## General Approach--can 3D methods be generalized?

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Re: General Approach--can 3D methods be generalized?

I tried to make a simple program to enumerate possible tetrahedral vertices for CRF polychora (vertices with four edges). I considered triangles to decagons as faces and derived these 212 possibilities:

Code: Select all
`333-333, sides: 333, 333, 333, 333333-334, sides: 333, 334, 333, 334333-335, sides: 333, 335, 333, 335333-344, sides: 333, 334, 334, 344333-355, sides: 333, 335, 335, 355333-444, sides: 333, 344, 344, 344333-446, sides: 333, 346, 344, 346333-448, sides: 333, 348, 344, 348333-440, sides: 333, 340, 344, 340333-466, sides: 333, 346, 346, 366333-488, sides: 333, 348, 348, 388333-400, sides: 333, 340, 340, 300333-555, sides: 333, 355, 355, 355333-550, sides: 333, 350, 355, 350333-500, sides: 333, 350, 350, 300333-666, sides: 333, 366, 366, 366333-888, sides: 333, 388, 388, 388333-000, sides: 333, 300, 300, 300334-344, sides: 334, 334, 334, 444334-345, sides: 334, 335, 334, 445334-433, sides: 334, 343, 343, 433334-434, sides: 334, 344, 343, 434334-436, sides: 334, 346, 343, 436334-438, sides: 334, 348, 343, 438334-430, sides: 334, 340, 343, 430334-444, sides: 334, 344, 344, 444334-446, sides: 334, 346, 344, 446334-448, sides: 334, 348, 344, 448334-440, sides: 334, 340, 344, 440334-466, sides: 334, 346, 346, 466334-468, sides: 334, 348, 346, 468334-460, sides: 334, 340, 346, 460334-533, sides: 334, 353, 353, 433334-530, sides: 334, 350, 353, 430334-550, sides: 334, 350, 355, 450334-644, sides: 334, 364, 364, 444334-646, sides: 334, 366, 364, 446334-666, sides: 334, 366, 366, 466334-844, sides: 334, 384, 384, 444334-848, sides: 334, 388, 384, 448334-044, sides: 334, 304, 304, 444334-045, sides: 334, 305, 304, 445334-040, sides: 334, 300, 304, 440334-050, sides: 334, 300, 305, 450335-355, sides: 335, 335, 335, 555335-444, sides: 335, 344, 344, 544335-440, sides: 335, 340, 344, 540335-466, sides: 335, 346, 346, 566335-533, sides: 335, 353, 353, 533335-535, sides: 335, 355, 353, 535335-530, sides: 335, 350, 353, 530335-555, sides: 335, 355, 355, 555335-644, sides: 335, 364, 364, 544335-666, sides: 335, 366, 366, 566335-844, sides: 335, 384, 384, 544335-044, sides: 335, 304, 304, 544335-040, sides: 335, 300, 304, 540335-055, sides: 335, 305, 305, 555344-434, sides: 344, 344, 443, 434344-436, sides: 344, 346, 443, 436344-438, sides: 344, 348, 443, 438344-430, sides: 344, 340, 443, 430344-444, sides: 344, 344, 444, 444344-446, sides: 344, 346, 444, 446344-448, sides: 344, 348, 444, 448344-440, sides: 344, 340, 444, 440344-454, sides: 344, 344, 445, 454344-450, sides: 344, 340, 445, 450344-464, sides: 344, 344, 446, 464344-466, sides: 344, 346, 446, 466344-468, sides: 344, 348, 446, 468344-460, sides: 344, 340, 446, 460344-474, sides: 344, 344, 447, 474344-484, sides: 344, 344, 448, 484344-486, sides: 344, 346, 448, 486344-494, sides: 344, 344, 449, 494344-404, sides: 344, 344, 440, 404344-406, sides: 344, 346, 440, 406344-545, sides: 344, 355, 454, 445344-540, sides: 344, 350, 454, 440344-505, sides: 344, 355, 450, 405344-636, sides: 344, 366, 463, 436344-646, sides: 344, 366, 464, 446344-666, sides: 344, 366, 466, 466344-686, sides: 344, 366, 468, 486344-606, sides: 344, 366, 460, 406344-838, sides: 344, 388, 483, 438344-848, sides: 344, 388, 484, 448344-868, sides: 344, 388, 486, 468344-030, sides: 344, 300, 403, 430344-040, sides: 344, 300, 404, 440344-050, sides: 344, 300, 405, 450344-060, sides: 344, 300, 406, 460346-446, sides: 346, 346, 444, 646346-448, sides: 346, 348, 444, 648346-440, sides: 346, 340, 444, 640346-456, sides: 346, 346, 445, 656346-540, sides: 346, 350, 454, 640346-634, sides: 346, 364, 463, 634346-636, sides: 346, 366, 463, 636346-644, sides: 346, 364, 464, 644346-646, sides: 346, 366, 464, 646346-664, sides: 346, 364, 466, 664346-684, sides: 346, 364, 468, 684346-604, sides: 346, 364, 460, 604346-834, sides: 346, 384, 483, 634346-844, sides: 346, 384, 484, 644346-848, sides: 346, 388, 484, 648346-864, sides: 346, 384, 486, 664346-034, sides: 346, 304, 403, 634346-044, sides: 346, 304, 404, 644346-040, sides: 346, 300, 404, 640346-064, sides: 346, 304, 406, 664346-065, sides: 346, 305, 406, 665348-646, sides: 348, 366, 464, 846348-834, sides: 348, 384, 483, 834348-838, sides: 348, 388, 483, 838348-844, sides: 348, 384, 484, 844348-864, sides: 348, 384, 486, 864348-034, sides: 348, 304, 403, 834348-044, sides: 348, 304, 404, 844348-064, sides: 348, 304, 406, 864340-545, sides: 340, 355, 454, 045340-646, sides: 340, 366, 464, 046340-034, sides: 340, 304, 403, 034340-035, sides: 340, 305, 403, 035340-030, sides: 340, 300, 403, 030340-044, sides: 340, 304, 404, 044340-045, sides: 340, 305, 404, 045340-054, sides: 340, 304, 405, 054340-064, sides: 340, 304, 406, 064355-535, sides: 355, 355, 553, 535355-530, sides: 355, 350, 553, 530355-555, sides: 355, 355, 555, 555355-666, sides: 355, 366, 566, 566355-030, sides: 355, 300, 503, 530355-040, sides: 355, 300, 504, 540350-035, sides: 350, 305, 503, 035350-030, sides: 350, 300, 503, 030350-045, sides: 350, 305, 504, 045366-636, sides: 366, 366, 663, 636366-646, sides: 366, 366, 664, 646366-656, sides: 366, 366, 665, 656366-848, sides: 366, 388, 684, 648366-040, sides: 366, 300, 604, 640388-838, sides: 388, 388, 883, 838300-030, sides: 300, 300, 003, 030444-444, sides: 444, 444, 444, 444444-445, sides: 444, 445, 444, 445444-446, sides: 444, 446, 444, 446444-447, sides: 444, 447, 444, 447444-448, sides: 444, 448, 444, 448444-449, sides: 444, 449, 444, 449444-440, sides: 444, 440, 444, 440444-450, sides: 444, 440, 445, 450444-466, sides: 444, 446, 446, 466444-468, sides: 444, 448, 446, 468444-460, sides: 444, 440, 446, 460444-666, sides: 444, 466, 466, 466444-668, sides: 444, 468, 466, 468444-660, sides: 444, 460, 466, 460445-455, sides: 445, 445, 445, 555445-466, sides: 445, 446, 446, 566445-544, sides: 445, 454, 454, 544445-540, sides: 445, 450, 454, 540445-644, sides: 445, 464, 464, 544445-640, sides: 445, 460, 464, 540445-666, sides: 445, 466, 466, 566445-744, sides: 445, 474, 474, 544445-844, sides: 445, 484, 484, 544445-866, sides: 445, 486, 486, 566445-944, sides: 445, 494, 494, 544445-044, sides: 445, 404, 404, 544445-055, sides: 445, 405, 405, 555445-066, sides: 445, 406, 406, 566446-644, sides: 446, 464, 464, 644446-646, sides: 446, 466, 464, 646446-648, sides: 446, 468, 464, 648446-640, sides: 446, 460, 464, 640446-744, sides: 446, 474, 474, 644446-844, sides: 446, 484, 484, 644446-846, sides: 446, 486, 484, 646446-944, sides: 446, 494, 494, 644446-044, sides: 446, 404, 404, 644446-046, sides: 446, 406, 404, 646446-056, sides: 446, 406, 405, 656447-744, sides: 447, 474, 474, 744447-844, sides: 447, 484, 484, 744447-944, sides: 447, 494, 494, 744447-044, sides: 447, 404, 404, 744448-844, sides: 448, 484, 484, 844448-846, sides: 448, 486, 484, 846448-944, sides: 448, 494, 494, 844448-044, sides: 448, 404, 404, 844448-046, sides: 448, 406, 404, 846449-944, sides: 449, 494, 494, 944449-044, sides: 449, 404, 404, 944440-044, sides: 440, 404, 404, 044440-045, sides: 440, 405, 404, 045440-046, sides: 440, 406, 404, 046450-045, sides: 450, 405, 504, 045450-046, sides: 450, 406, 504, 046466-646, sides: 466, 466, 664, 646466-648, sides: 466, 468, 664, 648466-640, sides: 466, 460, 664, 640466-656, sides: 466, 466, 665, 656468-846, sides: 468, 486, 684, 846468-046, sides: 468, 406, 604, 846460-046, sides: 460, 406, 604, 046555-555, sides: 555, 555, 555, 555555-666, sides: 555, 566, 566, 566566-656, sides: 566, 566, 665, 656`

Here, decagons are shortened to "0" to keep them as one character. Now, some of these "tetrahedra" are degenerate -- there's for example 334-433 which ends up coplanar and some work out to have too large diameter (566-656 corresponds to four truncated icosahedra in bitruncated (5,3,5) which is hyperbolic), but this is the first pass of the list.
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

I started checking these tetrahedra to eliminate the invalid ones. I'm also trying to assign a known CRF polychoron to each combination.

Notation note: If we label the tetrahedron abc-def, the faces are abc, adf, bde and cef.

First where I can't find anything is the tetrahedron 333-550. This is a vertex that has one tetrahedron (or an extension of one, such as triangular bipyramid), one metabidiminished icosahedron (or similar) and two pentagonal rotundas.

I can imagine a requisite CRF polychoron as two pentagonal rotundas joined in "ortho" manner, with metabidiminished icosahedra fitting between each pair of pentagons and tetrahedra between each pair of triangles. But this is not the complete shape and I'm not sure how to close it.

334-530: Similar case, but we have pentagonal rotunda and pentagonal cupola joined in "ortho" manner with triangle/pentagon gaps filled with pentagonal pyramids and triangle/square gaps filled with square pyramids.
334-550: Here we have pentagonal rotunda joined with diminished rhombicosidodecahedron. The pentagon/pentagon gaps are filled with metabidiminished icosahedra and the triangle/square gaps are filled with square pyramids.
334-644: Here we have one square pyramid, one cube and two triangular cupolas joining at a vertex. No idea what that would make.
334-646: Here we have four different cells: square pyramid, truncated tetrahedron, triangular cupola and hexagonal prism. Again, no idea.
334-848: Analogical to 334-646, the cells are square pyramid, truncated cube, square cupola and octagonal prism.
334-045: Another weird one with square pyramid, pentagonal rotunda, pentagonal cupola and pentagonal prism.
334-040: Square pyramid, truncated dodecahedron, pentagonal cupola and decagonal prism.
334-050: Square pyramid, truncated dodecahedron, pentagonal rotunda and diminished rhombicosidodecahedron.

335-530: Start would be two pentagonal rotundas joined in "gyro" orientation with all triangle/pentagon gaps filled with pentagonal pyramids.
335-555: Pentagonal pyramid, two metabidiminished icosahedra and dodecahedron.
335-644: Pentagonal pyramid, two triangular cupolas and pentagonal prism.
335-040: Pentagonal pyramid, truncated dodecahedron, pentagonal cupola and diminished rhombicosidodecahedron.
335-055: Pentagonal pyramid, two pentagonal rotundas and dodecahedron.

334-436: Two triangular cupolas in gyro orientation with all triangle/square gaps filled by triangular prisms.
334-438: Analogue with square cupolas.
344-450: Diminished rhombicosidodecahedron with pentagonal cupola in ortho orientation, triangle/square gaps filled by triangular prisms, square/pentagon gaps filled by pentagonal prisms.
344-505: Two diminished rhombicosidodecahedra in ortho orientation, square/square gaps filled by triangular prisms, pentagon/pentagon gaps filled by metabidiminished icosahedra.
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

And here's first part of the analysis:

Tetrahedra that are possible (with examples, if I know of any):

