be continued. So tid and teddi are out.
mr_e_man wrote:Also I noticed that most Archimedean solids in the cubic family can be decomposed into smaller polyhedra while maintaining symmetry:
tut = tet + 4 oct + 6 tet
tricu = tet + 3 squippy + 3 tet
(These break each hexagon into 6 triangles. The following decompositions preserve the external faces.)
tic = cube + 6 squacu + 8 tet
toe = oct + 8 tricu + 6 squippy
co = 2 tricu = 6 squippy + 8 tet
sirco = op + 2 squacu
girco = sirco + 6 squacu + 8 tricu + 12 cube
Another way to describe this is that these segmentochora are degenerate:
tic || cube
toe || oct
co || point
girco || sirco
It seems that no such decompositions are possible with the dodecahedral family.
On Wed 25 Aug 2004 (01:44:23), [andrew weimholt] wrote:
>Here's an interesting 3-honeycomb I came across while playing with
>Great Stella. It consists of Dodecahedra, Cubes, and Bilunabirotundas
>(J91s)
(N→∞)
12N | 2 4 1 2 | 3 6 4 3 1 2 | 1 2 5 3 1
----+----------------+-----------------------+---------------
2 | 12N * * * | 2 2 0 0 0 0 | 1 2 1 0 0
2 | * 24N * * | 0 2 1 1 0 0 | 0 1 2 1 0
2 | * * 6N * | 2 0 2 0 0 2 | 1 0 2 2 1
2 | * * * 12N | 0 0 1 1 1 1 | 0 0 1 2 1
----+----------------+-----------------------+---------------
3 | 2 0 1 0 | 12N * * * * * | 1 0 1 0 0
3 | 1 2 0 0 | * 24N * * * * | 0 1 1 0 0
4 | 0 2 1 1 | * * 12N * * * | 0 0 1 1 0
3 | 0 2 0 1 | * * * 12N * * | 0 0 1 1 0
3 | 0 0 0 3 | * * * * 4N * | 0 0 0 1 1
6 | 0 0 3 3 | * * * * * 4N | 0 0 0 1 1
----+----------------+-----------------------+---------------
4 | 4 0 2 0 | 4 0 0 0 0 0 | 3N * * * * tet (T) 1
4 | 2 4 0 0 | 0 4 0 0 0 0 | * 6N * * * tet (T) 2
5 | 2 4 1 1 | 1 2 1 1 0 0 | * * 12N * * squippy (Y4)
9 | 0 6 3 6 | 0 0 3 3 1 1 | * * * 4N * tricu (Q3)
12 | 0 0 6 12 | 0 0 0 0 4 4 | * * * * N tut (T3)
Klitzing wrote:When I read your starting article here I can get the followings
- 2D convex tilings with regular polygons only can use 3-, 4-, 6-, 8-, and 12-gons,
as any other being used regular convex polygon ultimatly results in a non-continuable configuration- morover there is only a single 2D convex tiling with regular polygons only using 8-gons: the uniform 4.8.8-tiling
- occurances of 6-gons variously might be decomposed into 6 3-gons each (6-pyramid)
- occurances of 12-gons variously might be decomposed into 6 3-gons, 6 4-gons, and 1 6-gon each (6-cupola),
in fact within 2 different orientations.About further restrictions on the usage of 10-gons - at least therein - you still seem unfixed.
- 3D convex tilings with regular polygons should similarily be restricted to use 3-, 4-, 5-, 6-, 8-, 10-, and 12-gons only,
as any other being used regular convex polygon ultimatly results in a non-continuable configuration- morover usages of 12-gons only can occur within infinite stacks of according 2D tilings.
- sevaral CRF cells can also be excluded for similar reasons:
4-ap, n-ap with n>5, 7-p, 9-p, 10-p, 11-p, n-p with n>12, J52-53, J84-90, J92, snic, snid, grid
Within a later post of that very thread you show up a configuration with a tid, but I don't get exactly why it isn't continuable.
Still, that one alone doesn't rule out any other 10-gon usage.
A bit later in that thread you further observe the (non-new) facts of degenerate segmentochoraas well as the obvious decomposition of sirco into the lace tower squacu || op || squacu.
- tic || cube
- toe || oct
- co || point
- girco || sirco
From that you now deduce that ANY 3D convex tiling with regular polygons should be decomposable into P3 (trip), P4 (cube), P8 (op), T (tet), Y4 (squippy), and Q4 (squacu)
- except for the single case of Weimholt's cube-doe-bilbiro honeycomb.
First of all I don't see the final 10-gon exclusion. But esp. I don't see why that mentioned exception should be the only possible 5-gon usage.
--- rk
Klitzing wrote:A bit later in that thread you further observe the (non-new) facts of degenerate segmentochoraas well as the obvious decomposition of sirco into the lace tower squacu || op || squacu.
- tic || cube
- toe || oct
- co || point
- girco || sirco
From that you now deduce that ANY 3D convex tiling with regular polygons should be decomposable into P3 (trip), P4 (cube), P8 (op), T (tet), Y4 (squippy), and Q4 (squacu)
- except for the single case of Weimholt's cube-doe-bilbiro honeycomb.
Klitzing wrote:About further restrictions on the usage of 10-gons - at least therein - you still seem unfixed.
Within a later post of that very thread you show up a configuration with a tid, but I don't get exactly why it isn't continuable.
Still, that one alone doesn't rule out any other 10-gon usage.
mr_e_man wrote:Now I've also ruled out [...] indeed anything with a 5.4.3.4 vertex; the two large angles 159.09°, at the triangle, conspire so they can't both be surrounded by polyhedra, though one can be surrounded.
Klitzing wrote:Within a later post of that very thread you show up a configuration with a tid, but I don't get exactly why it isn't continuable.
Klitzing wrote:You still don't show, that tid-10-tid is impossible
mr_e_man wrote:Two tids, sharing a decagon, don't fit with anything else.
Klitzing wrote:that pero-10-pero is impossible - in either orientation
mr_e_man wrote:In fact pobro can't be used either. The 5.5.3.3 vertex can only be completed with a pair of dodecahedra (attaching to the two pentagons) and a pair of tridiminished icosahedra (with their sharp ends attaching to the two triangles). But a teddi, with its three 5.3.3.3 vertices, needs to attach to three dodecahedra, not another teddi.
mr_e_man wrote:The only thing left with decagons is pero. Thus, one pero must be paired with another, effectively forming an icosidodecahedron. As before, the pentagons can't be covered by id or pero or bilbiro and must be covered by doe's. Again we get these triangular depressions which can only be filled by teddis. So id and pero are out.
Klitzing wrote:The argument about the orange teddis also misses mentioning of further reseach.
