quickfur wrote:But if 4D vertices are always rigid, then doesn't that mean we can just brute-force enumerate all possible vertex configurations, and thereby find all 4D CRFs? We don't even need to precompute any verfs; just enumerate all possible combinations of 3D CRFs around a vertex, and discard those that don't close up locally.
Indeed. Formerly, the absence of such calculations made that I didn't consider such an approach as easy. However, now that you have made these, I think we can finally investigate some CVP3-polytopes (or prove their non-existence)Marek14 wrote:quickfur wrote:But if 4D vertices are always rigid, then doesn't that mean we can just brute-force enumerate all possible vertex configurations, and thereby find all 4D CRFs? We don't even need to precompute any verfs; just enumerate all possible combinations of 3D CRFs around a vertex, and discard those that don't close up locally.
Haven't I been mentioning the possibility of something like that for quite some time? But it was just a hunch, only today I actually tried to calculate the conditions. The result was surprising
I think you don't have to worry about that much. let's take a central vertex. now we add some vertices. let's say these vertices are all part of a surtope. This means the added vertices should make the verf of the central vertex. This verf can be seen in matrix context: vertex 1 has distance x to vertex 2, vertex 2 has distance y to vertex 3, vertex 1 has distance z to vertex 3, vertex 3 has distance w to vertex 2, etc, nicely placed in a matrix.Of course, it IS possible that for some vertices the equations will turn out to be dependent and they won't be rigid, but it should be the exception, not the rule.
So, let's look at dihedral angles. Stella is a big help here since its net mode computes dihedral angles automatically.
I include all prisms and antiprisms up to 10, for completeness.
BTW, you shouldn't automatically assume that there can be only 5 faces to an edge -- dihedral angles of some Johnson solids can get quite small.
Now, let's talk "possibilities". Basically, in some Johnson solids, the same dihedral angle exists at several nonequivalent places. I note this. Some of those places might be also assymetrical (i.e. joining two identical polygons, but without an axis of symmetry passing through the edge, so you have to try to fit it in both orientations).
The exact number of possiblities for CRF purposes will have to be checked, some possibilities might be chiral.
For the weird solids (sphenocorona and up) I've given up on describing the edges and counting possibilities...
31.7175 (dpc4)
4-10 in pentagonal cupola
37.3774 (dpp)
3-5 in pentagonal pyramid
3-10 in pentagonal cupola
45
4-8 in square cupola
54.7356 (90 - d4/2)
3-4 in square pyramid
4-6 in triangular cupola
3-8 in square cupola
60
4-4 in triangular prism
4-4 in elongated triangular pyramid (asymmetrical)
4-4 in elongated triangular dipyramid
4-4 in augmented triangular prism
63.4349 (2*dpc4)
5-10 in pentagonal rotunda
4-4 in pentagonal orthobicupola
5-5 in metabidiminished icosahedron
5-5 in tridiminished icosahedron (asymmetrical)
5-5 in augmented tridiminished icosahedron (asymmetrical)
5-5 in bilunabirotunda
69.0948 (dpc4 + dpp = d20/2)
3-4 in pentagonal gyrobicupola (equatorial)
70.5288 (let's call it d4 because it's an important number)
3-3 in tetrahedron
6-6 in truncated tetrahedron
3-6 in triangular cupola
3-3 in elongated triangular pyramid (asymmetrical)
3-3 in triangular dipyramid (apex) (asymmetrical)
3-3 in elongated triangular dipyramid (asymmetrical)
3-3 in augmented tridiminished icosahedron (augment) (asymmetrical)
6-6 in augmented truncated tetrahedron
72.973
4-4 in sphenomegacorona
74.7547 (2*dpp)
3-3 in pentagonal dipyramid (equatorial)
3-3 in pentagonal orthobicupola
79.1877 (dpr3; dpr3 + 2*dpc4 = did)
3-10 in pentagonal rotunda