Code: Select all
`333-333, sides: 333, 333, 333, 333 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.50000  0.28868  0.20412 0.79057 example: pentachoron (K4.1)333-334, sides: 333, 334, 333, 334 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.57735  0.40825 0.70711 example: square duopyramid (K4.4)333-335, sides: 333, 335, 333, 335 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902  0.75576  0.53440 0.21851 example: pentagonal duopyramid (K4.86)333-344, sides: 333, 334, 334, 344 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.57735 -0.20412 0.79057 example: triangular prismatic pyramid (K4.7)333-355, sides: 333, 335, 335, 355 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902  0.75576 -0.45644 0.35355 example: metabidiminished icosahedral pyramid (K4.87)333-444, sides: 333, 344, 344, 344 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.00000  0.00000 1.00000 example: tetrahedral prism (K4.9)333-446, sides: 333, 346, 344, 346 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000  0.28868  0.20412 0.79057 example: trigon || trigonal cupola (K4.25)333-448, sides: 333, 348, 344, 348 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.70711  0.40825  0.28868 0.50000 example: square || square cupola (K4.73 -- by the way, there seems to be a mistake in the segmentochoron paper since it doesn't list square cupolas as cells)333-440, sides: 333, 340, 344, 340 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.46709  0.33028 0.13505 example: pentagon || pentagonal cupola (K4.154)333-466, sides: 333, 346, 346, 366 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000  0.28868 -0.40825 0.70711 example: tetrahedron || truncated tetrahedron (K4.56)333-555, sides: 333, 355, 355, 355 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902 -0.17841 -0.12616 0.92561 example: tetrahedral ursachoron333-550, sides: 333, 350, 355, 350 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.11026 -0.07797 0.57206 example: not sure333-500, sides: 333, 350, 350, 300 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.11026 -0.53440 0.21851 example: rhodomesohedral rotunda333-666, sides: 333, 366, 366, 366 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000 -0.28868 -0.20412 0.79057 example: truncated pentachoron333-888, sides: 333, 388, 388, 388 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.70711 -0.40825 -0.28868 0.50000 example: truncated tesseract333-000, sides: 333, 300, 300, 300 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902 -0.46709 -0.33028 0.13505 example: truncated 120-cell334-344, sides: 334, 334, 334, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.57735 -0.40825 0.70711 example: cubic pyramid (K4.26)334-345, sides: 334, 335, 334, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.30902  0.75576 -0.53440 0.21851 example: pentagonal prismatic pyramid (K4.141)334-434, sides: 334, 344, 343, 434 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.00000  0.61237 0.79057 example: digonal gyrobicupolic ring (K4.8)334-436, sides: 334, 346, 343, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000  0.28868  0.40825 0.70711 example: trigonal gyrobicupolic ring (K4.27)334-438, sides: 334, 348, 343, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711  0.40825  0.32370 0.47807 example: square gyrobicupolic ring (K4.64)334-430, sides: 334, 340, 343, 430 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.46709 -0.28209 0.21851 example: pentagonal gyrobicupolic ring (K4.133)334-444, sides: 334, 344, 344, 444 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.00000  0.00000 1.00000 example: square pyramidal prism (K4.12)334-446, sides: 334, 346, 344, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000  0.28868 -0.20412 0.79057 example: hexagon || trigonal cupola (K4.51)334-448, sides: 334, 348, 344, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711  0.40825 -0.28868 0.50000 example: octagon || square cupola (K4.105)334-440, sides: 334, 340, 344, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.46709  0.33028 0.13505 example: decagon || pentagonal cupola (K4.165)334-530, sides: 334, 350, 353, 430 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.11026  0.53440 0.21851 example: not sure334-550, sides: 334, 350, 355, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.11026 -0.45644 0.35355 example: not sure334-644, sides: 334, 364, 364, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000 -0.57735  0.40825 0.70711 example: not sure334-646, sides: 334, 366, 364, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000 -0.28868  0.20412 0.79057 example: not sure334-666, sides: 334, 366, 366, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000 -0.28868 -0.40825 0.70711 example: octahedral rotunda334-848, sides: 334, 388, 384, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711 -0.40825  0.28868 0.50000 example: not sure334-045, sides: 334, 305, 304, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.30902 -0.75576  0.53440 0.21851 example: not sure334-040, sides: 334, 300, 304, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902 -0.46709  0.33028 0.13505 example: not sure334-050, sides: 334, 300, 305, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902 -0.46709  0.04819 0.35355 example: not sure335-444, sides: 335, 344, 344, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000 0.00000  0.00000  0.00000 1.00000 example: pentagonal pyramidal prism (K4.38)335-530, sides: 335, 350, 353, 530 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.80902  0.11026  0.28868 0.50000 example: not sure335-555, sides: 335, 355, 355, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.30902 -0.17841  0.46709 0.80902 example: not sure335-644, sides: 335, 364, 364, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000 0.00000 -0.57735  0.75576 0.30902 example: not sure335-666, sides: 335, 366, 366, 566 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.50000 -0.28868 -0.75576 0.30902 example: icosahedral rotunda335-040, sides: 335, 300, 304, 540 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.80902 -0.46709 -0.17841 0.30902 example: not sure335-055, sides: 335, 305, 305, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.30902 -0.75576  0.28868 0.50000 example: not sure344-434, sides: 344, 344, 443, 434 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.50000 0.86603 example: triangular duoprism (K4.10)344-436, sides: 344, 346, 443, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868  0.50000 0.64550 example: not sure344-438, sides: 344, 348, 443, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825  0.50000 0.28868 example: not sure344-444, sides: 344, 344, 444, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.00000 1.00000 example: triangular-square duoprism (K4.18)344-446, sides: 344, 346, 444, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868  0.00000 0.81650 example: triangular cupolaic prism (K4.45)344-448, sides: 344, 348, 444, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825  0.00000 0.57735 example: square cupolaic prism (4.69)344-440, sides: 344, 340, 444, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.00000 0.35682 example: pentagonal cupolaic prism (4.117)344-454, sides: 344, 344, 445, 454 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.30902 0.95106 example: triangular-pentagonal duoprism (K4.34)344-450, sides: 344, 340, 445, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.30902 0.17841 example: not sure344-464, sides: 344, 344, 446, 464 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.50000 0.86603 example: triangular-hexagonal duoprism (K4.47)344-466, sides: 344, 346, 446, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.50000 0.64550    example: truncated tetrahedron || truncated octahedron (K4.76)344-468, sides: 344, 348, 446, 468 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825 -0.50000 0.28868 example: truncated octahedron || truncated cuboctahedron (4.149)344-484, sides: 344, 344, 448, 484 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.70711 0.70711 example: triangular-octagonal duoprism (K4.59)344-486, sides: 344, 346, 448, 486 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.70711 0.40825    example: truncated cube || truncated cuboctahedron (K4.128)344-404, sides: 344, 344, 440, 404 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.80902 0.58779 example: triangular-decagonal duoprism (K4.94)344-4n4, sides: 344, 344, 44n, 4n4 - valid example: triangular-n-gonal duoprism (K4.175)344-406, sides: 344, 346, 440, 406 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.80902 0.11026    example: truncated dodecahedron || truncated icosidodecahedron (K4.174)344-545, sides: 344, 355, 454, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.30902  0.17841  0.00000 0.93417    example: metabidiminished icosahedral prism (K4.40)344-540, sides: 344, 350, 454, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.11026  0.00000 0.57735 example: pentagonal rotundaic prism (K4.92)344-505, sides: 344, 355, 450, 405 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.30902  0.17841  0.80902 0.46709    example: metabidiminished icosahedral prism (K4.40)344-636, sides: 344, 366, 463, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868  0.50000 0.64550    example: octahedron || truncated tetrahedron (K4.52)344-646, sides: 344, 366, 464, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868  0.00000 0.81650    example: truncated tetrahedral prism (K4.57)344-666, sides: 344, 366, 466, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.50000 0.64550    example: great rhombated pentachoron344-686, sides: 344, 366, 468, 486 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.70711 0.40825    example: great rhombated tesseract344-606, sides: 344, 366, 460, 406 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.80902 0.11026    example: great rhombated 120-cell344-838, sides: 344, 388, 483, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825  0.50000 0.28868 example: cuboctahedron || truncated cube (4.129)344-848, sides: 344, 388, 484, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825  0.00000 0.57735 example: truncated cubic prism (4.99)344-868, sides: 344, 388, 486, 468 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825 -0.50000 0.28868 example: great rhombated 24-cell344-040, sides: 344, 300, 404, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.00000 0.35682 example: truncated dodecahedral prism (K4.130)344-050, sides: 344, 300, 405, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709 -0.30902 0.17841 example: diminished small rhombated 120-cell`

And tetrahedra that are invalid:
Code: Select all
`333-488, sides: 333, 348, 348, 388 - invalid; planar triangle with central pointThis corresponds to the fact that cube || truncated cube is flat.333-400, sides: 333, 340, 340, 300 - invalid; hyperbolicThis corresponds to the fact that dodecahedron || truncated dodecahedron is not a segmentochoron.334-433, sides: 334, 343, 343, 433 - invalid; planar quadrangleThis corresponds to the fact that vertex of octahedron can be sliced in two square pyramids in two different ways.334-466, sides: 334, 346, 346, 466 - invalid; planar triangle with central pointThis corresponds to the fact that octahedron || truncated octahedron is flat.334-468, sides: 334, 348, 346, 468 - invalid; hyperbolicThis corresponds to the fact that cuboctahedron || truncated cuboctahedron is not a segmentochoron.334-460, sides: 334, 340, 346, 460 - invalid; hyperbolicThis corresponds to the fact that icosidodecahedron || truncated icosidodecahedron is not a segmentochoron.334-533, sides: 334, 353, 353, 433 - invalid; hyperbolic334-844, sides: 334, 384, 384, 444 - invalid; planar quadrangleThis corresponds to the fact that a vertex of square orthobicupola can be sliced in two square cupolas while identically shaped vertex of elongated square pyramid can be sliced into square pyramid and cube.334-044, sides: 334, 304, 304, 444 - invalid; hyperbolic335-355, sides: 335, 335, 335, 555 - invalid; hyperbolicThis corresponds to the fact that dodecahedral pyramid isn't a segmentochoron.335-440, sides: 335, 340, 344, 540 - invalid; hyperbolic335-466, sides: 335, 346, 346, 566 - invalid; hyperbolic335-533, sides: 335, 353, 353, 533 - invalid; hyperbolic335-535, sides: 335, 355, 353, 535 - invalid; planar quadrangleThis corresponds to a fact that vertex of gyroelongated pentagonal prism can be sliced into pentagonal pyramid and metabidiminished icosahedron in two different ways.335-844, sides: 335, 384, 384, 544 - invalid; hyperbolic335-044, sides: 335, 304, 304, 544 - invalid; hyperbolic344-430, sides: 344, 340, 443, 430 - invalid; hyperbolic344-460, sides: 344, 340, 446, 460 - invalid; hyperbolicThis corresponds to the fact that truncated icosahedron || truncated icosidodecahedron isn't a segmentochoron.344-030, sides: 344, 300, 403, 430 - invalid; hyperbolicThis corresponds to the fact that icosidodecahedron || truncated dodecahedron isn't a segmentochoron.344-060, sides: 344, 300, 406, 460 - invalid; hyperbolic`
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

Other unknown tetrahedra:

346-636, two triangular cupolas and two truncated tetrahedra
346-644, two triangular cupolas and two hexagonal prisms
346-664, two triangular cupolas and two truncated octahedra
346-684, two triangular cupolas and two truncated cuboctahedra
346-604, two triangular cupolas and two truncated icosidodecahedra
346-844, triangular cupola, square cupola, octagonal prism and hexagonal prism
346-864, triangular cupola, square cupola, truncated cuboctahedron and truncated octahedron
346-044, triangular cupola, pentagonal cupola, decagonal prism and hexagonal prism
346-064, triangular cupola, pentagonal cupola, truncated icosidodecahedron and truncated octahedron
346-065, triangular cupola, pentagonal rotunda, truncated icosidodecahedron and truncated icosahedron
348-838, two square cupolas and two truncated cubes
348-864, two square cupolas and two truncated cuboctahedra
348-064, square cupola, pentagonal cupola, truncated icosidodecahedron and truncated cuboctahedron

340-545, pentagonal cupola, metabidiminished icosahedron, pentagonal prism and diminished rhombicosidodecahedron: if we use tridiminished icosahedra, we could join diminished rhombicosidodecahedron and pentagonal cupola in gyro orientation and fill square/square gaps with pentagonal prisms and triangle/pentagon gaps with tridiminished icosahedra.

340-030, two pentagonal cupolas and two truncated dodecahedra
340-045, pentagonal cupola, pentagonal rotunda, decagonal prism and diminished rhombicosidodecahedron
340-064, two pentagonal cupolas and two truncated icosidodecahedra

355-535, four metabidiminished icosahedra: looks like a hypothetical "ursachoric" member of the sequence pentachoron - triangular duoprism - ??? - decachoron.

355-530, two metabidiminished icosahedra and two pentagonal rotundas: if we use tridiminished icosahedra, we could join two pentagonal rotundas in gyro orientation and fill all triangle/pentagon gaps with tridiminished icosahedra.

355-555, two metabidiminished icosahedra and two dodecahedra: looks like a hypothetical "ursachoric" member of the sequence (pentagonal duopyramid) - triangular-pentagonal duoprism - ??? - bitruncated 120-cell.

355-030, metabidiminished icosahedron, truncated dodecahedron and two pentagonal rotundas: if we use tridiminished icosahedra, we could join pentagonal rotunda to each decagonal face of truncated dodecahedron and tridiminished icosahedron to each triangular face.

350-030, two pentagonal rotundas and two truncated dodecahedra

350-045, two pentagonal rotundas and two diminished rhombicosidodecahedra

445-540, two pentagonal prisms and two diminished rhombicosidodecahedra: we could join two diminished rhombicosidodecahedra in gyro orientation and fill all square/pentagon gaps with pentagonal prisms.

445-055, pentagonal prism, two diminished rhombicosidodecahedra and dodecahedron: we could join two diminished rhombicosidodecahedra in ortho orientation, fill square/square gaps with pentagonal prisms and fill pentagon/pentagon gaps with dodecahedra.