So eg. why can't you insert into those light green / dark green gaps ids? Do you really need further peroes?
mr_e_man wrote:A 5.3.3.3 vertex can only be completed in one way: The middle triangle attaches to a truncated dodecahedron (10.10.3), whose two decagons attach to pentagonal rotundas (10.5.3), and the three exposed pentagons are covered by a dodecahedron (5.5.5).
int N = 109; // total number of dihedral angles, plus one
int a[N] = { // list of angles, in units of 0.0001 degree
0, 317175, 373774, 450000, 547356, 600000, 634349, 690948, 705288, 729730,
747547, 791877, 867268, 900000, 951524, 952466, 961983, 965945, 974555, 988994,
997356, 1001939, 1008123, 1025238, 1038362, 1080000, 1094712, 1095240, 1109052, 1117348,
1146452, 1147356, 1165651, 1170190, 1173556, 1188922, 1200000, 1217175, 1217432, 1247019,
1252644, 1268699, 1269641, 1273774, 1275516, 1284960, 1294446, 1314416, 1326240, 1335912,
1339728, 1350000, 1359915, 1363359, 1372401, 1381897, 1410576, 1413411, 1415945, 1426226,
1429834, 1434787, 1437383, 1440000, 1441436, 1447356, 1452219, 1454406, 1482825, 1484340,
1495648, 1500000, 1513301, 1521911, 1529299, 1529756, 1532346, 1534349, 1536350, 1539424,
1539624, 1544188, 1547223, 1571481, 1583754, 1585718, 1586816, 1590948, 1591865, 1598924,
1605288, 1614828, 1627356, 1641754, 1642068, 1642574, 1642596, 1664406, 1668114, 1691877,
1694282, 1694712, 1702644, 1713411, 1716457, 1717546, 1743401, 1744343, 1747356
};
float b[N]; // list of angles, in units of 1 degree
for(int i = 0; i < N; i++) {
b[i] = (float)a[i] * 1.0e-4;
}
int k = 3; // number of angles being added
int S0 = 0; // initial value of the sum
int S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, S11; // partial sums; only the first k will be used
int lo = 3600000 - ((k>>1)+1), hi = 3600000 + ((k>>1)+1); // error bounds for the sum
for(int i1 = N-1; i1 > 0; i1--) {
S1 = S0 + a[i1];
if(S1 + (k-1)*a[i1] < lo) break; // sum will be too small, for this and later values of i1; end the loop
if(S1 + (k-1)*a[1] > hi) continue; // sum will be too large; skip to next value of i1
for(int i2 = i1; i2 > 0; i2--) {
S2 = S1 + a[i2];
if(S2 + (k-2)*a[i2] < lo) break; // sum will be too small, for this and later values of i2; skip to next value of i1
if(S2 + (k-2)*a[1] > hi) continue; // sum will be too large; skip to next value of i2
for(int i3 = i2; i3 > 0; i3--) {
S3 = S2 + a[i3];
if(S3 + (k-3)*a[i3] < lo) break; // sum is too small, for this and later values of i3; skip to next value of i2
if(S3 + (k-3)*a[1] > hi) continue; // sum is too large; skip to next value of i3
printf("%8.4f + %8.4f + %8.4f\n", b[i1], b[i2], b[i3]);
}
}
}
printf("Done");
int N = 109; // total number of dihedral angles, plus one
int a[N] = { // list of angles, in units of 0.0001 degree
0, 317175, 373774, 450000, 547356, 600000, 634349, 690948, 705288, 729730,
747547, 791877, 867268, 900000, 951524, 952466, 961983, 965945, 974555, 988994,
997356, 1001939, 1008123, 1025238, 1038362, 1080000, 1094712, 1095240, 1109052, 1117348,
1146452, 1147356, 1165651, 1170190, 1173556, 1188922, 1200000, 1217175, 1217432, 1247019,
1252644, 1268699, 1269641, 1273774, 1275516, 1284960, 1294446, 1314416, 1326240, 1335912,
1339728, 1350000, 1359915, 1363359, 1372401, 1381897, 1410576, 1413411, 1415945, 1426226,
1429834, 1434787, 1437383, 1440000, 1441436, 1447356, 1452219, 1454406, 1482825, 1484340,
1495648, 1500000, 1513301, 1521911, 1529299, 1529756, 1532346, 1534349, 1536350, 1539424,
1539624, 1544188, 1547223, 1571481, 1583754, 1585718, 1586816, 1590948, 1591865, 1598924,
1605288, 1614828, 1627356, 1641754, 1642068, 1642574, 1642596, 1664406, 1668114, 1691877,
1694282, 1694712, 1702644, 1713411, 1716457, 1717546, 1743401, 1744343, 1747356
};
float b[N]; // list of angles, in units of 1 degree
for(int i = 0; i < N; i++) {
b[i] = (float)a[i] * 1.0e-4;
}
int k = 3; // number of angles being added
int S0 = 0; // initial value of the sum
int S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, S11; // partial sums; only the first k will be used
int lo = 3600000 - ((k>>1)+1), hi = 3600000 + ((k>>1)+1); // error bounds for the sum
for(int i1 = 1; i1 < N; i1++) {
S1 = S0 + a[i1];
if(S1 + (k-1)*a[i1] > hi) break; // sum will be too large, for this and later values of i1; end the loop
if(S1 + (k-1)*a[N-1] < lo) continue; // sum will be too small; skip to next value of i1
for(int i2 = i1; i2 < N; i2++) {
S2 = S1 + a[i2];
if(S2 + (k-2)*a[i2] > hi) break; // sum will be too large, for this and later values of i2; skip to next value of i1
if(S2 + (k-2)*a[N-1] < lo) continue; // sum will be too small; skip to next value of i2
for(int i3 = i2; i3 < N; i3++) {