86.7268
3-3 in sphenomegacorona
90
4-4 in cube
8-8 in truncated cube
3-4 in triangular prism
4-5 in pentagonal prism
4-6 in hexagonal prism
4-7 in heptagonal prism
4-8 in octagonal prism
4-9 in enneagonal prism
4-10 in decagonal prism
3-4 in elongated triangular pyramid (base)
4-4 in elongated square pyramid (2 possibilities) (both asymmetrical)
4-5 in elongated pentagonal pyramid
4-4 in elongated square dipyramid
4-6 in elongated triangular cupola (2 possibilities)
4-8 in elongated square cupola (2 possibilities)
4-10 in elongated pentagonal cupola (2 possibilities)
4-10 in elongated pentagonal rotunda (2 possibilities)
3-4 in gyrobifastigium (individual prisms)
4-4 in square orthobicupola (equatorial)
3-4 in augmented triangular prism (base/side)
3-4 in biaugmented triangular prism (base/side)
4-5 in augmented pentagonal prism (2 possibilities)
4-5 in biaugmented pentagonal prism (2 possibilities)
4-6 in augmented hexagonal prism (3 possibilities)
4-6 in parabiaugmented hexagonal prism
4-6 in metabiaugmented hexagonal prism (3 possibilities)
4-6 in triaugmented hexagonal prism
8-8 in augmented truncated cube (2 possibilities) (both asymmetrical)
8-8 in biaugmented truncated cube
95.1524 (3*dpc4)
4-5 in pentagonal gyrocupolarotunda (equatorial)
95.2466 (da10)
3-10 in decagonal antiprism
3-10 in gyroelongated pentagonal cupola
3-10 in gyroelongated pentagonal rotunda
95.843
3-9 in enneagonal antiprism
96.1983
3-3 in snub disphenoid (type 1) (2 possibilities) (1 asymmetrical)
96.5945 (da8)
3-8 in octagonal antiprism
3-8 in gyroelongated square cupola
97.4555
3-4 in sphenocorona
3-4 in augmented sphenocorona
97.5723
3-7 in heptagonal antiprism
98.8994 (da6)
3-6 in hexagonal antiprism
3-6 in gyroelongated triangular cupola
99.7356 (135 - d4/2)
3-4 in square gyrobicupola (equatorial)
100.194
4-4 in disphenocingulum
100.812 (d20 - dpp = dpp + 2*dpc4)
3-5 in pentagonal antiprism
3-5 in gyroelongated pentagonal pyramid
3-5 in pentagonal orthocupolarotunda (equatorial)
3-5 in metabidiminished icosahedron (2 possibilities)
3-5 in tridiminished icosahedron (2 possibilities)
3-5 in augmented tridiminished icosahedron (main body)
3-5 in bilunabirotunda
3-5 in triangular hebesphenorotunda
102.524
4-4 in hebesphenomegacorona
103.836 (da4)
3-4 in square antiprism
3-4 in gyroelongated square pyramid
108
4-4 in pentagonal prism
4-4 in elongated pentagonal pyramid (asymmetrical)
4-4 in elongated pentagonal dipyramid
4-4 in augmented pentagonal prism (2 possibilities) (1 asymmetrical)
4-4 in biaugmented pentagonal prism
109.471 (180 - d4)
3-3 in octahedron
3-6 in truncated tetrahedron
6-6 in truncated octahedron
3-3 in square pyramid
3-3 in elongated square pyramid (asymmetrical)
3-3 in gyroelongated square pyramid (apex) (asymmetrical)
3-3 in elongated square dipyramid (asymmetrical)
3-3 in gyroelongated square dipyramid (apex) (asymmetrical)
4-4 in triangular orthobicupola
3-3 in square orthobicupola
3-3 in augmented triangular prism (augment) (asymmetrical)
3-3 in biaugmented triangular prism (augment) (2 possibilities) (both asymmetrical)
3-3 in triaugmented triangular prism (augment) (asymmetrical)
3-3 in augmented pentagonal prism (asymmetrical)
3-3 in biaugmented pentagonal prism (2 possibilities) (both asymmetrical)
3-3 in augmented hexagonal prism (asymmetrical)
3-3 in parabiaugmented hexagonal prism (asymmetrical)
3-3 in metabiaugmented hexagonal prism (2 possibilities) (both asymmetrical)
3-3 in triaugmented hexagonal prism (asymmetrical)
3-6 in augmented truncated tetrahedron (main body) (2 possibilities)
3-3 in augmented sphenocorona
109.524
3-4 in sphenocorona
3-4 in augmented sphenocorona
110.905 (dpc4 + dpr3)
3-4 in pentagonal orthocupolarotunda (equatorial)
3-4 in bilunabirotunda
4-6 in triangular hebesphenorotunda
3-4 in triangular hebesphenorotunda
111.735
3-3 in hebesphenomegacorona