440-045, two decagonal prisms and two diminished rhombicosidodecahedra

Now, I finished the analysis. The complete list of valid tetrahedra is:
Code: Select all
`333-333, sides: 333, 333, 333, 333 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.50000  0.28868  0.20412 0.79057 example: pentachoron (K4.1)333-334, sides: 333, 334, 333, 334 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.57735  0.40825 0.70711 example: square duopyramid (K4.4)333-335, sides: 333, 335, 333, 335 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902  0.75576  0.53440 0.21851 example: pentagonal duopyramid (K4.86)333-344, sides: 333, 334, 334, 344 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.57735 -0.20412 0.79057 example: triangular prismatic pyramid (K4.7)333-355, sides: 333, 335, 335, 355 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902  0.75576 -0.45644 0.35355 example: metabidiminished icosahedral pyramid (K4.87)333-444, sides: 333, 344, 344, 344 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000 0.00000  0.00000  0.00000 1.00000 example: tetrahedral prism (K4.9)333-446, sides: 333, 346, 344, 346 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000  0.28868  0.20412 0.79057 example: trigon || trigonal cupola (K4.25)333-448, sides: 333, 348, 344, 348 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.70711  0.40825  0.28868 0.50000 example: square || square cupola (K4.73 -- by the way, there seems to be a mistake in the segmentochoron paper since it doesn't list square cupolas as cells)333-440, sides: 333, 340, 344, 340 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.46709  0.33028 0.13505 example: pentagon || pentagonal cupola (K4.154)333-466, sides: 333, 346, 346, 366 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000  0.28868 -0.40825 0.70711 example: tetrahedron || truncated tetrahedron (K4.56)333-555, sides: 333, 355, 355, 355 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.30902 -0.17841 -0.12616 0.92561 example: tetrahedral ursachoron333-550, sides: 333, 350, 355, 350 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.11026 -0.07797 0.57206 example: not sure333-500, sides: 333, 350, 350, 300 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902  0.11026 -0.53440 0.21851 example: rhodomesohedral rotunda333-666, sides: 333, 366, 366, 366 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.50000 -0.28868 -0.20412 0.79057 example: truncated pentachoron333-888, sides: 333, 388, 388, 388 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.70711 -0.40825 -0.28868 0.50000 example: truncated tesseract333-000, sides: 333, 300, 300, 300 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.50000  0.28868  0.81650 0.00000-0.80902 -0.46709 -0.33028 0.13505 example: truncated 120-cell334-344, sides: 334, 334, 334, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.57735 -0.40825 0.70711 example: cubic pyramid (K4.26)334-345, sides: 334, 335, 334, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.30902  0.75576 -0.53440 0.21851 example: pentagonal prismatic pyramid (K4.141)334-434, sides: 334, 344, 343, 434 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.00000  0.61237 0.79057 example: digonal gyrobicupolic ring (K4.8)334-436, sides: 334, 346, 343, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000  0.28868  0.40825 0.70711 example: trigonal gyrobicupolic ring (K4.27)334-438, sides: 334, 348, 343, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711  0.40825  0.32370 0.47807 example: square gyrobicupolic ring (K4.64)334-430, sides: 334, 340, 343, 430 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.46709 -0.28209 0.21851 example: pentagonal gyrobicupolic ring (K4.133)334-444, sides: 334, 344, 344, 444 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000  0.00000  0.00000 1.00000 example: square pyramidal prism (K4.12)334-446, sides: 334, 346, 344, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000  0.28868 -0.20412 0.79057 example: hexagon || trigonal cupola (K4.51)334-448, sides: 334, 348, 344, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711  0.40825 -0.28868 0.50000 example: octagon || square cupola (K4.105)334-440, sides: 334, 340, 344, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.46709  0.33028 0.13505 example: decagon || pentagonal cupola (K4.165)334-530, sides: 334, 350, 353, 430 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.11026  0.53440 0.21851 example: not sure334-550, sides: 334, 350, 355, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902  0.11026 -0.45644 0.35355 example: not sure334-644, sides: 334, 364, 364, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000 0.00000 -0.57735  0.40825 0.70711 example: not sure334-646, sides: 334, 366, 364, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000 -0.28868  0.20412 0.79057 example: not sure334-666, sides: 334, 366, 366, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.50000 -0.28868 -0.40825 0.70711 example: octahedral rotunda334-848, sides: 334, 388, 384, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.70711 -0.40825  0.28868 0.50000 example: not sure334-045, sides: 334, 305, 304, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.30902 -0.75576  0.53440 0.21851 example: not sure334-040, sides: 334, 300, 304, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902 -0.46709  0.33028 0.13505 example: not sure334-050, sides: 334, 300, 305, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.57735  0.81650 0.00000-0.80902 -0.46709  0.04819 0.35355 example: not sure335-444, sides: 335, 344, 344, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000 0.00000  0.00000  0.00000 1.00000 example: pentagonal pyramidal prism (K4.38)335-530, sides: 335, 350, 353, 530 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.80902  0.11026  0.28868 0.50000 example: not sure335-555, sides: 335, 355, 355, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.30902 -0.17841  0.46709 0.80902 example: not sure335-644, sides: 335, 364, 364, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000 0.00000 -0.57735  0.75576 0.30902 example: not sure335-666, sides: 335, 366, 366, 566 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.50000 -0.28868 -0.75576 0.30902 example: icosahedral rotunda335-040, sides: 335, 300, 304, 540 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.80902 -0.46709 -0.17841 0.30902 example: not sure335-055, sides: 335, 305, 305, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902  0.75576  0.57735 0.00000-0.30902 -0.75576  0.28868 0.50000 example: not sure344-434, sides: 344, 344, 443, 434 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.50000 0.86603 example: triangular duoprism (K4.10)344-436, sides: 344, 346, 443, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868  0.50000 0.64550 example: not sure344-438, sides: 344, 348, 443, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825  0.50000 0.28868 example: not sure344-444, sides: 344, 344, 444, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.00000 1.00000 example: triangular-square duoprism (K4.18)344-446, sides: 344, 346, 444, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868  0.00000 0.81650 example: triangular cupolaic prism (K4.45)344-448, sides: 344, 348, 444, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825  0.00000 0.57735 example: square cupolaic prism (4.69)344-440, sides: 344, 340, 444, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.00000 0.35682 example: pentagonal cupolaic prism (4.117)344-454, sides: 344, 344, 445, 454 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.30902 0.95106 example: triangular-pentagonal duoprism (K4.34)344-450, sides: 344, 340, 445, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.30902 0.17841 example: not sure344-464, sides: 344, 344, 446, 464 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.50000 0.86603 example: triangular-hexagonal duoprism (K4.47)344-466, sides: 344, 346, 446, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.50000 0.64550    example: truncated tetrahedron || truncated octahedron (K4.76)344-468, sides: 344, 348, 446, 468 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.40825 -0.50000 0.28868 example: truncated octahedron || truncated cuboctahedron (4.149)344-484, sides: 344, 344, 448, 484 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.70711 0.70711 example: triangular-octagonal duoprism (K4.59)344-486, sides: 344, 346, 448, 486 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.70711 0.40825    example: truncated cube || truncated cuboctahedron (K4.128)344-404, sides: 344, 344, 440, 404 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000 -0.80902 0.58779 example: triangular-decagonal duoprism (K4.94)344-406, sides: 344, 346, 440, 406 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.28868 -0.80902 0.11026    example: truncated dodecahedron || truncated icosidodecahedron (K4.174)344-4n4, sides: 344, 344, 44n, 4n4 - valid example: triangular-n-gonal duoprism (K4.175)344-545, sides: 344, 355, 454, 445 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.30902  0.17841  0.00000 0.93417    example: metabidiminished icosahedral prism (K4.40)344-540, sides: 344, 350, 454, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.11026  0.00000 0.57735 example: pentagonal rotundaic prism (K4.92)344-505, sides: 344, 355, 450, 405 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.30902  0.17841  0.80902 0.46709    example: metabidiminished icosahedral prism (K4.40)344-636, sides: 344, 366, 463, 436 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868  0.50000 0.64550    example: octahedron || truncated tetrahedron (K4.52)344-646, sides: 344, 366, 464, 446 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868  0.00000 0.81650    example: truncated tetrahedral prism (K4.57)344-666, sides: 344, 366, 466, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.50000 0.64550    example: great rhombated pentachoron344-686, sides: 344, 366, 468, 486 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.70711 0.40825    example: great rhombated tesseract344-606, sides: 344, 366, 460, 406 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.28868 -0.80902 0.11026    example: great rhombated 120-cell344-838, sides: 344, 388, 483, 438 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825  0.50000 0.28868 example: cuboctahedron || truncated cube (4.129)344-848, sides: 344, 388, 484, 448 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825  0.00000 0.57735 example: truncated cubic prism (K4.99)344-868, sides: 344, 388, 486, 468 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711 -0.40825 -0.50000 0.28868 example: great rhombated 24-cell344-040, sides: 344, 300, 404, 440 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709  0.00000 0.35682 example: truncated dodecahedral prism (K4.130)344-050, sides: 344, 300, 405, 450 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902 -0.46709 -0.30902 0.17841 example: diminished small rhombated 120-cell346-446, sides: 346, 346, 444, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.50000  0.28868 -0.40825 0.70711    example: cuboctahedron || truncated octahedron (K4.95)346-456, sides: 346, 346, 445, 656 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.50000  0.28868 -0.78672 0.21851    example: icosidodecahedron || truncated icosahedron (K4.158)346-540, sides: 346, 350, 454, 640 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.80902  0.11026 -0.53440 0.21851    example: expanded/truncated rhodomesohedral rotunda346-636, sides: 346, 366, 463, 636 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.50000 -0.28868  0.40825 0.70711    example: unknown346-644, sides: 346, 364, 464, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.57735  0.20412 0.79057    example: unknown346-646, sides: 346, 366, 464, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.50000 -0.28868 -0.20412 0.79057    example: diminished prismatotruncated pentachoron (pyroperihedral rotunda)346-664, sides: 346, 364, 466, 664 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.57735  0.20412 0.79057    example: unknown346-684, sides: 346, 364, 468, 684 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.57735  0.66190 0.47807    example: unknown346-604, sides: 346, 364, 460, 604 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.57735 -0.78672 0.21851    example: unknown346-844, sides: 346, 384, 484, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.81650  0.28868 0.50000    example: unknown346-848, sides: 346, 388, 484, 648 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.70711 -0.40825 -0.28868 0.50000    example: diminished prismatotruncated 16-cell346-864, sides: 346, 384, 486, 664 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.81650 -0.32370 0.47807    example: unknown346-044, sides: 346, 304, 404, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.93417 -0.33028 0.13505    example: unknown346-040, sides: 346, 300, 404, 640 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.80902 -0.46709 -0.33028 0.13505    example: diminished prismatotruncated 600-cell346-064, sides: 346, 304, 406, 664 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000 0.00000 -0.93417  0.28209 0.21851    example: unknown346-065, sides: 346, 305, 406, 665 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000  0.28868  0.81650 0.00000-0.30902 -0.75576 -0.53440 0.21851    example: unknown348-646, sides: 348, 366, 464, 846 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.70711  0.40825  0.57735 0.00000-0.50000 -0.28868 -0.40825 0.70711    example: stauroperihedral rotunda348-838, sides: 348, 388, 483, 838 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.70711  0.40825  0.57735 0.00000-0.70711 -0.40825  0.28868 0.50000    example: unknown348-864, sides: 348, 384, 486, 864 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.70711  0.40825  0.57735 0.00000 0.00000 -0.81650 -0.28868 0.50000    example: unknown348-064, sides: 348, 304, 406, 864 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.70711  0.40825  0.57735 0.00000 0.00000 -0.93417  0.20547 0.29173    example: unknown340-545, sides: 340, 355, 454, 045 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902 -0.46709  0.35682 0.00000-0.30902  0.17841 -0.46709 0.80902    example: unknown340-646, sides: 340, 366, 464, 046 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902 -0.46709  0.35682 0.00000-0.50000  0.28868 -0.75576 0.30902    example: rhodoperihedral rotunda340-030, sides: 340, 300, 403, 030 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902 -0.46709  0.35682 0.00000-0.80902 -0.46709  0.17841 0.30902    example: unknown340-045, sides: 340, 305, 404, 045 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902 -0.46709  0.35682 0.00000-0.30902 -0.75576  0.28868 0.50000    example: unknown340-064, sides: 340, 304, 406, 064 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902 -0.46709  0.35682 0.00000 0.00000 -0.93417 -0.17841 0.30902    example: unknown355-535, sides: 355, 355, 553, 535 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902 -0.17841  0.93417 0.00000-0.30902 -0.17841  0.39894 0.84470    example: unknown355-530, sides: 355, 350, 553, 530 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902 -0.17841  0.93417 0.00000-0.80902  0.11026  0.28868 0.50000    example: unknown355-555, sides: 355, 355, 555, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902 -0.17841  0.93417 0.00000-0.30902 -0.17841 -0.46709 0.80902    example: unknown355-666, sides: 355, 366, 566, 566 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902 -0.17841  0.93417 0.00000-0.50000 -0.28868 -0.75576 0.30902    example: various diminishings of truncated 600-cell355-030, sides: 355, 300, 503, 530 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.30902 -0.17841  0.93417 0.00000-0.80902 -0.46709  0.17841 0.30902    example: unknown350-030, sides: 350, 300, 503, 030 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902  0.11026  0.57735 0.00000-0.80902 -0.46709 -0.17841 0.30902    example: unknown350-045, sides: 350, 305, 504, 045 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.80902  0.11026  0.57735 0.00000-0.30902 -0.75576 -0.28868 0.50000    example: unknown366-636, sides: 366, 366, 663, 636 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000 -0.28868  0.81650 0.00000-0.50000 -0.28868  0.20412 0.79057    example: decachoron366-646, sides: 366, 366, 664, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000 -0.28868  0.81650 0.00000-0.50000 -0.28868 -0.40825 0.70711    example: bitruncated tesseract366-656, sides: 366, 366, 665, 656 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.50000 -0.28868  0.81650 0.00000-0.50000 -0.28868 -0.78672 0.21851    example: bitruncated 120-cell388-838, sides: 388, 388, 883, 838 - valid 1.00000  0.00000  0.00000 0.00000 0.50000  0.86603  0.00000 0.00000-0.70711 -0.40825  0.57735 0.00000-0.70711 -0.40825 -0.28868 0.50000    example: 48-cell444-444, sides: 444, 444, 444, 444 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000 0.00000  0.00000  0.00000 1.00000    example: tesseract (K4.20)444-445, sides: 444, 445, 444, 445 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.30902  0.00000  0.00000 0.95106    example: square-pentagonal duoprism (K4.42)444-446, sides: 444, 446, 444, 446 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.00000  0.00000 0.86603    example: square-hexagonal duoprism (K4.54)444-448, sides: 444, 448, 444, 448 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.00000  0.00000 0.70711    example: square-octagonal duoprism (K4.70)444-440, sides: 444, 440, 444, 440 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902  0.00000  0.00000 0.58779    example: square-decagonal duoprism (K4.97)444-44n, sides: 444, 44n, 444, 44n example: square-n-gonal duoprism (K4.177)444-450, sides: 444, 440, 445, 450 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902  0.00000 -0.30902 0.50000    example: diminished rhombicosidodecahedral prism444-466, sides: 444, 446, 446, 466 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000  0.00000 -0.50000 0.70711    example: truncated octahedral prism (K4.89)444-468, sides: 444, 448, 446, 468 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.70711  0.00000 -0.50000 0.50000    example: truncated cuboctahedral prism (K4.125)444-460, sides: 444, 440, 446, 460 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.80902  0.00000 -0.50000 0.30902    example: truncated icosidodecahedral prism (K4.150)444-666, sides: 444, 466, 466, 466 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000 0.00000  0.00000  1.00000 0.00000-0.50000 -0.50000 -0.50000 0.50000    example: truncated 24-cell445-455, sides: 445, 445, 445, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.30902  0.00000 -0.42533 0.85065    example: dodecahedral prism (K4.74)445-466, sides: 445, 446, 446, 566 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.50000  0.00000 -0.68819 0.52573    example: truncated icosahedral prism (K4.127)445-544, sides: 445, 454, 454, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000 0.00000  0.30902  0.00000 0.95106    example: pentagonal duoprism445-540, sides: 445, 450, 454, 540 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.80902 -0.30902 -0.26287 0.42533    example: unknown445-644, sides: 445, 464, 464, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000 0.00000 -0.50000  0.00000 0.86603    example: pentagonal-hexagonal duoprism445-640, sides: 445, 460, 464, 540 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.80902 -0.50000 -0.26287 0.16246    example: diminished prismatotruncated 120-cell445-666, sides: 445, 466, 466, 566 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.50000 -0.50000 -0.68819 0.16246    example: great rhombated 600-cell445-844, sides: 445, 484, 484, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000 0.00000 -0.70711  0.00000 0.70711    example: pentagonal-octagonal duoprism445-044, sides: 445, 404, 404, 544 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000 0.00000 -0.80902  0.00000 0.58779    example: pentagonal-decagonal duoprism445-055, sides: 445, 405, 405, 555 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.30902  0.00000  0.95106 0.00000-0.30902 -0.80902 -0.42533 0.26287 example: unknown   445-n44, sides: 445, 4n4, 4n4, 544 - valid example: pentagonal-n-gonal duoprism446-644, sides: 446, 464, 464, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000 0.00000 -0.50000  0.00000 0.86603    example: hexagonal duoprism446-646, sides: 446, 466, 464, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000-0.50000 -0.50000 -0.28868 0.64550    example: great prismated decachoron446-648, sides: 446, 468, 464, 648 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000-0.70711 -0.50000 -0.40825 0.28868    example: great prismated 48-cell446-844, sides: 446, 484, 484, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000 0.00000 -0.70711  0.00000 0.70711    example: hexagonal-octagonal duoprism446-846, sides: 446, 486, 484, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000-0.50000 -0.70711 -0.28868 0.40825    example: great prismated tesseract446-044, sides: 446, 404, 404, 644 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000 0.00000 -0.80902  0.00000 0.58779    example: hexagonal-decagonal duoprism446-046, sides: 446, 406, 404, 646 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.50000  0.00000  0.86603 0.00000-0.50000 -0.80902 -0.28868 0.11026    example: great prismated 120-cell446-n44, sides: 446, 4n4, 4n4, 644 - valid example: hexagonal-n-gonal duoprism448-844, sides: 448, 484, 484, 844 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.70711  0.00000  0.70711 0.00000 0.00000 -0.70711  0.00000 0.70711    example: octagonal duoprism448-044, sides: 448, 404, 404, 844 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.70711  0.00000  0.70711 0.00000 0.00000 -0.80902  0.00000 0.58779    example: octagonal-decagonal duoprism448-n44, sides: 448, 4n4, 4n4, 844 - valid example: octagonal-n-gonal duoprism440-044, sides: 440, 404, 404, 044 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.80902  0.00000  0.58779 0.00000 0.00000 -0.80902  0.00000 0.58779    example: decagonal duoprism440-045, sides: 440, 405, 404, 045 - valid 1.00000  0.00000  0.00000 0.00000 0.00000  1.00000  0.00000 0.00000-0.80902  0.00000  0.58779 0.00000-0.30902 -0.80902 -0.42533 0.26287 example: unknown   440-n44, sides: 440, 4n4, 4n4, 044 example: decagonal-n-gonal duoprism44n-n44, sides: 44n, 4n4, 4n4, n44 - valid example: n-gonal duoprism44m-n44, sides: 44m, 4n4, 4n4, m44 - valid example: m-gonal-n-gonal duoprism555-555, sides: 555, 555, 555, 555 - valid 1.00000  0.00000  0.00000 0.00000-0.30902  0.95106  0.00000 0.00000-0.30902 -0.42533  0.85065 0.00000-0.30902 -0.42533 -0.68819 0.50000 example: 120-cell`