S3 = S2 + a[i3];
if(S3 + (k-3)*a[i3] > hi) break; // sum is too large, for this and later values of i3; skip to next value of i2
if(S3 + (k-3)*a[N-1] < lo) continue; // sum is too small; skip to next value of i3
printf("%8.4f + %8.4f + %8.4f\n", b[i1], b[i2], b[i3]);
}
}
}
printf("Done");
174.7356 + 125.2644 + 60.0000
174.7356 + 114.7356 + 70.5288
174.3401 + 153.9424 + 31.7175
174.3401 + 148.2825 + 37.3774
174.3401 + 116.5651 + 69.0948
174.3401 + 110.9052 + 74.7547
171.3411 + 109.4712 + 79.1877
170.2644 + 144.7356 + 45.0000
170.2644 + 135.0000 + 54.7356
170.2644 + 99.7356 + 90.0000
169.4712 + 120.0000 + 70.5288
169.1877 + 159.0948 + 31.7175
169.1877 + 153.4349 + 37.3774
169.1877 + 127.3774 + 63.4349
169.1877 + 121.7175 + 69.0948
169.1877 + 100.8123 + 90.0000
164.2068 + 141.0576 + 54.7356
164.2068 + 125.2644 + 70.5288
160.5288 + 144.7356 + 54.7356
160.5288 + 109.4712 + 90.0000
160.5288 + 99.7356 + 99.7356
159.0948 + 121.7175 + 79.1877
159.0948 + 110.9052 + 90.0000
158.3754 + 138.1897 + 63.4349
158.3754 + 126.8699 + 74.7547
158.3754 + 100.8123 + 100.8123
153.9424 + 142.6226 + 63.4349
153.9424 + 126.8699 + 79.1877
153.9424 + 110.9052 + 95.1524
153.4349 + 127.3774 + 79.1877
153.4349 + 116.5651 + 90.0000
150.0000 + 150.0000 + 60.0000
150.0000 + 120.0000 + 90.0000
148.2825 + 148.2825 + 63.4349
148.2825 + 142.6226 + 69.0948
148.2825 + 121.7175 + 90.0000
148.2825 + 116.5651 + 95.1524
148.2825 + 110.9052 + 100.8123
144.7356 + 144.7356 + 70.5288
144.7356 + 125.2644 + 90.0000
144.0000 + 108.0000 + 108.0000
142.6226 + 142.6226 + 74.7547
142.6226 + 138.1897 + 79.1877
142.6226 + 127.3774 + 90.0000
142.6226 + 116.5651 + 100.8123
141.0576 + 109.4712 + 109.4712
138.1897 + 110.9052 + 110.9052
135.0000 + 135.0000 + 90.0000
135.0000 + 125.2644 + 99.7356
127.3774 + 121.7175 + 110.9052
126.8699 + 116.5651 + 116.5651
125.2644 + 125.2644 + 109.4712
125.2644 + 120.0000 + 114.7356
121.7175 + 121.7175 + 116.5651
120.0000 + 120.0000 + 120.0000
174.7356 + 70.5288 + 60.0000 + 54.7356
174.3401 + 116.5651 + 37.3774 + 31.7175
174.3401 + 110.9052 + 37.3774 + 37.3774
174.3401 + 79.1877 + 74.7547 + 31.7175
174.3401 + 79.1877 + 69.0948 + 37.3774
171.3411 + 79.1877 + 54.7356 + 54.7356
170.2644 + 99.7356 + 45.0000 + 45.0000
170.2644 + 90.0000 + 54.7356 + 45.0000
169.4712 + 70.5288 + 60.0000 + 60.0000
169.1877 + 127.3774 + 31.7175 + 31.7175
169.1877 + 121.7175 + 37.3774 + 31.7175
169.1877 + 100.8123 + 45.0000 + 45.0000
169.1877 + 90.0000 + 69.0948 + 31.7175
169.1877 + 90.0000 + 63.4349 + 37.3774
164.2068 + 70.5288 + 70.5288 + 54.7356
160.5288 + 109.4712 + 45.0000 + 45.0000
160.5288 + 99.7356 + 54.7356 + 45.0000
160.5288 + 90.0000 + 54.7356 + 54.7356
159.0948 + 110.9052 + 45.0000 + 45.0000
159.0948 + 90.0000 + 79.1877 + 31.7175
158.3754 + 138.1897 + 31.7175 + 31.7175
158.3754 + 126.8699 + 37.3774 + 37.3774
158.3754 + 100.8123 + 69.0948 + 31.7175
158.3754 + 100.8123 + 63.4349 + 37.3774
158.3754 + 95.1524 + 74.7547 + 31.7175
158.3754 + 95.1524 + 69.0948 + 37.3774
158.3754 + 74.7547 + 63.4349 + 63.4349
158.3754 + 69.0948 + 69.0948 + 63.4349
153.9424 + 142.6226 + 31.7175 + 31.7175
153.9424 + 110.9052 + 63.4349 + 31.7175
153.9424 + 95.1524 + 79.1877 + 31.7175
153.9424 + 79.1877 + 63.4349 + 63.4349
153.4349 + 116.5651 + 45.0000 + 45.0000
153.4349 + 90.0000 + 79.1877 + 37.3774
150.0000 + 120.0000 + 45.0000 + 45.0000
150.0000 + 90.0000 + 60.0000 + 60.0000
148.2825 + 148.2825 + 31.7175 + 31.7175
148.2825 + 142.6226 + 37.3774 + 31.7175
148.2825 + 135.0000 + 45.0000 + 31.7175
148.2825 + 125.2644 + 54.7356 + 31.7175
148.2825 + 121.7175 + 45.0000 + 45.0000
148.2825 + 120.0000 + 60.0000 + 31.7175
148.2825 + 116.5651 + 63.4349 + 31.7175
148.2825 + 110.9052 + 69.0948 + 31.7175
148.2825 + 110.9052 + 63.4349 + 37.3774
148.2825 + 109.4712 + 70.5288 + 31.7175
148.2825 + 100.8123 + 79.1877 + 31.7175
148.2825 + 95.1524 + 79.1877 + 37.3774
148.2825 + 90.0000 + 90.0000 + 31.7175
148.2825 + 79.1877 + 69.0948 + 63.4349
144.7356 + 125.2644 + 45.0000 + 45.0000
144.7356 + 99.7356 + 70.5288 + 45.0000
144.7356 + 90.0000 + 70.5288 + 54.7356
142.6226 + 142.6226 + 37.3774 + 37.3774
142.6226 + 135.0000 + 45.0000 + 37.3774
142.6226 + 127.3774 + 45.0000 + 45.0000
142.6226 + 125.2644 + 54.7356 + 37.3774
142.6226 + 120.0000 + 60.0000 + 37.3774
142.6226 + 116.5651 + 69.0948 + 31.7175
142.6226 + 116.5651 + 63.4349 + 37.3774
142.6226 + 110.9052 + 74.7547 + 31.7175
142.6226 + 110.9052 + 69.0948 + 37.3774
142.6226 + 109.4712 + 70.5288 + 37.3774
142.6226 + 100.8123 + 79.1877 + 37.3774
142.6226 + 90.0000 + 90.0000 + 37.3774
142.6226 + 79.1877 + 74.7547 + 63.4349
142.6226 + 79.1877 + 69.0948 + 69.0948
141.0576 + 109.4712 + 54.7356 + 54.7356