114.645
3-3 in snub square antiprism (middle edges) (2 possibilities)
114.736 (150 - d4/2)
3-4 in augmented triangular prism (augment/side)
3-4 in biaugmented triangular prism (augment/side)
116.565 (d12 = dpp + dpr3)
5-5 in dodecahedron
10-10 in truncated dodecahedron
3-3 in pentagonal gyrocupolarotunda
5-5 in augmented dodecahedron (4 possibilities) (all asymmetrical)
5-5 in parabiaugmented dodecahedron (2 possibilities) (1 asymmetrical)
5-5 in metabiaugmented dodecahedron (7 possibilities) (5 asymmetrical)
5-5 in triaugmented dodecahedron (4 possibilities) (all asymmetrical)
10-10 in augmented truncated dodecahedron (4 possibilities) (all asymmetrical)
10-10 in parabiaugmented truncated dodecahedron (2 possibilities) (1 asymmetrical)
10-10 in metabiaugmented truncated dodecahedron (7 possibilities) (5 asymmetrical)
10-10 in triaugmented truncated dodecahedron (4 possibilities) (all asymmetrical)
5-10 in diminished rhombicosidodecahedron
5-10 in paragyrate diminished rhombicosidodecahedron
5-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
5-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)
5-10 in parabidiminished rhombicosidodecahedron
5-10 in metabidiminished rhombicosidodecahedron (3 possibilities)
5-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)
5-10 in tridiminished rhombicosidodecahedron (3 possibilities)
117.019 (dsp)
4-4 in sphenocorona
117.356
3-3 in sphenomegacorona
118.892
3-3 in sphenocorona
3-3 in augmented sphenocorona
120
4-4 in hexagonal prism
4-4 in elongated triangular cupola (prism) (asymmetrical)
4-4 in elongated triangular orthobicupola (prism) (asymmetrical)
4-4 in elongated triangular gyrobicupola (prism)
4-4 in augmented hexagonal prism (2 possibilities) (both asymmetrical)
4-4 in parabiaugmented hexagonal prism
4-4 in metabiaugmented hexagonal prism (asymmetrical)
121.717 (90 + dpc4)
4-4 in elongated pentagonal cupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal orthobicupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal gyrobicupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal orthocupolarotunda (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal gyrocupolarotunda (cupola/prism) (asymmetrical)
4-10 in diminished rhombicosidodecahedron
4-10 in paragyrate diminished rhombicosidodecahedron
4-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
4-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)
4-10 in parabidiminished rhombicosidodecahedron
4-10 in metabidiminished rhombicosidodecahedron (3 possibilities)
4-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)
4-10 in tridiminished rhombicosidodecahedron (3 possibilities)
121.743
3-3 in snub disphenoid (type 2) (2 possibilities) (both asymmetrical)
124.702
3-3 in disphenocingulum
125.264 (90 + d4/2)
3-4 in cuboctahedron
4-6 in truncated octahedron
3-8 in truncated cube
6-8 in truncated cuboctahedron
3-4 in triangular cupola (2 possibilities)
3-4 in elongated triangular cupola (cupola) (2 possibilities)
3-4 in gyroelongated triangular cupola (cupola) (2 possibilities)
3-4 in triangular orthobicupola (2 possibilities)
3-4 in elongated triangular orthobicupola (cupola) (2 possibilities)
3-4 in elongated triangular gyrobicupola (cupola) (2 possibilities)
3-4 in gyroelongated triangular bicupola (cupola) (2 possibilities)
3-4 in augmented truncated tetrahedron (augment) (2 possibilities)
3-8 in augmented truncated cube (3 possibilities)
3-8 in biaugmented truncated cube
126.87 (4*dpc4)
5-5 in pentagonal orthobirotunda
126.964 (da10 + dpc)
3-4 in gyroelongated pentagonal cupola (cupola/antiprism)
3-4 in gyroelongated pentagonal bicupola (cupola/antiprism)
3-4 in gyroelongated pentagonal cupolarotunda (cupola/antiprism)
127.377 (90 + dpp)
3-4 in elongated pentagonal pyramid
3-4 in elongated pentagonal dipyramid
3-4 in elongated pentagonal cupola (cupola/prism)