And the invalid ones are:
Code: Select all
`333-488, sides: 333, 348, 348, 388 - invalid; planar triangle with central pointThis corresponds to the fact that cube || truncated cube is flat.333-400, sides: 333, 340, 340, 300 - invalid; hyperbolicThis corresponds to the fact that dodecahedron || truncated dodecahedron is not a segmentochoron.334-433, sides: 334, 343, 343, 433 - invalid; planar quadrangleThis corresponds to the fact that vertex of octahedron can be sliced in two square pyramids in two different ways.334-466, sides: 334, 346, 346, 466 - invalid; planar triangle with central pointThis corresponds to the fact that octahedron || truncated octahedron is flat.334-468, sides: 334, 348, 346, 468 - invalid; hyperbolicThis corresponds to the fact that cuboctahedron || truncated cuboctahedron is not a segmentochoron.334-460, sides: 334, 340, 346, 460 - invalid; hyperbolicThis corresponds to the fact that icosidodecahedron || truncated icosidodecahedron is not a segmentochoron.334-533, sides: 334, 353, 353, 433 - invalid; hyperbolic334-844, sides: 334, 384, 384, 444 - invalid; planar quadrangleThis corresponds to the fact that a vertex of square orthobicupola can be sliced in two square cupolas while identically shaped vertex of elongated square pyramid can be sliced into square pyramid and cube.334-044, sides: 334, 304, 304, 444 - invalid; hyperbolic335-355, sides: 335, 335, 335, 555 - invalid; hyperbolicThis corresponds to the fact that dodecahedral pyramid isn't a segmentochoron.335-440, sides: 335, 340, 344, 540 - invalid; hyperbolic335-466, sides: 335, 346, 346, 566 - invalid; hyperbolic335-533, sides: 335, 353, 353, 533 - invalid; hyperbolic335-535, sides: 335, 355, 353, 535 - invalid; planar quadrangleThis corresponds to the fact that vertex of gyroelongated pentagonal prism can be sliced into pentagonal pyramid and metabidiminished icosahedron in two different ways.335-844, sides: 335, 384, 384, 544 - invalid; hyperbolic335-044, sides: 335, 304, 304, 544 - invalid; hyperbolic344-430, sides: 344, 340, 443, 430 - invalid; hyperbolic344-460, sides: 344, 340, 446, 460 - invalid; hyperbolicThis corresponds to the fact that truncated icosahedron || truncated icosidodecahedron isn't a segmentochoron.344-030, sides: 344, 300, 403, 430 - invalid; hyperbolicThis corresponds to the fact that icosidodecahedron || truncated dodecahedron isn't a segmentochoron.344-060, sides: 344, 300, 406, 460 - invalid; hyperbolic346-448, sides: 346, 348, 444, 648 - invalid; planar triangle with central pointThis corresponds to the fact that rhombicuboctahedron || truncated cuboctahedron is flat.346-440, sides: 346, 340, 444, 640 - invalid; hyperbolicThis corresponds to the fact that rhombicosidodecahedron || truncated icosidodecahedron isn't a segmentochoron.346-634, sides: 346, 364, 463, 634 - invalid; planar quadrangleThis corresponds to the fact that vertex of cuboctahedron can be sliced into two triangular cupolas in two different ways.346-834, sides: 346, 384, 483, 634 - invalid; hyperbolic346-034, sides: 346, 304, 403, 634 - invalid; hyperbolic348-834, sides: 348, 384, 483, 834 - invalid; hyperbolic348-844, sides: 348, 384, 484, 844 - invalid; planar quadrangleThis corresponds to the fact that vertex of rhombicuboctahedron can be sliced into square cupola and elongated square cupola in two different ways.348-034, sides: 348, 304, 403, 834 - invalid; hyperbolic348-044, sides: 348, 304, 404, 844 - invalid; hyperbolic340-034, sides: 340, 304, 403, 034 - invalid; hyperbolic340-035, sides: 340, 305, 403, 035 - invalid; hyperbolic340-044, sides: 340, 304, 404, 044 - invalid; hyperbolic340-054, sides: 340, 304, 405, 054 - invalid; planar quadrangleThis corresponds to the fact that vertex of rhombicosidodecahedron can be sliced intopentagonal cupola and diminished rhombicosidodecahedron in two different ways.355-040, sides: 355, 300, 504, 540 - invalid; EuclideanThis corresponds to the fact that truncated dodecahedron can be surrounded in Euclidean space with diminished rhombicosidodecahedra and tridiminished icosahedra.350-035, sides: 350, 305, 503, 035 - invalid; planar quadrangleThis corresponds to the fact that vertex of icosidodecahedron can be sliced into two pentagonal rotundas in two different ways.366-848, sides: 366, 388, 684, 648 - invalid; EuclideanThis corresponds to the verf of x3x3o *b4x.366-040, sides: 366, 300, 604, 640 - invalid; hyperbolicThis corresponds to the verf of x3x3o *b5x.300-030, sides: 300, 300, 003, 030 - invalid; hyperbolicThis corresponds to the verf of o3x5x3o.444-668, sides: 444, 468, 466, 468 - invalid; EuclideanThis corresponds to the verf of x3x3x *b4x.444-660, sides: 444, 460, 466, 460 - invalid; hyperbolicThis corresponds to the verf of x3x3x *b5x.445-866, sides: 445, 486, 486, 566 - invalid; hyperbolicThis corresponds to the verf of x4x3x5o.445-066, sides: 445, 406, 406, 566 - invalid; hyperbolicThis corresponds to the verf of x5x3x5o.446-640, sides: 446, 460, 464, 640 - invalid; hyperbolicThis corresponds to the verf of x3x5x3x.446-056, sides: 446, 406, 405, 656 - invalid; hyperbolic448-846, sides: 448, 486, 484, 846 - invalid; EuclideanThis corresponds to the verf of x4x3x4x.448-046, sides: 448, 406, 404, 846 - invalid; hyperbolicThis corresponds to the verf of x5x3x4x.440-046, sides: 440, 406, 404, 046 - invalid; hyperbolicThis corresponds to the verf of x5x3x5x.450-045, sides: 450, 405, 504, 045 - invalid; hyperbolic450-046, sides: 450, 406, 504, 046 - invalid; hyperbolic466-646, sides: 466, 466, 664, 646 - invalid; EuclideanThis corresponds to the verf of o4x3x4o.466-648, sides: 466, 468, 664, 648 - invalid; hyperbolicThis corresponds to the verf of x4x3x3x3*a.466-640, sides: 466, 460, 664, 640 - invalid; hyperbolicThis corresponds to the verf of x5x3x3x3*a.466-656, sides: 466, 466, 665, 656 - invalid; hyperbolicThis corresponds to the verf of o5x3x4o.468-846, sides: 468, 486, 684, 846 - invalid; hyperbolicThis corresponds to the verf of x4x3x4x3*a.468-046, sides: 468, 406, 604, 846 - invalid; hyperbolicThis corresponds to the verf of x5x3x4x3*a.460-046, sides: 460, 406, 604, 046 - invalid; hyperbolicThis corresponds to the verf of x5x3x5x3*a.555-666, sides: 555, 566, 566, 566 - invalid; hyperbolicThis corresponds to the verf of x3x5o3o.566-656, sides: 566, 566, 665, 656 - invalid; hyperbolicThis corresponds to the verf of o5x3x5o.`
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

So, all in all there are 44 possible vertices with 4 edges where we don't know of any explicit CRF polychoron. Some of them are various "decagonal blends" where we join two polyhedra with decagonal faces together and fill the gaps.

What would be next step in analysis? I guess it would be 5-edge vertices. This is more complicated, though.

I solved tetrahedra by first running a combinatorics program to enumerate possibilities and then solving to find explicit points on unit 4-sphere with the required distances (required angular distances, actually, but this is equivalent).

There are two kinds of vertices of this type, triangular dipyramids and square pyramids. The problem with square pyramid is that the base doesn't have to be planar and I need precise definition of its shape for every skew polygon that fits some Johnson solid:

The 3,3,3,3 polygon has a planar version corresponding to octahedron, and four different skew versions corresponding to triangular dipyramid, pentagonal dipyramid, snub disphenoid and sphenomegacorona.
The 3,3,3,4 polygon has a planar version corresponding to square antiprism, and two skew versions, one corresponding to augmentations of triangular prism and the other corresponding to sphenocorona.
Other 3,3,3,n polygons only appear in planar version.
The 3,3,4,4 polygon has a planar version corresponding to triangular orthobicupola and eight different skew versions corresponding to elongated triangular pyramid, elongated square pyramid (and square orthobicupola), elongated pentagonal pyramid, pentagonal orthobicupola, sphenocorona, sphenomegacorona, hebesphenomegacorona and disphenocingulum.
The 3,4,3,4 polygon has a planar version corresponding to cuboctahedron and three different skew versions corresponding to gyrobifastigium, square gyrobicupola and pentagonal gyrobicupola.
The 3,3,4,5 polygon only appears in two skew versions, one corresponding to pentagonal gyrocupolarotunda, the other corresponding to augments of pentagonal prism.
The 3,4,3,5 polygon only appears in skew version corresponding to pentagonal orthocupolarotunda, and also bilunabirotunda and triangular hebesphenorotunda.
The 3,3,4,6 polygon only appears in two skew versions, one corresponding to augments of hexagonal prism, the other corresponding to triangular hebesphenorotunda.
The 3,4,3,6 polygon only appears in skew version corresponding to augmented truncated tetrahedron.
The 3,4,3,8 polygon only appears in skew version corresponding to augments of truncated cube.
The 3,4,3,10 polygon only appears in skew version corresponding to augments of truncated dodecahedron.
The 3,3,5,5 polygon has a planar version corresponding to pentagonal orthobirotunda and two skew versions, one corresponding to augments of dodecahedron, the other corresponding to augmented tridiminished icosahedron.
The 3,5,3,5 polygon only appears in planar version.
The 3,4,4,4 polygon has a planar version corresponding to rhombicuboctahedron and two skew versions, one corresponding to elongated triangular cupola, the other corresponding to elongated pentagonal cupola.
The 3,4,4,5 polygon has a planar version corresponding to gyrate rhombicosidodecahedra and a skew version corresponding to elongated pentagonal rotunda.
The 3,4,5,4 polygon only appears in planar version.

As for triangular dipyramid verfs, they could come out skew, impossible to fit on a hyperplane. This is not a problem -- after all, Johnson solids have a lot of skew verfs.
The only problem is that the verfs should be convex. I guess that since the apexes of the two pyramids are the only vertices not joined, I could have a check whether a line joining them intersects the plane formed by equatorial vertices inside the triangle. The possibility of verf with a dimple should be removed simply by selecting the solution appropriately.

Note that lots of triangular dipyramid would be simply two tetrahedral verfs glued together along one face, but not all -- since the equatorial triangle is not required to be a verf of Johnson solid.

I can run the combinatorics program for these two, but I would need some input how to make the proper equations for them.
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

I tried looking at the quadragonal pyramids and the results are interesting.
First of all, while tetrahedra always had a solution (although it could be complex), here I frequently encounter combinations that have no solutions at all. This is most likely caused by the fact that the quadrangles are so restricted.

So, I started with pyramids derived from unit square, the verf of octahedron. 12 combinations have solutions -- plus there's an Euclidean solution for 3333-8888 (octahedron and four truncated cubes, x4x3o4o) and a hyperbolic solution for 3333-0000 (octahedron and four truncated dodecahedra, x5x3o4o).

3333-3333: octahedron and four tetrahedra. This is a simple solution for octahedral pyramid.

3333-3344: octahedron, two square pyramids, tetrahedron and triangular prism.

The squares can be changed to pentagons, leading to 3333-3355: octahedron, two pentagonal pyramids, tetrahedron and metabidiminished icosahedron.

3333-3464: octahedron, two square pyramids and two triangular cupolas.

3333-3405: octahedron, pentagonal pyramid, square pyramid, pentagonal cupola and pentagonal rotunda.

3333-4444: octahedron and four triangular prisms. This is a solution for octahedral prism.

3333-4466: octahedron, two triangular cupolas, triangular prism and truncated tetrahedron.

3333-4488: octahedron, two square cupolas, triangular prism and truncated cube.

3333-4400: octahedron. two pentagonal cupolas, triangular prism and truncated dodecahedron.

3333-5555: octahedron and five metabidiminished icosahedra. This is a solution for octahedral ursachoron.

3333-5500: octahedron, two pentagonal rotundas, metabidiminished icosahedron and truncated dodecahedron.

3333-6666: octahedron and four truncated tetrahedra. This is a solution for truncated 16-cell.
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

Now let's try joining triangular dipyramid. In case of octahedron, I put squares of both diagonals equal to 2. For triangular dipyramid, square of one diagonal is 1. Since height of unit tetrahedron is sqrt(2/3), double of that is sqrt(8/3) so square of the other diagonal is 8/3.

Only two solutions exist:

3333-3464: triangular dipyramid, two square pyramids and two triangular cupolas.
This combination apparently works even when the quadrangle is deformed. Now, it has two distinct forms -- either the faces "above" equatorial edges of triangular dipyramid are squares or triangle and hexagon. Only the first of these could be verf of a CRF, as the second one is flat, corresponding to slicing of triangular orthobicupola into square pyramids and triangular dipyramids.

3333-4444: triangular dipyramid and four triangular prisms -- this is a solution for triangular dipyramidal prism.

So, a working hypothesis is that when lateral edges have 3464 or 4444 configuration, the quadrangle can be arbitrarily skewed and the shape will still fit. The fact that sums of angles of both pairs of opposite faces is 180 degrees is probably related.
Marek14
Pentonian

Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

### Re: General Approach--can 3D methods be generalized?

Let's have a look at the remaining possible bases for the (3,3,3,3) base of the quadragonal pyramid. We examined octahedron and triangular dipyramid, so we have three left: pentagonal dipyramid, snub disphenoid and sphenomegacorona.