138.1897 + 110.9052 + 79.1877 + 31.7175
138.1897 + 79.1877 + 79.1877 + 63.4349
135.0000 + 135.0000 + 45.0000 + 45.0000
135.0000 + 125.2644 + 54.7356 + 45.0000
135.0000 + 120.0000 + 60.0000 + 45.0000
135.0000 + 116.5651 + 63.4349 + 45.0000
135.0000 + 110.9052 + 69.0948 + 45.0000
135.0000 + 109.4712 + 70.5288 + 45.0000
135.0000 + 100.8123 + 79.1877 + 45.0000
135.0000 + 99.7356 + 70.5288 + 54.7356
135.0000 + 90.0000 + 90.0000 + 45.0000
127.3774 + 121.7175 + 79.1877 + 31.7175
127.3774 + 110.9052 + 90.0000 + 31.7175
127.3774 + 90.0000 + 79.1877 + 63.4349
126.8699 + 116.5651 + 79.1877 + 37.3774
126.8699 + 79.1877 + 79.1877 + 74.7547
125.2644 + 125.2644 + 54.7356 + 54.7356
125.2644 + 120.0000 + 60.0000 + 54.7356
125.2644 + 116.5651 + 63.4349 + 54.7356
125.2644 + 114.7356 + 60.0000 + 60.0000
125.2644 + 110.9052 + 69.0948 + 54.7356
125.2644 + 109.4712 + 70.5288 + 54.7356
125.2644 + 100.8123 + 79.1877 + 54.7356
125.2644 + 99.7356 + 90.0000 + 45.0000
125.2644 + 90.0000 + 90.0000 + 54.7356
121.7175 + 121.7175 + 79.1877 + 37.3774
121.7175 + 116.5651 + 90.0000 + 31.7175
121.7175 + 110.9052 + 90.0000 + 37.3774
121.7175 + 90.0000 + 79.1877 + 69.0948
120.0000 + 120.0000 + 60.0000 + 60.0000
120.0000 + 116.5651 + 63.4349 + 60.0000
120.0000 + 114.7356 + 70.5288 + 54.7356
120.0000 + 110.9052 + 69.0948 + 60.0000
120.0000 + 109.4712 + 70.5288 + 60.0000
120.0000 + 100.8123 + 79.1877 + 60.0000
120.0000 + 90.0000 + 90.0000 + 60.0000
116.5651 + 116.5651 + 95.1524 + 31.7175
116.5651 + 116.5651 + 63.4349 + 63.4349
116.5651 + 110.9052 + 100.8123 + 31.7175
116.5651 + 110.9052 + 95.1524 + 37.3774
116.5651 + 110.9052 + 69.0948 + 63.4349
116.5651 + 109.4712 + 70.5288 + 63.4349
116.5651 + 100.8123 + 79.1877 + 63.4349
116.5651 + 95.1524 + 79.1877 + 69.0948
116.5651 + 90.0000 + 90.0000 + 63.4349
114.7356 + 114.7356 + 70.5288 + 60.0000
110.9052 + 110.9052 + 100.8123 + 37.3774
110.9052 + 110.9052 + 74.7547 + 63.4349
110.9052 + 110.9052 + 69.0948 + 69.0948
110.9052 + 109.4712 + 70.5288 + 69.0948
110.9052 + 100.8123 + 79.1877 + 69.0948
110.9052 + 95.1524 + 79.1877 + 74.7547
110.9052 + 90.0000 + 90.0000 + 69.0948
109.4712 + 109.4712 + 70.5288 + 70.5288
109.4712 + 100.8123 + 79.1877 + 70.5288
109.4712 + 90.0000 + 90.0000 + 70.5288
100.8123 + 100.8123 + 79.1877 + 79.1877
100.8123 + 90.0000 + 90.0000 + 79.1877
99.7356 + 99.7356 + 90.0000 + 70.5288
90.0000 + 90.0000 + 90.0000 + 90.0000
174.3401 + 79.1877 + 37.3774 + 37.3774 + 31.7175
170.2644 + 54.7356 + 45.0000 + 45.0000 + 45.0000
169.1877 + 90.0000 + 37.3774 + 31.7175 + 31.7175
169.1877 + 69.0948 + 45.0000 + 45.0000 + 31.7175
169.1877 + 63.4349 + 45.0000 + 45.0000 + 37.3774
160.5288 + 54.7356 + 54.7356 + 45.0000 + 45.0000
159.0948 + 79.1877 + 45.0000 + 45.0000 + 31.7175
158.3754 + 100.8123 + 37.3774 + 31.7175 + 31.7175
158.3754 + 95.1524 + 37.3774 + 37.3774 + 31.7175
158.3754 + 74.7547 + 63.4349 + 31.7175 + 31.7175
158.3754 + 69.0948 + 69.0948 + 31.7175 + 31.7175
158.3754 + 69.0948 + 63.4349 + 37.3774 + 31.7175
158.3754 + 63.4349 + 63.4349 + 37.3774 + 37.3774
153.9424 + 110.9052 + 31.7175 + 31.7175 + 31.7175
153.9424 + 79.1877 + 63.4349 + 31.7175 + 31.7175
153.4349 + 79.1877 + 45.0000 + 45.0000 + 37.3774
150.0000 + 60.0000 + 60.0000 + 45.0000 + 45.0000
148.2825 + 116.5651 + 31.7175 + 31.7175 + 31.7175
148.2825 + 110.9052 + 37.3774 + 31.7175 + 31.7175
148.2825 + 90.0000 + 45.0000 + 45.0000 + 31.7175
148.2825 + 79.1877 + 69.0948 + 31.7175 + 31.7175
148.2825 + 79.1877 + 63.4349 + 37.3774 + 31.7175
148.2825 + 70.5288 + 54.7356 + 54.7356 + 31.7175
148.2825 + 60.0000 + 60.0000 + 60.0000 + 31.7175
144.7356 + 70.5288 + 54.7356 + 45.0000 + 45.0000
142.6226 + 116.5651 + 37.3774 + 31.7175 + 31.7175
142.6226 + 110.9052 + 37.3774 + 37.3774 + 31.7175
142.6226 + 90.0000 + 45.0000 + 45.0000 + 37.3774
142.6226 + 79.1877 + 74.7547 + 31.7175 + 31.7175
142.6226 + 79.1877 + 69.0948 + 37.3774 + 31.7175
142.6226 + 79.1877 + 63.4349 + 37.3774 + 37.3774
142.6226 + 70.5288 + 54.7356 + 54.7356 + 37.3774
142.6226 + 60.0000 + 60.0000 + 60.0000 + 37.3774
141.0576 + 54.7356 + 54.7356 + 54.7356 + 54.7356
138.1897 + 79.1877 + 79.1877 + 31.7175 + 31.7175
135.0000 + 116.5651 + 45.0000 + 31.7175 + 31.7175
135.0000 + 110.9052 + 45.0000 + 37.3774 + 31.7175
135.0000 + 90.0000 + 45.0000 + 45.0000 + 45.0000
135.0000 + 79.1877 + 69.0948 + 45.0000 + 31.7175
135.0000 + 79.1877 + 63.4349 + 45.0000 + 37.3774
135.0000 + 70.5288 + 54.7356 + 54.7356 + 45.0000
135.0000 + 60.0000 + 60.0000 + 60.0000 + 45.0000
127.3774 + 110.9052 + 45.0000 + 45.0000 + 31.7175
127.3774 + 90.0000 + 79.1877 + 31.7175 + 31.7175
127.3774 + 79.1877 + 63.4349 + 45.0000 + 45.0000