3-4 in elongated pentagonal orthobicupola (cupola/prism)
3-4 in elongated pentagonal gyrobicupola (cupola/prism)
3-4 in elongated pentagonal orthocupolarotunda (cupola/prism)
3-4 in elongated pentagonal gyrocupolarotunda (cupola/prism)
127.552
3-3 in square antiprism
3-3 in gyroelongated square pyramid (antiprism band) (asymmetrical)
3-3 in gyroelongated square dipyramid (antiprism band)
128.496
3-3 in hebesphenomegacorona
128.571
4-4 in heptagonal prism
129.445
3-3 in sphenomegacorona
131.442
3-3 in augmented sphenocorona
132.624 (da10 + dpp)
3-3 in gyroelongated pentagonal cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal bicupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (cupola/antiprism) (asymmetrical)
133.591
3-3 in disphenocingulum
133.973
3-4 in hebesphenomegacorona
135
4-4 in rhombicuboctahedron (asymmetrical)
4-8 in truncated cuboctahedron
4-4 in octagonal prism
4-4 in square cupola
4-4 in elongated square cupola (3 possibilities) (all asymmetrical)
4-4 in gyroelongated square cupola (asymmetrical)
4-4 in square orthobicupola (cupola) (asymmetrical)
4-4 in square gyrobicupola (asymmetrical)
4-4 in elongated square gyrobicupola (3 possibilities) (2 of them asymmetrical)
4-4 in gyroelongated square bicupola (asymmetrical)
4-4 in augmented truncated cube (asymmetrical)
4-4 in biaugmented truncated cube (asymmetrical)
135.992
3-3 in sphenocorona
3-3 in augmented sphenocorona
136.336
3-4 in disphenocingulum
137.24
3-4 in sphenomegacorona
138.19 (d20 = 2*(dpc4 + dpp)
3-3 in icosahedron
6-6 in truncated icosahedron
3-3 in pentagonal antiprism
3-3 in pentagonal pyramid
3-3 in elongated pentagonal pyramid (asymmetrical)
3-3 in gyroelongated pentagonal pyramid (3 possibilities) (all asymmetrical)
3-3 in pentagonal dipyramid (apex) (asymmetrical)
3-3 in elongated pentagonal dipyramid (asymmetrical)
3-3 in augmented dodecahedron (asymmetrical)
3-3 in parabiaugmented dodecahedron (asymmetrical)
3-3 in metabiaugmented dodecahedron (3 possibilities) (all asymmetrical)
3-3 in triaugmented dodecahedron (3 possibilities) (all asymmetrical)
3-3 in metabidiminished icosahedron (4 possibilities) (3 asymmetrical)
3-3 in tridiminished icosahedron (asymmetrical)
3-3 in augmented tridiminished icosahedron (main body) (asymmetrical)
3-6 in triangular hebesphenorotunda
3-3 in triangular hebesphenorotunda
140
4-4 in enneagonal prism
141.058 (2*d4)
3-3 in triangular dipyramid (equatorial)
3-3 in triangular orthobicupola
3-6 in augmented truncated tetrahedron (augment/main body)
141.31
3-3 in hebesphenomegacorona
141.595 (45 + da8)
3-4 in gyroelongated square cupola (cupola/antiprism)
3-4 in gyroelongated square bicupola (cupola/antiprism)
142.623 (did)
3-5 in icosidodecahedron
5-6 in truncated icosahedron
3-10 in truncated dodecahedron
6-10 in truncated icosidodecahedron
3-5 in pentagonal rotunda (3 possibilities)
3-5 in elongated pentagonal rotunda (3 possibilities)
3-5 in gyroelongated pentagonal cupola (rotunda) (3 possibilities)
3-5 in pentagonal orthocupolarotunda (rotunda) (3 possibilities)
3-5 in pentagonal gyrocupolarotunda (3 possibilities)
3-5 in pentagonal orthobirotunda (3 possibilities)
3-5 in elongated pentagonal orthocupolarotunda (3 possibilities)
3-5 in elongated pentagonal gyrocupolarotunda (3 possibilities)
3-5 in elongated pentagonal orthobirotunda (3 possibilities)
3-5 in elongated pentagonal gyrobirotunda (3 possibilities)
3-5 in gyroelongated pentagonal cupolarotunda (3 possibilities)
3-5 in gyroelongated pentagonal birotunda (3 possibilities)
3-10 in augmented truncated dodecahedron (main body) (7 possibilities)
3-10 in parabiaugmented truncated dodecahedron (main body) (3 possibilities)
3-10 in metabiaugmented truncated dodecahedron (main body) (14 possibilities)
3-10 in triaugmented truncated dodecahedron (main body) (9 possibilities)
3-5 in bilunabirotunda