Pentagonal dipyramid: The diagonals of pentagonal dipyramid verf are (1+sqrt(5))/2 and twice the height of pentagonal pyramid, 2*sqrt((5-sqrt(5))/10).

As for snub disphenoid and sphenomegacorona, I eventually decided to try a numerical solution with values measured in Stella. The vertex of snub disphenoid we're interested in has diagonals 1.513 and 1.28917 with squares 2.28917 and 1.66196. Sphenomegacorona vertex has with diagonals 1.18927 and 1.56358 with squares 1.41436 and 2.44478.

Unfortunately, now I can't find any solutions for these despite the fact that I know that 3333-4444 must always have a solution (it's a prism) and 3333-3464 should have as well. Until someone helps me here, I'll assume that the non-quadratic polyhedra only have solutions that all others have...

So, what do we get? Well, pentagonal dipyramid (and presumably snub disphenoid and sphenomegacorona) have the same three solutions as triangular dipyramid: A solution for 3333-4444 (prism) and two solutions for 3333-3464 (surrounding of degree-4 vertex with two square pyramids and two triangular cupolas with two variants based on which pair of edges has squares built on them and which has triangle and hexagon).

But there's one anomaly as well, and that is a flat Euclidean solution for 3333-5000 on pentagonal dipyramid. Turns out that if you join two truncated dodecahedra by decagons in ortho configuration, and then put pentagonal rotundas on the decagons closest to the join, you can exactly fit pentagonal dipyramids in the gaps thus created. This is an analogue to the "flat" 3333-3464 solution for triangular dipyramid; both solutions of 3333-3464 for pentagonal dipyramid could be theoretically present in CRF polychora.
Marek14
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### Re: General Approach--can 3D methods be generalized?

Now let's investigate the (3,3,3,4) quadrangle.
This looks like square antiprism, and that is indeed one of the solutions, but not the only one. It can also be augmented triangular prism or sphenocorona. Let's just forget about the sphenocorona for now...
For square antiprism, the squares of both diagonals will be 1 + sqrt(2).

For augmented triangular prism, we'll have to compute them.

Vertices of triangular prism:
[0,0,0]
[1,0,0]
[1/2,sqrt(3)/2,0]
[0,0,1]
[1,0,1]
[1/2,sqrt(3)/2,1]

If we augment the [0,0,0]-[1,0,0]-[1,0,1]-[0,0,1] face with square pyramid, its vertex will be [1/2,-sqrt(2)/2,1/2]. The squares of diagonals are
[1/2,-sqrt(2)/2,1/2] - [1/2,sqrt(3)/2,0] = (sqrt(3)/2 + sqrt(2)/2)^2 + 1/4 = (3+sqrt(6))/2
and [0,0,0] - [1,0,1] = 2.

The solutions are:
3334-3333: antiprismatic solution with square antiprism, three tetrahedra and square pyramid. Corresponds to square antiprismatic pyramid.

3334-3344: antiprismatic solution with square antiprism, two square pyramids, tetrahedron and cube. Seems to correspond to square || gyrated cube (K4.14)

3334-3484: antiprismatic solution with square antiprism, two square pyramids, scquare cupola and octagonal prism.

3334-4433: antiprismatic solution with square antiprism, three square pyramids and triangular prism.

3334-4444: antiprismatic solution with square antiprism, three triangular prisms and cube. Corresponds to square antiprismatic prism.
3334-4444: augmented prismatic solution with augmented triangular prism, three triangular prisms and cube. Corresponds to augmented triangular prismatic prism.
There's a sphenocorona solution here as well, of course, corresponding to sphenocorona prism.

3334-4466: antiprismatic solution with square antiprism, two triangular cupolas, triangular prism and truncated octahedron. This can be envisioned as joining square antiprism and truncated octahedron in a square, filling the triangle/hexagon gaps with triangular cupolas and inserting triangular prisms to remaining triangles of the antiprism.

3334-5533: antiprismatic solution with square antiprism, two pentagonal pyramids, metabidiminished icosahedron and square pyramid.

3334-6644: antiprismatic solution with square antiprism, two triangular cupolas, truncated tetrahedron and cube. This can be envisioned as joining square antiprism and cube in a square, filling the triangle/square gaps with triangular cupolas and inserting truncated tetrahedra to remaining triangles of the antiprism.

3334-6666: antiprismatic solution with square antiprism, three truncated tetrahedra and truncated octahedron. A square antiprism can be completely surrounded by truncated tetrahedra and truncated octahedra like this. How far can this thing be built?

3334-8844: antiprismatic solution with square antiprism, two square cupolas, truncated cube and cube. This can be envisioned as joining square antiprism and cube in a square, filling the triangle/square gaps with square cupolas and inserting truncated cubes to remaining triangles of the antiprism.

3334-0044: antiprismatic solution with square antiprism, two pentagonal cupolas, truncated dodecahedron and cube. This can be envisioned as joining square antiprism and cube in a square, filling the triangle/square gaps with pentagonal cupolas and inserting truncated dodecahedra to remaining triangles of the antiprism.

So, we can see that there are 11 solutions for square antiprism, just one for augmented triangular prism (prismatic) and my best guess is that there will be only the prismatic solution for sphenocorona.
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### Re: General Approach--can 3D methods be generalized?

Let's try (3,3,3,5) base. This can be only pentagonal antiprism or one of the Johnson solids with this vertex configuration (like thawro). This is a simple vertex since it has no skew forms and both diagonals are (1+sqrt(5))/2, so their squares are (3+sqrt(5))/2.

The solutions are:

3335-3333: pentagonal antiprism, three tetrahedra and pentagonal pyramid. Corresponds to pentagonal antiprismatic pyramid, probably occurs in thawro pseudopyramid.

3335-3344: pentagonal antiprism, two square pyramids, tetrahedron and pentagonal prism. Corresponds to pentagon || gyrated pentagonal prism (K4.22).

3335-3355: pentagonal antiprism, two pentagonal pyramids, tetrahedron and dodecahedron. This can be envisioned as joining pentagonal antiprism and dodecahedron, filling the triangle/pentagon gaps with pentagonal pyramids and adding tetrahedra to remaining triangles in the pentagonal antiprism.

3335-3404: pentagonal antiprism, two square pyramids, pentagonal cupola and diminished rhombicosidodecahedron.

3335-3503: pentagonal antiprism, tetrahedron, pentagonal pyramid and two pentagonal rotundas.

3335-4433: pentagonal antiprism, two square pyramids, triangular prism and pentagonal pyramid. This can be envisioned as joining pentagonal antiprism and pentagonal pyramid in a pentagon, filling the triangle/triangle gaps with square pyramids and adding triangular prisms to remaining triangles in the pentagonal antiprism.

3335-4444: pentagonal antiprism, three triangular prisms and pentagonal prism. Corresponds to pentagonal antiprismatic prism.

3335-4466: pentagonal antiprism, two triangular cupolas, triangular prism and truncated icosahedron. This can be envisioned as joining pentagonal antiprism and truncated icosahedron, filling the triangle/hexagon gaps with triangular cupolas and adding triangular prisms to remaining triangles in the pentagonal antiprism.

3335-4003: pentagonal antiprism, square pyramid, pentagonal cupola, truncated dodecahedron and pentagonal rotunda.

3335-5533: This is a flat configuration where everything folds into a pentagon.

3335-5555: pentagonal antiprism, three metabidiminished icosahedra and dodecahedron. This can be envisioned as attaching dodecahedra on pentagonal faces of pentagonal antiprism and tridiminished icosahedra on the triangular faces.

3335-5053: pentagonal antiprism, pentagonal pyramid, two pentagonal rotundas and metabidiminished icosahedron.

3335-6644: pentagonal antiprism, two triangular cupolas, truncated tetrahedron and pentagonal prism. This can be envisioned as joining pentagonal antiprism and pentagonal prism, filling the triangle/square gaps with triangular cupolas and adding truncated tetrahedra to remaining triangles in the pentagonal antiprism.

3335-6666: pentagonal antiprism, three truncated tetrahedra and truncated icosahedron. This can be envisioned as attaching truncated icosahedra on pentagonal faces of pentagonal antiprism and truncated tetrahedra on the triangular faces.

3335-8844: pentagonal antiprism, two square cupolas, truncated cube and pentagonal prism. This can be envisioned as joining pentagonal antiprism and pentagonal prism, filling the triangle/square gaps with square cupolas and adding truncated cubes to remaining triangles in the pentagonal antiprism.

3335-0044: pentagonal antiprism, two pentagonal cupolas, truncated dodecahedron and pentagonal prism. This can be envisioned as joining pentagonal antiprism and pentagonal prism, filling the triangle/square gaps with pentagonal cupolas and adding truncated dodecahedra to remaining triangles in the pentagonal antiprism.

3335-0055: This is a flat configuration. It corresponds to the anomalous Euclidean edge I've found before. A pentagonal antiprism is joined to dodecahedron and space around one vertex is filled with two pentagonal rotundas and one truncated dodecahedron.

All in all, 15 possible vertex configurations.
Last edited by Marek14 on Mon Nov 10, 2014 1:26 pm, edited 1 time in total.
Marek14
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### Re: General Approach--can 3D methods be generalized?

OK, so what about the remaining antiprisms?
My results indicate that there are 9 combinatoric configurations that have to be tackled for any antiprism, regardless of size. In order to find them, I have to find general formula for the diagonal of antiprism verf.

There are two methods I use in these computations for constraining lengths of edges/distances of vertices. One of them is Euclidean metric where (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 + (w1-w2)^2 equals to computed square of a diagonal. The other one is a dot product x1*x2 + y1*y2 + z1*z2 + w1*w2. If this product is equal to cos((n-2)*pi/n), then the vertices are at a distance of n-gon shortchord. This uses the fact that all vertices are restricted to the unit 3-sphere and so their distance from origin is 1.

So, let's solve a simple 3D case that will find coordinates of an antiprism verf:

Vertices will be A:[x1,y1,z], B:[x2,y2,z], C:[-x2,y2,z] and D:[-x1,y1,z]. The identical z-coordinate will ensure that the figure will be planar while the mirrored coordinates ensure the reflection symmetry.
First of all, all of these are on the unit sphere:
x1^2 + y1^2 + z^2 == 1
x2^2 + y2^2 + z^2 == 1
AB, CD and AD are at distance, corresponding to dot product 1/2.
AB (and also CD): x1*x2 + y1*y2 + z^2 == 1/2
AD: -x1^2 + y1^2 + z^2 == 1/2

BC, then, will have dot product equal to the shortchord s:
-x2^2 + y2^2 + z^2 == s

And finally, we input diagonal equation we're interested in, also in the form of dot product:
-x1*x2 + y1*y2 + z^2 == ddp

And, just to be sure, we'll try this again with d in form of diagonal square:
(x1 + x2)^2 + (y1 - y2)^2 == dds

Now, the result is very complicated, but it can be fortunately simplified into a very simple expression:
ddp = 1/2 +/- sqrt(1 - s)/sqrt(2)
dds = 1 +/- sqrt(2 - 2s)

Let's test this for the antiprisms we already know. For triangular antiprism, a.k.a. octahedron, we should get ddp equal to 0 and dds equal to 2. s is 1/2 here:
ddp = 1/2 +/- sqrt(1 - 1/2)/sqrt(2) = 1/2 +/- 1/2 = 1 / 0
dds = 1 +/- sqrt(2 - 1) = 1 +/- sqrt(1) = 1 +/- 1 = 2 / 0
For square antiprism, s is 0 and dds should be equal to 1 + sqrt(2):
dds = 1 +/- sqrt(2 - 0) = 1 +/- sqrt(2) = 1 + sqrt(2) / 1 - sqrt(2)

So it looks like we should take - in ddp and + in dds. Good to know. Let's try for pentagonal antiprism with s = cos(3*pi/5) = (1 - sqrt(5))/4. dds should be (3 + sqrt(5))/2, and it also comes right, though the process is a bit more complicated.
dds = 1 + sqrt(2 - (1 - sqrt(5))/2)
dds = 1 + sqrt((4 + sqrt(5) - 1)/2)
dds = 1 + sqrt((3 + sqrt(5))/2)

But we know that (3 + sqrt(5))/2 = ((1 + sqrt(5))/2)^2, so

dds = 1 + (1 + sqrt(5))/2
dds = (3 + sqrt(5))/2

So I'll now examine the rest of the antiprisms using this dds value.
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### Re: General Approach--can 3D methods be generalized?

General antiprisms (the ones with no general solutions could still have specific solutions, but it doesn't seem likely):

333n-3344: generally valid. This corresponds to n-gon || gyrated n-gonal prism and the vertex is formed by n-gonal antiprism, two square pyramids, tetrahedron and n-gonal prism.
333n-3444: no general solution
333n-4444: generally valid. This corresponds to antiprismatic prism and the vertex is formed by n-gonal antiprism, three triangular prisms and n-gonal prism.
333n-4644: no general solution
333n-4844: no general solution
333n-4044: no general solution

The last three are generally valid and they all correspond to n-gonal antiprism joined to n-gonal prism with cupolas and truncated polyhedra filling the gaps.
333n-6644: n-gonal antiprism, two triangular cupolas, truncated tetrahedron, n-gonal prism.
333n-8844: n-gonal antiprism, two square cupolas, truncated cube, n-gonal prism.
333n-0044: n-gonal antiprism, two pentagonal cupolas, truncated dodecahedron, n-gonal prism.

After the general cases, what is left? Some specific verfs using hexagonal, octagonal or decagonal antiprisms. And here's the thing -- while they do have some additional combinatoric options, none of them work out computationally. So these three, too, only have the five general solutions -- square antiprism and pentagonal antiprism are the only ones with more than five options for this verf geometry.
Marek14
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### Re: General Approach--can 3D methods be generalized?

One of the limits generalising polytopes is that you can create ever-larger polygons, but this does not apply higher up. For example, you can convert a decagon to a 20-gon, but the similar process does not exist for polyhedra.
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### Re: General Approach--can 3D methods be generalized?

wendy wrote:One of the limits generalising polytopes is that you can create ever-larger polygons, but this does not apply higher up. For example, you can convert a decagon to a 20-gon, but the similar process does not exist for polyhedra.

What I'm doing at this point actually reminds me my research into uniform tilings years ago. So far, the classification of vertices showed a few surprises, but a theoretical existence of a vertex doesn't, sadly, mean that there must be a CRF polychoron that actually has that vertex. But I think that the knowledge of such possibility might lead to searching in new directions.

Take those general vertices involving antiprisms, for example. They show a way to surround an antiprism/prism join with cupolas and other polyhedra, and it's possible that they could be filled in by other cells to eventually obtain something CRF.
Marek14
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### Re: General Approach--can 3D methods be generalized?

OK, today I'll try to tackle quadragonal vertices with the (3,3,4,4) base.

The (3,3,4,4) base is a nasty one -- it can appear in a whooping 9 configurations. I'll only research five, since the remaining 4 are nasty crown jewels -- sphenocorona, sphenomegacorona, hebesphenomegacorona and disphenocingulum.