126.8699 + 79.1877 + 79.1877 + 37.3774 + 37.3774
125.2644 + 116.5651 + 54.7356 + 31.7175 + 31.7175
125.2644 + 110.9052 + 54.7356 + 37.3774 + 31.7175
125.2644 + 99.7356 + 45.0000 + 45.0000 + 45.0000
125.2644 + 90.0000 + 54.7356 + 45.0000 + 45.0000
125.2644 + 79.1877 + 69.0948 + 54.7356 + 31.7175
125.2644 + 79.1877 + 63.4349 + 54.7356 + 37.3774
125.2644 + 70.5288 + 54.7356 + 54.7356 + 54.7356
125.2644 + 60.0000 + 60.0000 + 60.0000 + 54.7356
121.7175 + 116.5651 + 45.0000 + 45.0000 + 31.7175
121.7175 + 110.9052 + 45.0000 + 45.0000 + 37.3774
121.7175 + 90.0000 + 79.1877 + 37.3774 + 31.7175
121.7175 + 79.1877 + 69.0948 + 45.0000 + 45.0000
120.0000 + 116.5651 + 60.0000 + 31.7175 + 31.7175
120.0000 + 110.9052 + 60.0000 + 37.3774 + 31.7175
120.0000 + 90.0000 + 60.0000 + 45.0000 + 45.0000
120.0000 + 79.1877 + 69.0948 + 60.0000 + 31.7175
120.0000 + 79.1877 + 63.4349 + 60.0000 + 37.3774
120.0000 + 70.5288 + 60.0000 + 54.7356 + 54.7356
120.0000 + 60.0000 + 60.0000 + 60.0000 + 60.0000
116.5651 + 116.5651 + 63.4349 + 31.7175 + 31.7175
116.5651 + 110.9052 + 69.0948 + 31.7175 + 31.7175
116.5651 + 110.9052 + 63.4349 + 37.3774 + 31.7175
116.5651 + 109.4712 + 70.5288 + 31.7175 + 31.7175
116.5651 + 100.8123 + 79.1877 + 31.7175 + 31.7175
116.5651 + 95.1524 + 79.1877 + 37.3774 + 31.7175
116.5651 + 90.0000 + 90.0000 + 31.7175 + 31.7175
116.5651 + 90.0000 + 63.4349 + 45.0000 + 45.0000
116.5651 + 79.1877 + 69.0948 + 63.4349 + 31.7175
116.5651 + 79.1877 + 63.4349 + 63.4349 + 37.3774
116.5651 + 70.5288 + 63.4349 + 54.7356 + 54.7356
116.5651 + 63.4349 + 60.0000 + 60.0000 + 60.0000
114.7356 + 70.5288 + 60.0000 + 60.0000 + 54.7356
110.9052 + 110.9052 + 74.7547 + 31.7175 + 31.7175
110.9052 + 110.9052 + 69.0948 + 37.3774 + 31.7175
110.9052 + 110.9052 + 63.4349 + 37.3774 + 37.3774
110.9052 + 109.4712 + 70.5288 + 37.3774 + 31.7175
110.9052 + 100.8123 + 79.1877 + 37.3774 + 31.7175
110.9052 + 95.1524 + 79.1877 + 37.3774 + 37.3774
110.9052 + 90.0000 + 90.0000 + 37.3774 + 31.7175
110.9052 + 90.0000 + 69.0948 + 45.0000 + 45.0000
110.9052 + 79.1877 + 74.7547 + 63.4349 + 31.7175
110.9052 + 79.1877 + 69.0948 + 69.0948 + 31.7175
110.9052 + 79.1877 + 69.0948 + 63.4349 + 37.3774
110.9052 + 70.5288 + 69.0948 + 54.7356 + 54.7356
110.9052 + 69.0948 + 60.0000 + 60.0000 + 60.0000
109.4712 + 90.0000 + 70.5288 + 45.0000 + 45.0000
109.4712 + 79.1877 + 70.5288 + 69.0948 + 31.7175
109.4712 + 79.1877 + 70.5288 + 63.4349 + 37.3774
109.4712 + 70.5288 + 70.5288 + 54.7356 + 54.7356
109.4712 + 70.5288 + 60.0000 + 60.0000 + 60.0000
100.8123 + 90.0000 + 79.1877 + 45.0000 + 45.0000
100.8123 + 79.1877 + 79.1877 + 69.0948 + 31.7175
100.8123 + 79.1877 + 79.1877 + 63.4349 + 37.3774
100.8123 + 79.1877 + 70.5288 + 54.7356 + 54.7356
100.8123 + 79.1877 + 60.0000 + 60.0000 + 60.0000
99.7356 + 99.7356 + 70.5288 + 45.0000 + 45.0000
99.7356 + 90.0000 + 70.5288 + 54.7356 + 45.0000
95.1524 + 79.1877 + 79.1877 + 74.7547 + 31.7175
95.1524 + 79.1877 + 79.1877 + 69.0948 + 37.3774
90.0000 + 90.0000 + 90.0000 + 45.0000 + 45.0000
90.0000 + 90.0000 + 79.1877 + 69.0948 + 31.7175
90.0000 + 90.0000 + 79.1877 + 63.4349 + 37.3774
90.0000 + 90.0000 + 70.5288 + 54.7356 + 54.7356
90.0000 + 90.0000 + 60.0000 + 60.0000 + 60.0000
79.1877 + 79.1877 + 74.7547 + 63.4349 + 63.4349
79.1877 + 79.1877 + 69.0948 + 69.0948 + 63.4349
169.1877 + 45.0000 + 45.0000 + 37.3774 + 31.7175 + 31.7175
158.3754 + 74.7547 + 31.7175 + 31.7175 + 31.7175 + 31.7175
158.3754 + 69.0948 + 37.3774 + 31.7175 + 31.7175 + 31.7175
158.3754 + 63.4349 + 37.3774 + 37.3774 + 31.7175 + 31.7175
153.9424 + 79.1877 + 31.7175 + 31.7175 + 31.7175 + 31.7175
148.2825 + 79.1877 + 37.3774 + 31.7175 + 31.7175 + 31.7175
148.2825 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 31.7175
142.6226 + 79.1877 + 37.3774 + 37.3774 + 31.7175 + 31.7175
142.6226 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 37.3774
135.0000 + 79.1877 + 45.0000 + 37.3774 + 31.7175 + 31.7175
135.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
127.3774 + 79.1877 + 45.0000 + 45.0000 + 31.7175 + 31.7175
125.2644 + 79.1877 + 54.7356 + 37.3774 + 31.7175 + 31.7175
125.2644 + 54.7356 + 45.0000 + 45.0000 + 45.0000 + 45.0000
121.7175 + 79.1877 + 45.0000 + 45.0000 + 37.3774 + 31.7175
120.0000 + 79.1877 + 60.0000 + 37.3774 + 31.7175 + 31.7175
120.0000 + 60.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
116.5651 + 116.5651 + 31.7175 + 31.7175 + 31.7175 + 31.7175
116.5651 + 110.9052 + 37.3774 + 31.7175 + 31.7175 + 31.7175
116.5651 + 90.0000 + 45.0000 + 45.0000 + 31.7175 + 31.7175
116.5651 + 79.1877 + 69.0948 + 31.7175 + 31.7175 + 31.7175
116.5651 + 79.1877 + 63.4349 + 37.3774 + 31.7175 + 31.7175
116.5651 + 70.5288 + 54.7356 + 54.7356 + 31.7175 + 31.7175