3-5 in triangular hebesphenorotunda
142.983
3-4 in snub cube
143.479
3-3 in sphenocorona
3-3 in augmented sphenocorona
143.738
3-3 in sphenomegacorona
144
4-4 in decagonal prism
4-4 in elongated pentagonal cupola (prism) (asymmetrical)
4-4 in elongated pentagonal rotunda (asymmetrical)
4-4 in elongated pentagonal orthobicupola (prism) (asymmetrical)
4-4 in elongated pentagonal gyrobicupola (prism)
4-4 in elongated pentagonal orthocupolarotunda (prism) (asymmetrical)
4-4 in elongated pentagonal gyrocupolarotunda (prism) (asymmetrical)
4-4 in elongated pentagonal orthobirotunda (asymmetrical)
4-4 in elongated pentagonal gyrobirotunda
144.144
3-3 in snub square antiprism (other edges of triangles adjacent to squares) (2 possibilities) (both asymmetrical)
144.736 (180 - d4/2)
3-4 in rhombicuboctahedron
4-6 truncated cuboctahedron
3-4 in square cupola
3-4 in elongated square pyramid
3-4 in elongated square dipyramid
4-4 in elongated triangular cupola (cupola/prism) (asymmetrical)
3-4 in elongated square cupola (2 possibilities)
3-4 in gyroelongated square cupola (cupola)
3-4 in square orthobicupola
3-4 in square gyrobicupola (cupola)
4-4 in elongated triangular orthobicupola (cupola/prism) (asymmetrical)
4-4 in elongated triangular gyrobicupola (cupola/prism) (asymmetrical)
3-4 in elongated square gyrobicupola (2 possibilities)
3-4 in gyroelongated square bicupola (cupola)
3-3 in augmented triangular prism (augment/base) (asymmetrical)
3-3 in biaugmented triangular prism (augment/base) (asymmetrical)
3-3 in triaugmented triangular prism (augment/base) (asymmetrical)
3-5 in augmented pentagonal prism
3-5 in biaugmented pentagonal prism
3-6 in augmented hexagonal prism
3-6 in parabiaugmented hexagonal prism
3-6 in metabiaugmented hexagonal prism
3-6 in triaugmented hexagonal prism
3-4 in augmented truncated cube (augment)
3-8 in augmented truncated cube (augment/main body)
3-4 in biaugmented truncated cube (augment)
3-8 in biaugmented truncated cube (augment/main body)
145.222
3-3 in hexagonal antiprism
3-3 in gyroelongated triangular cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in in gyroelongated triangular bicupola (antiprism) (3 possibilities) (1 asymmetrical)
145.441
3-4 in snub square antiprism
148.283
4-5 in rhombicosidodecahedron
4-10 in truncated icosidodecahedron
4-5 in pentagonal cupola
4-5 in elongated pentagonal cupola
4-5 in gyroelongated pentagonal cupola
4-5 in pentagonal orthobicupola
4-5 in pentagonal gyrobicupola
4-5 in pentagonal orthocupolarotunda
4-5 in pentagonal gyrocupolarotunda (cupola)
4-5 in elongated pentagonal orthobicupola
4-5 in elongated pentagonal gyrobicupola
4-5 in elongated pentagonal orthocupolarotunda (cupola)
4-5 in elongated pentagonal gyrocupolarotunda (cupola)
4-5 in gyroelongated pentagonal bicupola
4-5 in augmented truncated dodecahedron
4-5 in parabiaugmented truncated dodecahedron
4-5 in metabiaugmented truncated dodecahedron (3 possibilities)
4-5 in triaugmented truncated dodecahedron (3 possibilities)
4-5 in gyrate rhombicosidodecahedron (7 possibilities)
4-5 in parabigyrate rhombicosidodecahedron (3 possibilities)
4-5 in metabigyrate rhombicosidodecahedron (15 possibilities)
4-5 in trigyrate rhombicosidodecahedron (10 possibilities)
4-5 in diminished rhombicosidodecahedron (6 possibilities)
4-5 in paragyrate diminished rhombicosidodecahedron (5 possibilities)
4-5 in metagyrate diminished rhombicosidodecahedron (22 possibilities)
4-5 in bigyrate diminished rhombicosidodecahedron (21 possibilities)
4-5 in parabidiminished rhombicosidodecahedron (2 possibilities)
4-5 in metabidiminished rhombicosidodecahedron (11 possibilities)
4-5 in gyrate bidiminished rhombicosidodecahedron (20 possibilities)
4-5 in tridiminished rhombicosidodecahedron (6 possibilities)
148.434
3-3 in disphenocingulum
149.565
3-3 in hebesphenomegacorona
150
3-4 in gyrobifastigium (blend)