The five I will look at today all belong to either elongated pyramids or to orthobicupolas -- or, in one interesting case, to both:
1. The planar version. This belongs to triangular orthobicupola. The main diagonal (between (3,3) and (4,4) vertex) has length sqrt(3), the shortchord of equatorial hexagon of triangular orthobicupola. The side diagonal (between both (3,4) vertices) can be computed:
A = [0,0]
B = [x,y]
C = [sqrt(3),0]
D = [x,-y]

x^2 + y^2 == 1
(x - sqrt(3))^2 + y^2 == 2

We get y = sqrt(2/3), so 2*y = 2*sqrt(2/3) and that's our side diagonal. The diagonal squares are therefore 3 and 8/3.

2. The triangular version. This belongs to elongated triangular pyramid/dipyramid.
The side diagonal is clearly 1 here -- it's the edge of the figure. What's the main diagonal? Well, a unit tetrahedron can have vertices [0,0,0],[1,0,0],[1/2,sqrt[3]/2,0] and [1/2, 1/(2 sqrt(3)), sqrt(2/3)], so we just have to find the square of distance between this fourth vertex and [0,0,-1]:

(1/2)^2 + (1/(2*Sqrt[3]))^2 + (Sqrt[2/3] + 1)^2 = 2/3 * (3 + Sqrt[6]).
So the diagonal squares are 2/3 * (3 + Sqrt[6]) and 1.

3. The square version. This is an interesting one since it appears not only in the elongated square pyramid/dipyramid, but also in the square orthobicupola. This duality allows us to easily compute both diagonal squares: The main diagonal square is 2 + Sqrt[2] (the diagonal is shortchord of octagon) while the side diagonal square is 2 (this diagonal is shortchord of square).

4. The pentagonal version. This appears in elongated pentagonal pyramid/dipyramid. The side diagonal square is the square of the shortchord of pentagon (3 + Sqrt[5])/2, but we have to compute the main diagonal.
From Pythagorean theorem, this seems to be the square of height of elongated pentagonal pyramid (which is 1 + height of pentagonal pyramid, Sqrt[(5 - Sqrt[5])/10]) and the distance from center of pentagonal face to a vertex, which is the circumscribed radius of the unit pentagon, Sqrt[(5 + Sqrt[5])/10]

So the diagonal square is:
(1 + Sqrt[(5 - Sqrt[5])/10])^2 + (5 + Sqrt[5])/10 = 2 + Sqrt[2 - 2/Sqrt[5]]

5. The pentagonal orthobicupola version. This appears in pentagonal orthobicupola. This is an easy one. The main diagonal is the shortchord of the decagon, so its square is (5 + Sqrt[5])/2. The side diagonal is double the height of pentagonal cupola, which is 2 * Sqrt[(5 - Sqrt[5])/10] (the same as double the height of pentagonal pyramid), which gives its square as 2 - 2/Sqrt[5]

So, the short version is:
1. Planar version: 3 and 8/3.
2. Triangular version: 2/3 * (3 + Sqrt[6]) and 1.
3. Square version: 2 + Sqrt[2] and 2.
4. Pentagonal version: 2 + Sqrt[2 - 2/Sqrt[5]] and (3 + Sqrt[5])/2.
5. Pentagonal orthobicupola version: (5 + Sqrt[5])/2 and 2 - 2/Sqrt[5].

Now that we have our five conditions, we can start looking checking the combinatorical possibilities:

3344-3333: There is a planar solution, but it's flat. It corresponds to cutting triangular orthobicupola into tetrahedra and octahedra.

3344-3343: Square solution. There are tetrahedra on triangular faces and triangular prisms on square faces. It corresponds to line || elongated square dipyramid which seems to be elongated square pyramidal dipyramid. Also corresponds to octagon || cube.

3344-3484: Square solution. Thre are square pyramids on triangular faces and octagonal prisms on square faces. Elongated square dipyramid can be completely surrounded like this, but square orthobicupola can't. It corresponds to omniaugmented (4,8)-duoprism -- a cube pyramid glued to a cubic cell of (4,8)-duoprism. Also, if a 334-848 tetrahedral vertex corresponds to a polychoron, two such polychora could be glued together through square cupolas, blending other square cupolas into square orthobicupolas.

3344-34(20)4: Pentagonal solution. There are square pyramids on triangular faces and 20-gonal prisms (yes, really) on square faces. This corresponds to the omniaugmented (5,20)-duoprism we already know about. You can see how it arises by gluing pentagonal prism pyramid to (5,20)-duoprism.

Basically, 3344-34n4 has a combinatoric solution for any n. I checked n where n-gon has an Euclidean construction up to 40, but only found these two.

3344-4346: Similarly to 3333-3464, the solution here exists for all five possibilities (and presumably for the four crown jewels as well). You add a triangular cupola and a square pyramid on the triangular faces, a triangular prism and a hexagonal prism on square faces, and it somehow always fits together.
The most interesting is the triangular solution, which comes out flat. It corresponds to cutting of elongated triangular cupola into elongated triangular pyramids, square pyramids and triangular prisms.
The planar solution can be understood as triangle || hexagonal prism glued to a hypothetical polychoron with 346-644 tetrahedral vertex.
The square solution can be understood as elongated trigonal gyrobicupolic ring where cubes belonging to triangular cupolaic prism become augmented -- or it can be understood as square || octagonal prism glued to a hypothetical polychoron with 346-844 tetrahedral vertex.
The pentagonal orthobicupola solution can be understood as pentagon || decagonal prism glued to a hypothetical polychoron with 346-044 tetrahedral vertex.

3344-4444: The prismatic solution which exists for all possibilities and for the crown jewels as well. Triangular prisms on triangles, cubes on squares.

3344-4468: Square solution. There are square cupolas and triangular prisms on triangles and hexagonal prisms and truncated cuboctahedra on squares. The version with elongated square pyramid/dipyramid appears to be augmentation of truncated cuboctahedral prism with square || octagonal prism. I *think* we didn't know this before. Or, we get this vertex when we augment a truncated cuboctahedron in hypothetical polychoron with 348-864 tetrahedral vertex with truncated octahedron || truncated cuboctahedron.

3344-44n4 has a general combinatorical solution, but I wasn't able to find one that fits unless n = 4.

3344-4686: There is a square solution, but it's flat. It's x3x3x *b4x where a truncated octahedron is cut into an octahedron, triangular cupolas and square pyramids (which join with cubes to form elongated square pyramids). Or, you start with truncated square tiling of plane, and then build triangular cupolas on squares, truncated cuboctahedra on some octagons and square cupolas on the other octagons (which then blend into square orthobicupolas).

3344-5545: Square solution. The weird part about this solution is that the verf cannot be cut: you'd always get an impossible vertex (455 or 458 which don't correspond to any valid polyhedra). This solution has metabidiminished icosahedra built on triangles and pentagonal prisms built on squares.

3344-5050: There is a pentagonal orthobicupola solution, but it's flat. It can be imagined like this: start with a planar configuration of decagon and two pentagons. Then replace the decagon with pentagonal orthobicupola and attach two pentagonal rotundas to one pentagon and two diminished rhombicosidodecahedra to the other.

3344-64n4 has a general combinatorical solution, but I wasn't able to find any specific ones.

3344-6646: Square solution. It can be imagined as elongated octahedral rotunda where cubes from truncated octahedral prism blend with square pyramids. It also corresponds to stauroperihedral birotunda where equatorial square cupolas blend to square orthobicupolas.

3344-6666: There is a planar solution, but it's flat. You can imagine it like a triangular orthobicupola surrounded in its equatorial plane by truncated tetrahedra and truncated octahedra. A truncation of gytoh tesselation (O22).

3344-8438: Square solution. It corresponds to a hypothetic polychoron wih 334-848 tetrahedral vertex with octagonal prism augmented by square cupolaic prism (or cube || octagonal prism).The side cubes blend with square pyramids. Another possibility is a hypothetic polychoron with 348-838 tetrahedral vertex with truncated cube augmented by cuboctahedron || truncated cube, which would cause square cupolas to blend into square orthobicupolas.

3344-84n4 has a general combinatorical solution, but I wasn't able to find any specific ones.

3344-8848: There is a square solution, but it's flat. One example is elongation of truncated cubic honeycomb where octahedra become elongated square dipyramids and a layer of octagonal prisms is put between layers of truncated cubes. Another example is taking a prismatotruncated cubic honeycomb, cutting it to reveal truncated square tiling (this cuts rhombicuboctahedra into square cupolas and elongated square cupolas), and then reflecting the half that contains square cupolas which become square orthobicupolas.

3344-04n4 has a general combinatorical solution, but I wasn't able to find any specific ones.

3344-0540: There is a pentagonal solution, but it's flat. Take a decagonal prism and two pentagonal prisms fit together, then augment one pentagonal prism to elongated pentagonal pyramid, add pentagonal rotunda to the other pentagonal prism and add truncated dodecahedron to the decagonal prism. It's elongated version of the 3333-5000 configuration with pentagonal bipyramids.

3344-0040: There is a square solution, but it's hyperbolic. One possibility is to take the truncated (5,3,4) tiling, cut it in a plane, elongate the resulting square pyramids and inserting decagonal prisms. Another possibility is to tile the hyperbolic plane with squares, octagons and decagons, build truncated dodecahedra on decagons, decagonal prisms on squares and square cupolas on octagons, then reflecting it.
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### Re: General Approach--can 3D methods be generalized?

So we see that the (3,3,4,4) base has mostly square solutions, which is actually not that surprising -- since both diagonals are shortchords, it's, in a way, the "most natural" arrangement for this vertex.

But the trick of cutting of verf I tried to use now allows me to improve on the previous results for bases 3333 and 3335:

Octahedron:
3333-3333: octahedral pyramid.
3333-3344: triangular prismatic dipyramid where square pyramids blend into octahedra.
3333-3355: verf can't be cut because it would result in (3,4,5) vertices which don't occur.
3333-3464: two trigonal gyrobicupolic rings joined together in triangular cupola, square pyramids blend into octahedra. Or hypothetical 334-644 polychoron with a cube augmented by a pyramid, blending square pyramids into octahedra.
3333-3405: hypothetical 334-045 polychoron with a pentagonal prism augmented by a pyramid, blending square pyramids into octahedra.
3333-4444: octahedral prism.
3333-4466: two triangle || hexagonal prism joined together, blending square pyramids into octahedra.
3333-4488: two square || octagonal prism joined together, blending square pyramids into octahedra.
3333-4400: two pentagon || decagonal prism joined together, blending square pyramids into octahedra.
3333-5555: octahedral ursachoron.
3333-5500: two copies of a hypothetical 334-050 polychoron joined in a diminished rhombicosidodecahedron, blending square pyramids into octahedra.
3333-6666: truncated 16-cell.

Triangular dipyramid:
3333-3464a: hexagon || triangular prism augmented with triangular prismatic pyramid, blending tetrahedra into triangular dipyramids.
3333-4444: triangular dipyramidal prism.

Pentagonal dipyramid:
3333-3464a: vertex cannot be cut because there's no 356-acrohedron.
3333-3464b: hypothetical 335-644 polychoron with pentagonal prism augmented by pentagonal prismatic pyramid, blending pentagonal pyramids into pentagonal dipyramids.
3333-4444: pentagonal dipyramidal prism.

Pentagonal antiprism (or gyroelongated pentagonal pyramid):
3335-3333: pentagonal antiprismatic pyramid.
3335-3344: vertex cannot be cut because there's no 345-acrohedron.
3335-3355: hypothetical 335-555 polychoron augmented with metabidiminished icosahedral pyramid, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-3404: hypothetical 334-550 polychoron and hypothetical 334-530 polychoron glued together in pentagonal rotunda, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids. Or hypothetical 340-545 polychoron augmented with pentagonal prismatic pyramid, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-3503: hypothetical 335-530 polychoron and hypothetical 333-550 polychoron joined in pentagonal rotunda, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids. Or hypothetical 355-530 polychoron augmented with metabidiminished icosahedral pyramid, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-4433: vertex cannot be cut because there's no 345-acrohedron.
3335-4444: pentagonal antiprismatic prism.
3335-4466: vertex cannot be cut because there's no 456-acrohedron.
3335-4003: hypothetical 334-550 polychoron and hypothetical 335-040 polychoron joined in diminished rhombicosidodecahedron, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids. Or hypothetical 355-030 polychoron and hypothetical 334-530 polychoron joined in pentagonal rotunda, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-5555: hypothetical 355-555 polychoron and hypothetical 335-555 polychoron joined in dodecahedron, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-5053: hypothetical 335-555 polychoron and hypothetical 335-055 polychoron joined in dodecahedron, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids. Or hypothetical 355-530 polychoron and hypothetical 335-530 polychoron joined in pentagonal rotunda, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids.
3335-6644: vertex cannot be cut because there's no 456-acrohedron.
3335-6666: a diminishing of truncated 600-cell joined to icosahedral rotunda, blending pentagonal pyramids and metabidiminished icosahedra into gyroelongated pentagonal pyramids. Basically also a diminishing of truncated 600-cell.
3335-8844: vertex cannot be cut because there's no 458-acrohedron.
3335-0044: vertex cannot be cut because there's no 450-acrohedron.
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### Re: General Approach--can 3D methods be generalized?

OK, so the next quadragonal base to research is... (3,3,4,5). This base doesn't have a planar form, only two skew ones: one belongs to pentagonal gyrocupolarotunda, the other belongs to augmented/biaugmented pentagonal prism.

For pentagonal gyrocupolarotunda, one diagonal will be shortchord of decagon with square (5 + Sqrt[5])/2. The other diagonal is more tricky to compute, but its square comes out to be 3/10 * (5 + sqrt(5)).

For augmented pentagonal prism, one diagonal square will be shortchord of square -- 2. The other diagonal square comes out as a fairly nasty number of (7 + Sqrt[5] + 2*Sqrt[5 + Sqrt[5]])/4.

Solutions found:

3345-4340: Pentagonal gyrocupolarotunda solution. This would be a hypothetical 340-045 polychoron with a decagonal prism augmented by pentagon || decagonal prism; this would blend pentagonal rotunda and pentagonal cupola into pentagonal gyrocupolarotunda.

3345-4444: The prismatic solution, valid for both kinds.

3345-5055: A flat pentagonal gyrocupolarotunda solution. Can be imagined by making a pentagon-pentagon-decagon vertex in plane, then replacing the decagon with pentagonal gyrocupolarotunda. One pentagon gets pentagonal rotunda on the cupola side and metabidiminished icosahedron on the rotunda side, the other pentagon gets diminished rhombicosidodecahedron on the cupola side and dodecahedron on the rotunda side.

And that's all. We see that these shapes would be pretty rare as cells. But the pentagonal gyrocupolarotunda could occur as a cell of an augmented polychoron if 340-045 vertex exists.
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### Re: General Approach--can 3D methods be generalized?

Next is (3,3,4,6) base. There are two possibilities.

First possibility is augmented hexagonal prism (and other related Johnson solids: parabiaugmented, metabiaugmented and triaugmented hexagonal prism). Here, one diagonal square will be 2, like with augmented pentagonal prism; the other diagonal square is (5 + Sqrt[6])/2.

The second, and more interesting, possibility is triangular hebesphenorotunda. Here, one diagonal is a shortchord of decagon (because the triangle-square-triangle patches of thawro are identical to those in pentagonal cupola), and its square is (5 + Sqrt[5])/2. The other diagonal is a shortchord of pentagon (because the patches of three triangles and pentagon have geometry of pentagonal antiprism, and therefore of icosahedron) and its square is (3 + Sqrt[5])/2.
The weird thing with thawro is that it can't be actually split along its diagonals, despite them being valid shortchords. This means that thawro vertices tend to be related to others, but they are still their own specific things.