116.5651 + 63.4349 + 45.0000 + 45.0000 + 45.0000 + 45.0000
116.5651 + 60.0000 + 60.0000 + 60.0000 + 31.7175 + 31.7175
110.9052 + 110.9052 + 37.3774 + 37.3774 + 31.7175 + 31.7175
110.9052 + 90.0000 + 45.0000 + 45.0000 + 37.3774 + 31.7175
110.9052 + 79.1877 + 74.7547 + 31.7175 + 31.7175 + 31.7175
110.9052 + 79.1877 + 69.0948 + 37.3774 + 31.7175 + 31.7175
110.9052 + 79.1877 + 63.4349 + 37.3774 + 37.3774 + 31.7175
110.9052 + 70.5288 + 54.7356 + 54.7356 + 37.3774 + 31.7175
110.9052 + 69.0948 + 45.0000 + 45.0000 + 45.0000 + 45.0000
110.9052 + 60.0000 + 60.0000 + 60.0000 + 37.3774 + 31.7175
109.4712 + 79.1877 + 70.5288 + 37.3774 + 31.7175 + 31.7175
109.4712 + 70.5288 + 45.0000 + 45.0000 + 45.0000 + 45.0000
100.8123 + 79.1877 + 79.1877 + 37.3774 + 31.7175 + 31.7175
100.8123 + 79.1877 + 45.0000 + 45.0000 + 45.0000 + 45.0000
99.7356 + 70.5288 + 54.7356 + 45.0000 + 45.0000 + 45.0000
95.1524 + 79.1877 + 79.1877 + 37.3774 + 37.3774 + 31.7175
90.0000 + 90.0000 + 79.1877 + 37.3774 + 31.7175 + 31.7175
90.0000 + 90.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
90.0000 + 79.1877 + 69.0948 + 45.0000 + 45.0000 + 31.7175
90.0000 + 79.1877 + 63.4349 + 45.0000 + 45.0000 + 37.3774
90.0000 + 70.5288 + 54.7356 + 54.7356 + 45.0000 + 45.0000
90.0000 + 60.0000 + 60.0000 + 60.0000 + 45.0000 + 45.0000
79.1877 + 79.1877 + 74.7547 + 63.4349 + 31.7175 + 31.7175
79.1877 + 79.1877 + 69.0948 + 69.0948 + 31.7175 + 31.7175
79.1877 + 79.1877 + 69.0948 + 63.4349 + 37.3774 + 31.7175
79.1877 + 79.1877 + 63.4349 + 63.4349 + 37.3774 + 37.3774
79.1877 + 70.5288 + 69.0948 + 54.7356 + 54.7356 + 31.7175
79.1877 + 70.5288 + 63.4349 + 54.7356 + 54.7356 + 37.3774
79.1877 + 69.0948 + 60.0000 + 60.0000 + 60.0000 + 31.7175
79.1877 + 63.4349 + 60.0000 + 60.0000 + 60.0000 + 37.3774
70.5288 + 70.5288 + 54.7356 + 54.7356 + 54.7356 + 54.7356
70.5288 + 60.0000 + 60.0000 + 60.0000 + 54.7356 + 54.7356
60.0000 + 60.0000 + 60.0000 + 60.0000 + 60.0000 + 60.0000
158.3754 + 37.3774 + 37.3774 + 31.7175 + 31.7175 + 31.7175 + 31.7175
116.5651 + 79.1877 + 37.3774 + 31.7175 + 31.7175 + 31.7175 + 31.7175
116.5651 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 31.7175 + 31.7175
110.9052 + 79.1877 + 37.3774 + 37.3774 + 31.7175 + 31.7175 + 31.7175
110.9052 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 37.3774 + 31.7175
90.0000 + 79.1877 + 45.0000 + 45.0000 + 37.3774 + 31.7175 + 31.7175
90.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
79.1877 + 79.1877 + 74.7547 + 31.7175 + 31.7175 + 31.7175 + 31.7175
79.1877 + 79.1877 + 69.0948 + 37.3774 + 31.7175 + 31.7175 + 31.7175
79.1877 + 79.1877 + 63.4349 + 37.3774 + 37.3774 + 31.7175 + 31.7175
79.1877 + 70.5288 + 54.7356 + 54.7356 + 37.3774 + 31.7175 + 31.7175
79.1877 + 69.0948 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 31.7175
79.1877 + 63.4349 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 37.3774
79.1877 + 60.0000 + 60.0000 + 60.0000 + 37.3774 + 31.7175 + 31.7175
70.5288 + 54.7356 + 54.7356 + 45.0000 + 45.0000 + 45.0000 + 45.0000
60.0000 + 60.0000 + 60.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
79.1877 + 79.1877 + 37.3774 + 37.3774 + 31.7175 + 31.7175 + 31.7175 + 31.7175
79.1877 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 37.3774 + 31.7175 + 31.7175
45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000 + 45.0000
mr_e_man wrote:As you probably expected, the crown jewels J84-90 and snic and snid don't fit with anything. Also the augmented pentagonal prism (angle 162.74°) doesn't fit.
From the polyhedra with 3,4,5,6,8,10-gons, there are 55 combinations of 3 angles adding to 360°, 128 combinations of 4 angles, 112 of 5, 56 of 6, 16 of 7, 3 of 8, and no combinations of 9 or more angles.
int N = 76; // total number of dihedral angles, plus one
int a[N] = { // list of angles, in units of 0.0001 degree
0, 317175, 373774, 450000, 547356, 600000, 634349, 705288, 729730, 791877,
867268, 900000, 952466, 961983, 965945, 974555, 988994, 1001939, 1008123, 1025238,
1038362, 1080000, 1094712, 1095240, 1109052, 1117348, 1146452, 1165651, 1170190, 1173556,
1188922, 1200000, 1217175, 1217432, 1247019, 1252644, 1275516, 1284960, 1294446, 1314416,
1335912, 1339728, 1350000, 1359915, 1363359, 1372401, 1381897, 1413411, 1426226, 1429834,
1434787, 1437383, 1440000, 1441436, 1447356, 1452219, 1454406, 1482825, 1484340, 1495648,
1529299, 1529756, 1532346, 1539624, 1544188, 1547223, 1571481, 1590948, 1591865, 1598924,
1614828, 1641754, 1642574, 1664406, 1668114, 1716457
};
mr_e_man wrote:I started with large prisms and antiprisms (in pairs sharing a large face), systematically adding up angles with other CRF polyhedra.