150.222
3-3 in heptagonal antiprism
151.33 (90 + da8 - d4/2)
3-3 in gyroelongated square cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated square bicupola (cupola/antiprism) (asymmetrical)
152.191
3-3 in augmented sphenocorona
152.93
3-5 in snub dodecahedron
152.976
3-4 in hebesphenomegacorona
153.235
3-3 in snub cube (2 possibilities) (1 asymmetrical)
153.435 (90 + 2*dpc4)
4-5 in elongated pentagonal rotunda
4-5 in elongated pentagonal orthocupolarotunda (rotunda/prism)
4-5 in elongated pentagonal gyrocupolarotunda (rotunda/prism)
4-5 in elongated pentagonal orthobirotunda
4-5 in elongated pentagonal gyrobirotunda
4-4 in gyrate rhombicosidodecahedron (asymmetrical)
4-4 in parabigyrate rhombicosidodecahedron (asymmetrical)
4-4 in metabigyrate rhombicosidodecahedron (3 possibilities) (all asymetrical)
4-4 in trigyrate rhombicosidodecahedron (3 possibilities) (all asymmetrical)
4-4 in paragyrate diminished rhombicosidodecahedron (asymmetrical)
4-4 in metagyrate diminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)
4-4 in bigyrate diminished rhombicosidodecahedron (5 possibilities) (all asymmetrical)
4-4 in gyrate bidiminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)
153.635 (180 + da6 - d4/2)
3-4 in gyroelongated triangular cupola (cupola/antiprism)
3-4 in gyroelongated triangular bicupola (cupola/antiprism)
153.942 (dpp + d12)
3-5 in augmented dodecahedron
3-5 in parabiaugmented dodecahedron
3-5 in metabiaugmented dodecahedron (3 possibilities)
3-5 in triaugmented dodecahedron (3 possibilities)
3-10 in augmented truncated dodecahedron (augment/main body)
3-10 in parabiaugmented truncated dodecahedron (augment/main body)
3-10 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-10 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-5 in gyrate rhombicosidodecahedron
3-5 in parabigyrate rhombicosidodecahedron
3-5 in metabigyrate rhombicosidodecahedron (3 possibilities)
3-5 in trigyrate rhombicosidodecahedron (3 possibilities)
3-5 in paragyrate diminished rhombicosidodecahedron
3-5 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
3-5 in bigyrate diminished rhombicosidodecahedron (5 possibilities)
3-5 in gyrate bidiminished rhombicosidodecahedron (3 possibilities)
153.962
3-3 in octagonal antiprism
3-3 in gyroelongated square cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated square bicupola (antiprism) (3 possibilities) (1 asymmetrical)
154.419
3-4 in disphenocingulum
154.722
3-4 in sphenomegacorona
156.866
3-3 in enneagonal antiprism
157.148
3-3 in hebesphenomegacorona
158.375 (2*dpr3)
3-3 in pentagonal orthobirotunda
158.572 (90 + da4 - d4/2)
3-3 in gyroelongated square pyramid (apex/antiprism) (asymmetrical)
3-3 in in gyroelongated square dipyramid (apex/antiprism) (asymmetrical)
158.682 (da10 + 2*dpc4)
3-5 in gyroelongated pentagonal rotunda (rotunda/antiprism)
3-5 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)
3-5 in gyroelongated pentagonal birotunda (rotunda/antiprism)
159.095
3-4 in rhombicosidodecahedron
4-6 in truncated icosidodecahedron
3-4 in pentagonal cupola
3-4 in elongated pentagonal cupola (cupola)
3-4 in gyroelongated pentagonal cupola (cupola)
3-4 in pentagonal orthobicupola
3-4 in pentagonal gyrobicupola (cupola)
3-4 in pentagonal orthocupolarotunda (cupola)
3-4 in pentagonal gyrocupolarotunda
3-4 in elongated pentagonal orthobicupola (cupola)
3-4 in elongated pentagonal gyrobicupola (cupola)
3-4 in elongated pentagonal orthocupolarotunda (cupola)
3-4 in elongated pentagonal gyrocupolarotunda (cupola)
3-4 in gyroelongated pentagonal bicupola (cupola)
3-4 in gyroelongated pentagonal cupolarotunda (cupola)
3-4 in augmented truncated dodecahedron (augment)
3-4 in parabiaugmented truncated dodecahedron (augment)
3-4 in metabiaugmented truncated dodecahedron (augment) (5 possibilities)
3-4 in triaugmented truncated dodecahedron (augment) (5 possibilities)