Solutions are:

3346-3343: Thawro solution. Thawro, two tetrahedra, triangular prism and triangular cupola. Parts of the polychoron will be similar to decagon || pentagonal prism (fitting of one tetrahedron and triangular prism) and to pentagonal duopyramid (fitting of the two tetrahedra).

3346-3365: Thawro solution. Thawro, pentagonal pyramid, tetrahedron, triangular cupola and truncated icosahedron. Fitting of the pentagonal pyramid and tetrahedron will be similar to metabidiminished icosahedral pyramid. Appears in D4.7.

3346-3463: Thawro solution. Thawro, tetrahedron, square pyramid, hexagonal prism and truncated tetrahedron. Not similar to anything else.

3346-3404: Thawro solution. Thawro, two square pyramids, decagonal prism and truncated icosidodecahedron. The two square pyramids fit like in pentagonal prismatic pyramid. 334-040 polychoron would have square pyramid and decagonal prism fit the same way as this. Appears in D4.5.3 and D4.5.4.

3346-4334: Thawro solution. Thawro, triangular prism, two square pyramids and triangular cupola. The two square pyramids fit like in pentagonal gyrobicupolic ring.

3346-4340: Thawro solution. Thawro, pentagonal cupola, square pyramid, triangular prism and truncated icosidodecahedron. 334-530 polychoron would have the same configuration of pentagonal cupola and square pyramid. Pentagon || decagonal prism has the same configuration of square pyramid and triangular prism. Appears in D4.5.1.

3346-4444: Prismatic solution valid for both kinds.

3346-4456: Thawro solution. Thawro, triangular cupola, triangular prism, pentagonal prism and truncated icosahedron. Not similar to anything else.

3346-4664: Thawro solution. Thawro, triangular prism, triangular cupola and two truncated octahedra. Not similar to anything else.

3346-4063: Thawro solution. Thawro, square pyramid, pentagonal cupola, truncated icosidodecahedron and truncated tetrahedron. 334-530 polychoron would have the same configuration of square pyramid and pentagonal cupola.

3346-5043: Thawro solution. Thawro, pentagonal pyramid, pentagonal rotunda, decagonal prism and triangular cupola. 335-530 polychoron would have the same configuration of pentagonal pyramid and pentagonal rotunda. 340-045 polychoron would have the same configuration of pentagonal rotunda and decagonal prism.

3346-5065: Thawro solution that is actually hyperbolic.

3346-6436: Thawro solution. Thawro, two truncated tetrahedra, triangular cupola and triangular prism. Not similar to anything else.

3346-6646: Thawro solution. Thawro, two truncated tetrahedra, hexagonal prism and truncated octahedron. The two truncated tetrahedra have the same configuration as in icosahedral rotunda.

3346-8848: Another hyperbolic thawro solution.

3346-0034: Thawro solution. Thawro, two pentagonal cupolas, truncated dodecahedron and triangular cupola. 335-040 polychoron would have the same configuration of pentagonal cupola and truncated dodecahedron. 340-030 polychoron would also have the same configuration of pentagonal cupola and truncated dodecahedron (but different pentagonal cupola).

3346-0040: Third hyperbolic thawro solution.

All in all, we see 15 solutions for thawro and only one (the prismatic one) for augmented hexagonal prism.
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### Re: General Approach--can 3D methods be generalized?

Next base is (3,3,5,5) and this one has three options:

1. Pentagonal orthobirotunda. The base is planar here. Main diagonal square corresponds to a decagon, so it's (5 + Sqrt[5])/2. The side diagonal square is 2/5 * (5 + Sqrt[5]).
2. Augmented dodecahedron. Here the side diagonal is a shortchord of a pentagon, so we know the side diagonal square is (3 + Sqrt[5])/2. The main diagonal is 2*(1 + 2/Sqrt[5]).
3. Augmented tridiminished icosahedron. Here the side diagonal has length 1 so its square is also 1. The main diagonal square comes out to be a nasty number (5 + Sqrt[5] + 2*Sqrt[3 + Sqrt[5]])/3.

Testing these values with 3355-4444 combination gives correct results. If the two diagonals don't work out correctly, a prismatic solution won't be found.

Solutions:

3355-3333: Hyperbolic pentagonal orthobirotunda solution. Seems to be a gyration of x3o3o *o5.

3355-3404: Augmented dodecahedron solution. Seems to be a hypothetical 455-055 acrochoron augmented with pentagonal prism pyramid. This would blend pentagonal pyramids with dodecahedra.

3355-4444: General prismatic solution valid for all three options.

3355-5555: Flat pentagonal orthobirotunda solution. Place a decagon and two pentagons in a plane, replace decagon with pentagonal orthobirotunda, replace one pentagon with pair of metabidiminished icosahedra and the other with a pair of dodecahedra and you get this vertex.

3355-6666: Hyperbolic pentagonal orthobirotunda solution.

3355-0530: Flat augmented dodecahedron solution. Place a decagon and two pentagons in a plane. In one half of the plane, place pentagonal rotunda on decagon and metabidiminished icosahedron and dodecahedraon on pentagons. In the other half, place truncated dodecahedron on decagon and pentagonal rotunda and pentagonal pyramid on pentagons. Fuse dodecahedron and pentagonal pyramid and you get this vertex.

And that's all. So apart from the prisms, the only of these polyhedra that can theoretically occur as a base of a quadragonal pyramid vertex is augmented dodecahedron. Surprising.
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### Re: General Approach--can 3D methods be generalized?

Next base is (3,4,3,4). We need to explore four options:

1. Planar option which belongs to cuboctahedron and various Johnson solids derived from it: triangular cupola, elongated triangular cupola, gyroelongated triangular cupola, triangular orthobicupola, elongated triangular orthobicupola, elongated triangular gyrobicupola, gyroelongated triangular bicupola and augmented truncated tetrahedron. Both diagonal squares are 3 in this case.
2. Gyrobifastigium. One diagonal square is 2 while the other is 7/2.
3. Square gyrobicupola. One diagonal corresponds to shortchord of octagon so its square is 2 + Sqrt[2] while the other is (6 - Sqrt[2])/2.
4. Pentagonal gyropicupola. One diagonal corresponds to shortchord of decagon so its square is (5 + Sqrt[5])/2 while the other is (5 - Sqrt[5])/2.

The solutions are:

3434-3333: Flat cuboctahedral solution, corresponding to cutting a cuboctahedron into tetrahedra and square pyramids.

3434-3334: Gyrobifastigium solution. Can be imagined as digonal gyrobicupolic ring with one square pyramid augmented by triangular prismatic pyramid. Two triangular prisms would then blend into gyrobifastigium.

3434-3344: Cuboctahedral solution. Can be imagined as 334-644 acrochoron with a triangular cupola augmented by triangular gyrobicupolic ring. Two triangular cupolas would then blend into cuboctahedron.

3434-3345: Pentagonal gyrobicupola solution. Can be imagined as 334-045 acrochoron with a pentagonal rotunda augmented by 334-530 acrochoron. Two pentagonal cupolas would then blend into pentagonal gyrobicupola.

3434-3443: Cuboctahedral solution. Can be imagined as 344-436 acrochoron with a triangular cupola augmented by triangle || triangular cupola. Two triangular cupolas would then blend into cuboctahedron.

3434-3464: Once again, this solution works for anything. We get 7 solutions in total (since all options except for cuboctahedron lead to two distinct vertices).
Cuboctahedron: Can be imagined as 334-646 acrochoron with a truncated tetrahedron augmented by octahedron || truncated tetrahedron. Or as 346-644 acrochoron with hexagonal prism augmented by triangle || hexagonal prism. Two triangular cupolas would then blend into cuboctahedron.
Gyrobifastigium A: Can be imagined as triangular cupolaic prism with a cube augmented with square pyramid prism. Two triangular prisms would then blend into gyrobifastigium.
Gyrobifastigium B: Can be imagined as triangle || hexagonal prism with a triangular cupola augmented with 344-436 acrochoron. Two triangular prisms would then blend into gyrobifastigium.
Square gyrobicupola A: Can be imagined as 346-844 acrochoron with octagonal prism augmented by square || octagonal prism. Two square cupolas would then blend into square gyrobicupola.
Square gyrobicupola B: This verf cannot be split. A vertex of square gyrobicupola is simply surrounded by square pyramid, triangular prism, triangular cupola and hexagonal prism with triangle and hexagon above the equatorial edges.
Pentagonal gyrobicupola A: Can be imagined as 346-044 acrochoron with octagonal prism augmented by pentagon || decagonal prism. Two pentagonal cupolas would then blend into pentagonal gyrobicupola.
Pentagonal gyrobicupola B: This verf cannot be split. A vertex of pentagonal gyrobicupola is simply surrounded by square pyramid, triangular prism, triangular cupola and hexagonal prism with triangle and hexagon above the equatorial edges.

3434-3405: Cuboctahedral solution. The verf cannot be split. A vertex of cuboctahedron is surrounded by pentagonal pyramid, triangular prism, pentagonal cupola and diminished rhombicosidodecahedron.

3434-3663: Another solution that works for everything. Here we only get 4 solutions since two solutions for assymetrical verfs are equivalent.
Cuboctahedron: Can be imagined as 346-636 acrochoron with truncated tetrahedron augmented by tetrahedron || truncated tetrahedron. Two triangular cupolas would then blend into cuboctahedron.
Gyrobifastigium: Can be imagined as octahedron || truncated tetrahedron with a triangular cupola augmented by triangle || triangular cupola. Two triangular prisms would then blend into gyrobifastigium.
Square gyrobicupola: This verf cannot be split. A vertex of square gyrobicupola is simply surrounded by tetrahedron, triangular cupola, truncated tetrahedron and another triangular cupola.
Pentagonal gyrobicupola: This verf cannot be split. A vertex of pentagonal gyrobicupola is simply surrounded by tetrahedron, triangular cupola, truncated tetrahedron and another triangular cupola.

3434-3883: Cuboctahedral solution. This verf cannot be split. A vertex of cuboctahedron is simply surrounded by tetrahedron, square cupola, truncated cube and another square cupola.

3434-3003: Cuboctahedral solution. This verf cannot be split. A vertex of cuboctahedron is simply surrounded by tetrahedron, pentagonal cupola, truncated dodecahedron and another pentagonal cupola.

3434-4444: General prismatic solution valid for all options.

3434-4466: Cuboctahedral solution. Can be imagined as 346-664 acrochoron with truncated octahedron augmented with cuboctahedron || truncated octahedron. Two triangular cupolas would then blend into cuboctahedron.

3434-4554: Cuboctahedral solution. This verf cannot be split. A vertex of cuboctahedron is simply surrounded by triangular prism, pentagonal prism, metabidiminished icosahedron and another pentagonal prism.

3434-4504: Pentagonal gyrobicupola solution. Can be imagined as 340-045 acrochoron with diminished rhombicosidodecahedron augmented with 344-450 acrochoron. Two pentagonal cupolas would then blend into pentagonal gyrobicupola.

3434-4506: Cuboctahedral solution. Can be imagined as 346-065 acrochoron with truncated icosahedron augmented with icosidodecahedron || truncated icosahedron. Two triangular cupolas would then blend into cuboctahedron.

3434-4664: Cuboctahedral solution. This is the verf of prismatotruncated pentachoron.

3434-4666: Gyrobifastigium solution. Can be imagined as great rhombated pentachoron with truncated octahedron augmented with truncated tetrahedron || truncated octahedron. Two triangular prisms would then blend into gyrobifastigium.

3434-4660: Pentagonal gyrogicupola solution. Can be imagined as 340-064 acrochoron with truncated icosidodecahedron augmented with rhodoperihedral rodunta. Two pentagonal cupolas would then blend into pentagonal gyrobicupola.

3434-4884: Cuboctahedral solution. This is the verf of prismatotruncated tesseract.

3434-4886: Flat square gyrobicupola solution. To imagine it, start with square and two octagons in plane. Replace one octagon with square gyrobicupola. On one side, put octagonal prism on square and truncated cube on octagon, on the other side, put triangular cupola on square and truncated cuboctahedron on octagon.

3434-4004: Cuboctahedral solution. This is the verf of prismatotruncated 120-cell.

3434-5050 Flat pentagonal gyrobicupola solution. To imagine it, start with decagon and two pentagons in plane. Replace the decagon with pentagonal gyrobicupola. On one side, put pentagonal rotunda on one pentagon and diminished rhombicosidodecahedron on the other. On the other side, switch them.

3434-5005: Hyperbolic cuboctahedral solution.

3434-6666: Flat cuboctahedral solution. This is the verf of x3x3o3 *o4.

3434-6886: Hyperbolic cuboctahedral solution.

3434-6006: Hyperbolic cuboctahedral solution.

So, in total:
14 cuboctahedral solutions
6 gyrobifastigium solutions
4 square gyrobicupola solutions
7 pentagonal gyrobicupola solutions
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### Re: General Approach--can 3D methods be generalized?

Now let's have a look at (3,4,3,5) base. This vertex configuration occurs in three Johnson solids well-known to CRF efforts on the page: pentagonal orthocupolarotunda, bilunabirotunda and triangular hebesphenorotunda, and it has the same geometry for all three to boot.
One diagonal corresponds to decagon, so the diagonal square is (5 + Sqrt[5])/2. The other diagonal corresponds to pentagon, so its diagonal square is (3 + Sqrt[5])/2.

The verf can be theoretically split along either diagonal. Splitting along the decagon diagonal might lead to augmentations where pentagonal cupola and pentagonal rotunda blend into pentagonal orthocupolarotunda, but splitting along the pentagon diagonal or splitting if the base cell is bilbiro or thawro won't lead to augmentation. It's still useful since the surroundings will then be related to some tetrahedral vertices. Note that with the pentagonal split, we only get one valid tetrahedral vertex -- the other would have to contain 345-acrohedron.

Solutions:

3435-3355: Lateral cells: Pentagonal pyramid, square pyramid, pentagonal pyramid, dodecahedron. Decagonal split leads to 335-055 vertex and 334-530 vertex joined in pentagonal rotunda. Pentagonal split leads to 335-555 vertex.

3435-3433: Lateral cells: Tetrahedron, triangular prism, square pyramid, pentagonal pyramid. Decagonal split leads to 334-530 vertex and 333-440 (decagon || pentagonal prism) joined in pentagonal cupola. Pentagonal split leads to 333-355 vertex (metabidiminished icosahedral pyramid).

3435-3404: Lateral cells: Square pyramid, triangular prism, pentagonal cupola, diminished rhombicosidodecahedron. Decagonal split leads to 340-045 vertex and 334-440 vertex (pentagon || decagonal prism) joined in decagonal prism. Pentagonal split leads to 334-550 vertex.

3435-3003: Lateral cells: Tetrahedron, pentagonal cupola, truncated dodecahedron, pentagonal rotunda. Decagonal split leads to 333-500 vertex (rhodomesohedral rotunda) and 340-030 vertex joined in truncated dodecahedron. Pentagonal split leads to 350-030 vertex.