| vertex | edges | squares | n-gon | n-prisms
+--------+---------+---------+--------+----------
| * | 1 1 1 1 | 1 1 1 1 | 1 | 1 1
+--------+---------+---------+--------+----------
| 1 | * | 1 0 0 1 | 1 | 1 1
| 1 | * | 1 1 0 0 | 0 | 1 0
| 1 | * | 0 1 1 0 | 1 | 1 1
| 1 | * | 0 0 1 1 | 0 | 0 1
+--------+---------+---------+--------+----------
| 1 | 1 1 0 0 | * | | 1 0
| 1 | 0 1 1 0 | * | | 1 0
| 1 | 0 0 1 1 | * | | 0 1
| 1 | 1 0 0 1 | * | | 0 1
+--------+---------+---------+--------+----------
| 1 | 1 0 1 0 | | * | 1 1
+--------+---------+---------+--------+----------
| 1 | 1 1 1 0 | 1 1 0 0 | 1 | *
| 1 | 1 0 1 1 | 0 0 1 1 | 1 | *
| edges | {4} |{n}|{10}|{3}|{6}| n-prisms | J5 | grid
+---------+---------+---+----+---+---+----------+----+------
| * | 1 0 0 1 | 1 | 1 | 0 | 0 | 1 1 | 1 | 1
| * | 1 1 0 0 | 0 | 0 | 1 | 0 | 1 0 | 1 | 0
| * | 0 1 1 0 | 1 | 0 | 0 | 0 | 1 1 | 0 | 0
| * | 0 0 1 1 | 0 | 0 | 0 | 1 | 0 1 | 0 | 1
+---------+---------+---+----+---+---+----------+----+------
| 1 1 0 0 | * | | | | | 1 0 | 1 | 0
| 0 1 1 0 | * | | | | | 1 0 | 0 | 0
| 0 0 1 1 | * | | | | | 0 1 | 0 | 0
| 1 0 0 1 | * | | | | | 0 1 | 0 | 1
+---------+---------+---+----+---+---+----------+----+------
| 1 0 1 0 | | * | | | | 1 1 | 0 | 0
+---------+---------+---+----+---+---+----------+----+------
| 1 0 0 0 | | | * | | | 0 0 | 1 | 1
+---------+---------+---+----+---+---+----------+----+------
| 0 1 0 0 | | | | * | | 0 0 | 1 | 0
+---------+---------+---+----+---+---+----------+----+------
| 0 0 0 1 | | | | | * | 0 0 | 0 | 1
+---------+---------+---+----+---+---+----------+----+------
| 1 1 1 0 | 1 1 0 0 | 1 | 0 | 0 | 0 | * | |
| 1 0 1 1 | 0 0 1 1 | 1 | 0 | 0 | 0 | * | |
+---------+---------+---+----+---+---+----------+----+------
| 1 1 0 0 | 1 0 0 0 | 0 | 1 | 1 | 0 | | * |
+---------+---------+---+----+---+---+----------+----+------
| 1 0 0 1 | 0 0 0 1 | 0 | 1 | 0 | 1 | | | *
| vertices | edges | {20} | {5} | {4}
+----------+-------+------+-----+-----
| * | 1 1 0 | 1 | 1 | 1 0
| * | 1 0 1 | 1 | 1 | 0 1
| * | 0 1 0 | 0 | 1 | 1 0
| * | 0 0 1 | 0 | 1 | 0 1
+----------+-------+------+-----+-----
| 1 1 0 0 | * | 1 | 1 | 0 0
| 1 0 1 0 | * | 0 | 1 | 1 0
| 0 1 0 1 | * | 0 | 1 | 0 1
+----------+-------+------+-----+-----
| 1 1 0 0 | 1 0 0 | * | |
+----------+-------+------+-----+-----
| 1 1 1 1 | 1 1 1 | | * |
+----------+-------+------+-----+-----
| 1 0 1 0 | 0 1 0 | | | *
| 0 1 0 1 | 0 0 1 | | | *
| vertices | edges | {20} | {5} | {4}
+-----------+-----------+-------+-----+-----
| * | 1 1 0 0 0 | 1 0 0 | 1 | 1 0
| * | 1 0 1 0 0 | 1 0 0 | 1 | 0 1
| * | 0 1 0 1 0 | 0 1 0 | 1 | 1 0
| * | 0 0 1 0 1 | 0 0 1 | 1 | 0 1
| * | 0 0 0 1 1 | 0 1 1 | 1 | 0 0
+-----------+-----------+-------+-----+-----
| 1 1 0 0 0 | * | 1 0 0 | 1 | 0 0
| 1 0 1 0 0 | * | 0 0 0 | 1 | 1 0
| 0 1 0 1 0 | * | 0 0 0 | 1 | 0 1
| 0 0 1 0 1 | * | 0 1 0 | 1 | 0 0
| 0 0 0 1 1 | * | 0 0 1 | 1 | 0 0
+-----------+-----------+-------+-----+-----
| 1 1 0 0 0 | 1 0 0 0 0 | * | |
| 0 0 1 0 1 | 0 0 0 1 0 | * | |
| 0 0 0 1 1 | 0 0 0 0 1 | * | |
+-----------+-----------+-------+-----+-----
| 1 1 1 1 1 | 1 1 1 1 1 | | * |
+-----------+-----------+-------+-----+-----
| 1 0 1 0 0 | 0 1 0 0 0 | | | *
| 0 1 0 1 0 | 0 0 1 0 0 | | | *
| edges | {4} | {3} | {n} | n-prism, antiprism
+-----------+-----+-------+-----+--------------------
| * | 1 0 | 1 0 0 | 1 | 1 1
| * | 1 1 | 0 0 0 | 0 | 1 0
| * | 0 1 | 0 0 1 | 1 | 1 1
| * | 0 0 | 1 1 0 | 0 | 0 1
| * | 0 0 | 0 1 1 | 0 | 0 1
+-----------+-----+-------+-----+--------------------
| 1 1 0 0 0 | * | | | 1 0
| 0 1 1 0 0 | * | | | 1 0
+-----------+-----+-------+-----+--------------------
| 1 0 0 1 0 | | * | | 0 1
| 0 0 0 1 1 | | * | | 0 1
| 0 0 1 0 1 | | * | | 0 1
+-----------+-----+-------+-----+--------------------
| 1 0 1 0 0 | | | * | 1 1
+-----------+-----+-------+-----+--------------------
| 1 1 1 0 0 | 1 1 | 0 0 0 | 1 | *
| 1 0 1 1 1 | 0 0 | 1 1 1 | 1 | *
| | | | | other | n-p, |
| edges | {4} | {3} | {n} | faces | ap | A,B,C
+-----------+-----+-------+-----+-------+------+-------
| * | 1 0 | 1 0 0 | 1 | 1 1 | 1 1 | 1 1 1
| * | 1 1 | 0 0 0 | 0 | 0 0 | 1 0 | 1 0 0
| * | 0 1 | 0 0 1 | 1 | 0 0 | 1 1 | 0 0 0
| * | 0 0 | 1 1 0 | 0 | 0 0 | 0 1 | 0 1 0
| * | 0 0 | 0 1 1 | 0 | 0 0 | 0 1 | 0 0 0
+-----------+-----+-------+-----+-------+------+-------
| 1 1 0 0 0 | * | | | | 1 0 | 1 0 0
| 0 1 1 0 0 | * | | | | 1 0 | 0 0 0
+-----------+-----+-------+-----+-------+------+-------
| 1 0 0 1 0 | | * | | | 0 1 | 0 1 0
| 0 0 0 1 1 | | * | | | 0 1 | 0 0 0
| 0 0 1 0 1 | | * | | | 0 1 | 0 0 0
+-----------+-----+-------+-----+-------+------+-------
| 1 0 1 0 0 | | | * | | 1 1 | 0 0 0
+-----------+-----+-------+-----+-------+------+-------
| 1 0 0 0 0 | | | | * | 0 0 | 1 0 1
| 1 0 0 0 0 | | | | * | 0 0 | 0 1 1
+-----------+-----+-------+-----+-------+------+-------
| 1 1 1 0 0 | 1 1 | 0 0 0 | 1 | 0 0 | * |
| 1 0 1 1 1 | 0 0 | 1 1 1 | 1 | 0 0 | * |
+-----------+-----+-------+-----+-------+------+-------
| 1 1 0 0 0 | 1 0 | 0 0 0 | 0 | 1 0 | | *
| 1 0 0 1 0 | 0 0 | 1 0 0 | 0 | 0 1 | | *
| 1 0 0 0 0 | 0 0 | 0 0 0 | 0 | 1 1 | | *
| edges | {3} | {n} | n-antiprisms
+-------------+-------------+-----+--------------
| * | 1 0 0 0 0 1 | 1 | 1 1
| * | 1 1 0 0 0 0 | 0 | 1 0
| * | 0 1 1 0 0 0 | 0 | 1 0
| * | 0 0 1 1 0 0 | 1 | 1 1
| * | 0 0 0 1 1 0 | 0 | 0 1
| * | 0 0 0 0 1 1 | 0 | 0 1
+-------------+-------------+-----+--------------
| 1 1 0 0 0 0 | * | | 1 0
| 0 1 1 0 0 0 | * | | 1 0
| 0 0 1 1 0 0 | * | | 1 0
| 0 0 0 1 1 0 | * | | 0 1
| 0 0 0 0 1 1 | * | | 0 1
| 1 0 0 0 0 1 | * | | 0 1
+-------------+-------------+-----+--------------
| 1 0 0 1 0 0 | | * | 1 1
+-------------+-------------+-----+--------------
| 1 1 1 1 0 0 | 1 1 1 0 0 0 | 1 | *
| 1 0 0 1 1 1 | 0 0 0 1 1 1 | 1 | *
mr_e_man wrote:One difficult case to consider was two (n-gon) antiprisms meeting a third (m-gon) antiprism at two of its triangle faces: The first two dihedral angles, between a triangle and the n-gon, have a sum slightly greater than 180°, and the third angle, between the two triangles, is slightly less than 180°, so the sum is arbitrarily close to 360° (both below it and above it) when n and m are large enough. It should be possible to rule out this case using algebraic number theory, but I just noted that the m-gon antiprism must be paired with another m-gon antiprism or prism at the same vertex, and the four large solid angles (each slightly less than 180°) don't leave enough space for anything else at the vertex.