3-4 in gyrate rhombicosidodecahedron (7 possibilities)
3-4 in parabigyrate rhombicosidodecahedron (3 possibilities)
3-4 in metabigyrate rhombicosidodecahedron (14 possibilities)
3-4 in trigyrate rhombicosidodecahedron (9 possibilities)
3-4 in diminished rhombicosidodecahedron (6 possibilities)
3-4 in paragyrate diminished rhombicosidodecahedron (5 possibilities)
3-4 in metagyrate diminished rhombicosidodecahedron (21? possibilities -- I'm honestly not quite sure here)
3-4 in bigyrate diminished rhombicosidodecahedron (17? possibilities)
3-4 in parabidiminished rhombicosidodecahedron (2 possibilities)
3-4 in metabidiminished rhombicosidodecahedron (9 possibilities)
3-4 in gyrate bidiminished rhombicosidodecahedron (13 possibilities)
3-4 in tridiminished rhombicosidodecahedron (4 possibilities)
3-4 in bilunabirotunda
3-4 in triangular hebesphenorotunda
159.187
3-3 in decagonal antiprism
3-3 in gyroelongated pentagonal cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated pentagonal rotunda (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated pentagonal bicupola (antiprism) (3 possibilities) (1 asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (antiprism) (4 possibilities) (all asymmetrical)
3-3 in gyroelongated pentagonal birotunda (3 possibilities) (1 asymmetrical)
159.892
3-3 in sphenocorona
3-3 in augmented sphenocorona
160.529 (90 + d4)
3-4 in elongated triangular pyramid (apex)
3-4 in elongated triangular dipyramid
3-4 in elongated triangular cupola (cupola/prism)
3-4 in elongated triangular orthobicupola (cupola/prism)
3-4 in elongated triangular gyrobicupola (cupola/prism)
161.483
3-3 in sphenomegacorona
162.736 (198 - d4/2)
3-4 in augmented pentagonal prism (augment/side)
3-4 in biaugmented pentagonal prism (2 possibilities)
164.172
3-3 in snub dodecahedron (2 possibilities) (1 asymmetrical)
164.207 (270 - 3*d4/2)
3-4 in augmented truncated tetrahedron (augment/main body)
164.257
3-3 in snub square antiprism (vertical edges) (asymmetrical)
166.441
3-3 in snub disphenoid (type 3) (2 possibilities)
166.811
3-3 in disphenocingulum
169.188 (90 + dpr3)
3-4 in elongated pentagonal rotunda
3-4 in elongated pentagonal orthocupolarotunda (rotunda/prism)
3-4 in elongated pentagonal gyrocupolarotunda (rotunda/prism)
3-4 in elongated pentagonal orthobirotunda
3-4 in elongated pentagonal gyrobirotunda
169.428 (d4 + da6)
3-3 in gyroelongated triangular cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated triangular bicupola (cupola/antiprism) (asymmetrical)
169.471 (240 - d4)
3-3 in biaugmented triangular prism (augment/augment)
3-3 in triaugmented triangular prism (augment/augment)
170.264 (135 + d4/2)
3-4 in augmented truncated cube (augment/main body)
3-4 in biaugmented truncated cube (augment/main body)
171.341 (dpp + 2*dpc4 + d4)
3-5 in augmented tridiminished icosahedron (augment/main body)
171.646
3-3 in sphenomegacorona
171.755 (90 + dsp - d4/2)
3-4 in augmented sphenocorona
174.34 (dpc4 + did)
3-4 in augmented truncated dodecahedron (augment/main body)
3-4 in parabiaugmented truncated dodecahedron (augment/main body)
3-4 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-4 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)
174.434 (da10 + dpr3)
3-3 in gyroelongated pentagonal rotunda (rotunda/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)
3-3 in gyroelongated pentagonal birotunda (rotunda/antiprism)
174.736 (210 - d4/2)
3-4 in augmented hexagonal prism
3-4 in parabiaugmented hexagonal prism
3-4 in metabiaugmented hexagonal prism (2 possibilities)
3-4 in triaugmented hexagonal prism
Of course, you must also take into account chiral polyhedra:
snub cube, snub dodecahedron, gyroelongated triangular bicupola, gyroelongated square bicupola, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda. Those might count twice in all cases.
So, what about vertex figures that are NOT deltahedra? Actually, there's no such thing! If the vertex figure contains a polygon with more than three sides, that polygon must correspond to one of the finite list of valid polyhedra. That means that its chords CANNOT vary freely, they must take one of a discrete set of values.
quickfur wrote:Recently I looked over Marek's proof again. It seems pretty solid, in proving that 4D vertices must be rigid, once the surrounding cells have been determined. There are simply too many constraints on vertices for there to be any additional degrees of freedom to deform the vertex.
However, I'm not so sure that this proof alone gives us a feasible computer brute force search algorithm. One significant oversight in Marek's proof is in the following statement:So, what about vertex figures that are NOT deltahedra? Actually, there's no such thing! If the vertex figure contains a polygon with more than three sides, that polygon must correspond to one of the finite list of valid polyhedra. That means that its chords CANNOT vary freely, they must take one of a discrete set of values.
The conclusion is sound, but the statement that the polygon must correspond to a finite list of valid polyhedra is inaccurate, because there are an infinite number of CRFs in 3D! Namely, the prisms and antiprisms, which are surely CRF, even though they do not belong to the set of Johnson solids. What this means it that the set of possible chord lengths is countable, but not necessarily finite. This means it's not so simple for a brute-force algorithm to merely try to enumerate all possible vertices, because such an algorithm might not terminate!
So it's true that you won't have the continuously-deformable vertices you have in 3D, but there are still potentially an infinite (albeit countable) number of possibilities. Unless you're willing to make the assumption of excluding polygons past a certain size (e.g., no polygons greater than 20 vertices will be checked), that means the number of verfs will still be infinite, and will need careful representation in order to avoid an infinite loop in the algorithm.
And we already have proof that we must at least consider polygons up to degree 20, because there exist non-trivial CRF augmentations of the 10,20-duoprism that contain elongated pentagonal bipyramid cells. They may not be crown jewels per se, but they also suggest that non-trivial CRF constructions may exist at least up to a 20-gon. Norman Johnson included prism augmentations among the Johnson solids, so arguably duoprism augmentations should not be excluded from a theoretical list of all 4D CRFs, even if we were to exclude the infinite families of CRFs. The fact that they are possible at least up to 10,20-duoprisms suggests that we may miss some actual crown jewels if we were to cut off the search for CRFs with polygons anywhere less than 20-gons. That's a lot of polygons to take into consideration, which also exponentially increases the number of possibilities a brute-force algorithm would have to check.
OTOH, this means there is the slim but exciting possibility that there may exist 4D CRF crown jewels that contain more unusual polygons, perhaps like CVP-3 heptagons, bridging 3D crown jewel cells like snub disphenoids. I suspect this is unlikely to be the case, but Marek's proof does not exclude this possibility, and the potentially infinite list of polygons (resp. prisms and antiprisms) we could choose from in order to bridge the CVP ≥3 vertices of the 3D crown jewels means there's a remote possibility that such things might actually exist.
quickfur wrote:Side note: I wonder if we can include the snub cube and snub dodecahedron among the 3D crown jewels, because they are CVP 3 in spite of being uniform. I would really love to see what kinds of CRFs could be constructed that contained these two polyhedra as cells (besides their obvious prisms, of course). So far, all my attempts to build a CRF out of the snub cube besides its prism have failed.
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