| edges | {3} | {n},{m} | antiprisms
+-------------+---------------+---------+------------
| * | 1 0 0 0 0 1 0 | 1 0 | 1 1 1
| * | 1 1 0 0 0 0 1 | 0 0 | 1 0 1
| * | 0 1 1 0 0 0 0 | 0 0 | 1 0 0
| * | 0 0 1 1 0 0 0 | 1 0 | 1 1 0
| * | 0 0 0 1 1 0 0 | 0 0 | 0 1 0
| * | 0 0 0 0 1 1 0 | 0 1 | 0 1 1
+-------------+---------------+---------+------------
| 1 1 0 0 0 0 | * | | 1 0 1
| 0 1 1 0 0 0 | * | | 1 0 0
| 0 0 1 1 0 0 | * | | 1 0 0
| 0 0 0 1 1 0 | * | | 0 1 0
| 0 0 0 0 1 1 | * | | 0 1 0
| 1 0 0 0 0 1 | * | | 0 1 1
| 0 1 0 0 0 0 | * | | 0 0 1
+-------------+---------------+---------+------------
| 1 0 0 1 0 0 | | * | 1 1 0
| 0 0 0 0 0 1 | | * | 0 0 1
+-------------+---------------+---------+------------
| 1 1 1 1 0 0 | 1 1 1 0 0 0 0 | 1 0 | *
| 1 0 0 1 1 1 | 0 0 0 1 1 1 0 | 1 0 | *
| 1 1 0 0 0 1 | 1 0 0 0 0 1 1 | 0 1 | *
mr_e_man wrote:A 12-gon prism can only appear in a 2D tiling stacked on top of itself.
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| edges | faces |
| X,Y,Z,W | 6,4,3,3, 10,5 | J92,grid,J6
+---------+---------------+-------------
| * | 1 1 0 0 0 0 | 1 1 0
| * | 0 1 1 0 0 0 | 1 0 0
| * | 0 0 1 1 0 1 | 1 0 1
| * | 1 0 0 1 1 0 | 1 1 1
+---------+---------------+-------------
| 1 0 0 1 | * | 1 1 0
| 1 1 0 0 | * | 1 0 0
| 0 1 1 0 | * | 1 0 0
| 0 0 1 1 | * | 1 0 1
| 0 0 0 1 | * | 0 1 1
| 0 0 1 0 | * | 0 0 1
+---------+---------------+-------------
| 1 1 1 1 | 1 1 1 1 0 0 | *
| 1 0 0 1 | 1 0 0 0 1 0 | *
| 0 0 1 1 | 0 0 0 1 1 1 | *
| edges | faces |
| W,X,Y,Z | 5,3,3,3, 4,5 | (5.3.3.3), (5.3.4.3)
+---------+--------------+----------------------
| * | 1 1 0 0 0 0 | 1 0
| * | 0 1 1 0 1 0 | 1 1
| * | 0 0 1 1 0 1 | 1 1
| * | 1 0 0 1 0 0 | 1 0
+---------+--------------+----------------------
| 1 0 0 1 | * | 1 0
| 1 1 0 0 | * | 1 0
| 0 1 1 0 | * | 1 1
| 0 0 1 1 | * | 1 0
| 0 1 0 0 | * | 0 1
| 0 0 1 0 | * | 0 1
+---------+--------------+----------------------
| 1 1 1 1 | 1 1 1 1 0 0 | *
| 0 1 1 0 | 0 0 1 0 1 1 | *
| edges | faces |
| W,X,Y,Z,T | 5,3,3,3, 10, 5 | (5.3.3.3), (10.10.3), (10.5.3), (5.5.5)
+-----------+------------------+-----------------------------------------
| * | 1 1 0 0 0 0 1 0 | 1 0 1 0 1
| * | 0 1 1 0 1 0 0 0 | 1 1 1 0 0
| * | 0 0 1 1 0 1 0 0 | 1 1 0 1 0
| * | 1 0 0 1 0 0 0 1 | 1 0 0 1 1
| * | 0 0 0 0 1 1 1 1 | 0 1 1 1 1
+-----------+------------------+-----------------------------------------
| 1 0 0 1 0 | * | 1 0 0 0 1
| 1 1 0 0 0 | * | 1 0 1 0 0
| 0 1 1 0 0 | * | 1 1 0 0 0
| 0 0 1 1 0 | * | 1 0 0 1 0
| 0 1 0 0 1 | * | 0 1 1 0 0
| 0 0 1 0 1 | * | 0 1 0 1 0
| 1 0 0 0 1 | * | 0 0 1 0 1
| 0 0 0 1 1 | * | 0 0 0 1 1
+-----------+------------------+-----------------------------------------
| 1 1 1 1 0 | 1 1 1 1 0 0 0 0 | *
| 0 1 1 0 1 | 0 0 1 0 1 1 0 0 | *
| 1 1 0 0 1 | 0 1 0 0 1 0 1 0 | *
| 0 0 1 1 1 | 0 0 0 1 0 1 0 1 | *
| 1 0 0 1 1 | 1 0 0 0 0 0 1 1 | *
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