3435-4444: Lateral cells: Triangular prism, cube, triangular prism, pentagonal prism. Prismatic form.

3435-4533: Lateral cells: Pentagonal cupola, pentagonal prism, pentagonal pyramid, pentagonal rotunda. Decagonal split leads to 335-530 vertex and 334-045 vertex joined in pentagonal rotunda. Pentagonal split would result in (3,4,5) acrohedra on BOTH sides.

3435-4504: Lateral cells: Triangular prism, pentagonal prism, pentagonal rotunda, diminished rhombicosidodecahedron. Decagonal split leads to 350-045 vertex and 344-450 vertex joined in diminished rhombicosidodecahedron. Pentagonal split leads to 344-505 vertex.

3435-4666: Lateral cells: Triangular cupola, hexagonal prism, truncated tetrahedron, truncated icosahedron. Decagonal split leads to 346-065 vertex and 340-646 vertex (rhodoperihedral rotunda) joined in truncated icosidodecahedron. Pentagonal split leads to 355-666 vertex (a diminishing of truncated 600-cell).

3435-4053: Lateral cells: Square pyramid, decagonal prism, pentagonal rotunda, metabidiminished icosahedron. Decagonal split leads to 334-550 vertex and 340-045 vertex joined in diminished rhombicosidodecahedron. Pentagonal split leads to 355-530 vertex.

3435-4003: Lateral cells: Square pyramid, decagonal prism, truncated dodecahedron, pentagonal rotunda. Decagonal split leads to 350-030 vertex and 334-040 vertex joined in truncated dodecahedron. Pentagonal split leads to 334-550 vertex.

3435-5053: Lateral cells: Pentagonal pyramid, diminished rhombicosidodecahedron, pentagonal rotunda, metabidiminished icosahedron. Decagonal split leads to 355-030 vertex and 335-040 vertex joined in truncated dodecahedron. Pentagonal split leads to 335-555 vertex.

3435-5055: Flat configuration, lateral cells are metabidiminished icosahedron, diminished rhombicosidodecahedron, pentagonal rotunda and dodecahedron. You can imagine it as starting with decagon and two pentagons in plane and replacing the decagon with pentagonal orthocupolarotunda. Now put pentagonal rotunda and diminished rhombicosidodecahedron on the pentagons on cupola side and dodecahedron and metabidiminished icosahedron on the rotunda side.

3435-6044: Lateral cells: Triangular cupola, truncated icosidodecahedron, pentagonal cupola, pentagonal prism. Decagonal split leads to 346-540 vertex (expanded/truncated rhodomesohedral rotunda) and 340-064 vertex joined in truncated icosidodecahedron. Pentagonal split leads to 340-545 vertex.

All in all, 12 possible solutions.
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### Re: General Approach--can 3D methods be generalized?

well, this configuration [3,4,3,5] also occurs in srid (x3o5x) and thus likewise in J5, J24, J30 - J33, J38 - J41, J46 + J 47, J68 - J83 ...
--- rk
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### Re: General Approach--can 3D methods be generalized?

Klitzing wrote:well, this configuration [3,4,3,5] also occurs in srid (x3o5x) and thus likewise in J5, J24, J30 - J33, J38 - J41, J46 + J 47, J68 - J83 ...
--- rk

No, it doesn't -- you're thinking of (3,4,5,4).
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### Re: General Approach--can 3D methods be generalized?

OK, next base is (3,4,3,6). This is a simple one, since it only occurs in one Johnson solid, augmented truncated tetrahedron. This also means that when it occurs, it will generally be as a blend from join of a polychoron with truncated tetrahedral cell and a polychoron with triangular cupola cell. One diagonal square is clearly 3. The other diagonal square can be computed as 11/3.

Solutions are:

3436-3334: Flat solution. It corresponds to the augmented truncated tetrahedron being cut into triangular cupolas and square pyramids on the truncated tetrahedron side and tetrahedra and square pyramids on the triangular cupola side.

3436-3464: This corresponds to octahedral rotunda with truncated tetrahedron augmented with octahedron || truncated tetrahedron. This would blend truncated tetrahedron and triangular cupola together to form an augmented truncated tetrahedron.

3436-4444: The standard prismatic solution.

3436-4643: This corresponds to 334-646 acrochoron and 346-644 acrochoron joined in hexagonal prism. This would blend truncated tetrahedron and triangular cupola together to form an augmented truncated tetrahedron.

3436-6646: Flat solution. Take three hexagons in plance, replace one of them with augmented truncated tetrahedron, then add truncated octahedron and triangular cupola on the truncated tetrahedron side and truncated tetrahedron and truncated octahedron on the triangular cupola side.

And that's all -- only three solutions.
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### Re: General Approach--can 3D methods be generalized?

Now it's time for (3,4,3,8) base. This appears only on augmented and biaugmented truncated cube and the vertex would be generally a blend of vertex with truncated cube and vertex with square cupola. One diagonal square is 2 + Sqrt[2] (corresponding to octagon), the other diagonal square is (6 + Sqrt[2])/2.

Solutions are:

3438-3334: Hyperbolic solution, analogical to 3436-3334 flat solution.

3438-3464: A blend of 346-848 vertex (diminished prismatotruncated 16-cell) and 334-448 vertex (square || octagonal prism), joined in octagonal prism.

3438-4444: Standard prism solution.

3438-4643: A blend of 334-848 vertex and 346-844 vertex, joined in octagonal prism.

3438-4883: Flat solution. Start with square/octagon/octagon configuration in plane, replace one octagon with augmented truncated cube. On truncated cube side, augment the other octagon with truncated cube and the square with square pyramid, on square cupola side, augment the other octagon with truncated cube and the square with octagonal prism.

3438-6646: Hyperbolic solution analogical to 3436-6646.

3438-6844: Flat solution. Start with square/octagon/octagon configuration in plane, replace one octagon with augmented truncated cube. On truncated cube side, augment the other octagon with square cupola and the square with octagonal prism. On square cupola side, augment the other octagon with truncated cuboctahedron and the square with triangular cupola.

So there are only 3 spherical solutions usable for CRFs, just like in (3,4,3,6) case.
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### Re: General Approach--can 3D methods be generalized?

Next base is (3,4,3,10). It's found in the four Johnson solids that are augmentations of truncated dodecahedron, so the vertex will generally be a blend of vertex with truncated dodecahedron and vertex with pentagonal cupola. One diagonal square is (5 + Sqrt[5])/2 (corresponding to decagon), the other diagonal square is (35 + Sqrt[5])/10.

Solutions are:

3430-3334: Hyperbolic solution analogical to 3436-3334 and 3438-3334

3430-3464: A blend of 346-040 vertex (diminished prismatotruncated 600-cell) and 334-448 vertex (pentagon || decagonal prism), joined in decagonal prism.

3430-4444: Standard prismatic solution.

3430-4643: A blend of 334-040 vertex and 346-044 vertex, joined in decagonal prism.

3430-5053: Flat solution. Start with pentagon/pentagon/decagon configuration in plane, replace the decagon with augmented truncated dodecahedron. On the truncated dodecahedron side, put pentagonal rotunda and pentagonal pyramid on the pentagons. On the pentagonal cupola side, put pentagonal rotunda and diminished rhombicosidodecahedron on them.

3430-6646: Hyperbolic solution analogical to 3436-6646 and 3438-6646.

All in all, three spherical solutions.
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### Re: General Approach--can 3D methods be generalized?

Next, let's have a look at (3,4,4,4) base. There are three versions:

First version belongs to the rhombicuboctahedron. The same vertex also occurs with square cupola and all the Johnson solids that contain it: elongated square cupola, gyroelongated square cupola, square orthobicupola, square gyrobicupola, elongated square gyrobicupola, gyroelongated square bicupola, augmented truncated cube and biaugmented truncated cube. Since both diagonals correspond to octagons, their squares are 2 + Sqrt[2]. If rhombicuboctahedron, elongated square cupola or elongated square gyrobicupola are used, the vertices can be generally split in two, one involving square cupola, the other involving elongated square cupola or octagonal prism.

The second version belongs to the elongated triangular cupola, elongated triangular orthobicupola and elongated triangular gyrobicupola. One diagonal square is 3, the other is 2*(1 + Sqrt[2/3]). The vertices can be generally split in two, one involving triangular cupola, the other involving hexagonal prism or elongated triangular cupola.

The third version belongs to the elongated pentagonal cupola, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda and elongated pentagonal gyrocupolarotunda. One diagonal square is (5 + Sqrt[5])/2, the other is 2 + Sqrt[2 - 2/Sqrt[5]]. The vertices can be generally split in two, one involving pentagonal cupola, the other involving decagonal prism, elongated pentagonal cupola or elongated pentagonal rotunda.

Solutions:

3444-3333: Hyperbolic rhombicuboctahedral solution. Corresponds to cutting of rhombicuboctahedron into tetrahedra and square pyramids.

3444-3343: Flat elongated triangular cupola solution. Corresponds to cutting of elongated triangular cupola into tetrahedra, square pyramids and triangular prisms.

3444-3443: Rhombicuboctahedral solution. Corresponds to cube || rhombicuboctahedron.

3444-3484: Rhombicuboctahedral solution. Here, a vertex is surrounded by square pyramid, triangular prism, octagonal prism and another octagonal prism. One way to see it is as octagonal duoprism augmented with square || octagonal prism. Another is 334-848 polychoron elongated in truncated cube.

3444-34(20)4: Elongated pentagonal cupola solution. It can be understood as (10,20)-duoprism augmented with pentagon || decagonal prism.

3444-34n4 has general combinatoric solution, but as for specific solutions, I was only able to find one for n=8 and one for n=20.

3444-3663: Rhombicuboctahedral solution. The vertex cannot be cut (it would lead to nonexistent 368-acrohedron). The sequence of cells around it is tetrahedron-triangular cupola-truncated octahedron-triangular cupola.

3444-4334: Rhombicuboctahedral solution. Corresponds to octahedron || rhombicuboctahedron.

3444-4368: Rhombicuboctahedral solution. A vertex is surrounded by square cupola, triangular prism, triangular cupola and truncated cuboctahedron. One way to see it is as 346-848 acrochoron (diminished prismatotruncated 16-cell) with a truncated cube augmented by cuboctahedron || truncated cube. Another is 334-486 acrochoron (like truncated cube || truncated cuboctahedron) and 348-864 acrochoron joined in truncated cuboctahedron.

3444-4444: Standard prism solution valid for all three versions.

3444-4486: Flat rhombicuboctahedral solution. You can imagine it by taking x4x3x4x and excavating one truncated cuboctahedron with triangular cupolas, square cupolas and cubes, blending the square cupolas with octagonal prisms.

I found no solution for generic 3444-44n4 except for n=4.

3444-4664: Rhombicuboctahedral solution. Corresponds to prismatotruncated 24-cell.

I found no solutions for generic 3444-4n46, 3444-4n48 and 3444-4n40.

3444-5445: Rhombicuboctahedral solution. Allows to surround rhombicuboctahedron with cubes on main squares, tridiminished icosahedra on triangles and pentagonal prism on the other squares. Ursachoric form?

3444-5450: Flat elongated pentagonal cupola solution. You can start with pentagon/pentagon/decagon configuration, then replace the decagon with elongated pentagonal cupola. On pentagonal cupola side, put diminished rhombicosidodecahedron and pentagonal rotunda on the pentagons. On decagonal prism side, put two pentagonal prisms there.

3444-6336: Rhombicuboctahedral solution. The vertex can't be split, as it would lead to nonexistent 368-acrohedron. Vertex is surrounded by truncated tetrahedron, triangular cupola, square pyramid and triangular cupola.

3444-6446: Rhombicuboctahedral solution. Corresponds to elongated stauroperihedral rotunda.

3444-6466: Flat elongated triangular cupola solution. Start with hexagon/hexagon/hexagon configuration, replace one hexagon with elongated triangular cupola. On cupola side, put truncated octahedron and truncated tetrahedron on the hexagons. On prism side, put two hexagonal prisms there.

3444-6666: Hyperbolic rhombicuboctahedral solution.

3444-8338: Rhombicuboctahedral solution. Blend of 334-848 acrochoron and 348-838 acrochoron, joined in truncated cube.

3444-8448: Flat rhombicuboctahedral solution. Corresponds to x4x3o4x.

3444-8668: Hyperbolic rhombicuboctahedral solution.

3444-0330: Rhombicuboctahedral solution. The vertex can't be split, as it would lead to nonexistent 380-acrohedron. Vertex is surrounded by truncated dodecahedron, pentagonal cupola, square pyramid and pentagonal cupola.

3444-0440: Hyperbolic rhombicuboctahedral solution.

3444-0660: Hyperbolic rhombicuboctahedral solution.

In total, 12 usable rhombicuboctahedral solutions and 1 (prismatic) solution for elongated triangular cupola and elongated pentagonal cupola.
Marek14
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### Re: General Approach--can 3D methods be generalized?

wendy wrote:One of the limits generalising polytopes is that you can create ever-larger polygons, but this does not apply higher up. For example, you can convert a decagon to a 20-gon, but the similar process does not exist for polyhedra.

That's not strictly true. You can apply Stott-expansion to the faces of a polyhedron (i.e., radially expand each face outwards, and take the convex hull), for example, and always get another polyhedron with more faces. Or truncate the vertices of a polyhedron, and get new faces where the old vertices were. Or apply Conway's kis- operator, which erects pyramids on the existing faces. Or any of the various general polyhedron operations. Just don't expect the results to retain any sort of regularity or uniformity.
quickfur
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### Re: General Approach--can 3D methods be generalized?

OK, let's have a look at the (3,4,4,5) base.

This base has two forms. One is based in gyrate rhombicosidodecahedron. The diagonal squares are (5+Sqrt[5])/2 and 2/5 * (5 + 2*Sqrt[5]).

The second form is based in elongated pentagonal rotunda. The diagonal squares are 2+Sqrt[2/5 * (5+Sqrt[5])] and (5+Sqrt[5])/2.

Solutions:

3445-3333: hyperbolic gyrate rhombicosidodecahedron solution.

3445-3433: hyperbolic elongated pentagonal rotunda solution.

3445-3404: gyrate rhombicosidodecahedron solution. A blend of 440-045 acrochoron and decagon || pentagonal cupola joined in decagonal prism.

3445-4444: standard prismatic solution valid for both bases.

For 3445-4n44, only prismatic solution n=4 was found. For 3445-4n40, no solutions at all were found.

3445-5455: flat elongated pentagonal rotunda solution.

3445-6466: hyperbolic elongated pentagonal rotunda solution.

3445-6666: hyperbolic gyrate rhombicosidodecahedron solution

Basically, not many solutions here.
Marek14
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### Re: General Approach--can 3D methods be generalized?

Only 2 bases left. The (3,4,5,4) base belongs to rhombicosidodecahedron and its derivatives and has no alternate forms. Both diagonal squares are (5+Sqrt[5])/2.

Solutions:

3454-3333: hyperbolic solution.

3454-3354: Either a blend of 334-045 acrochoron and 334-550 acrochoron joined in pentagonal rotunda, or a blend of 340-545 acrochoron and pentagonal gyrobicupolic ring joined in pentagonal cupola.

3454-3305: Either a blend of 350-045 acrochoron and 334-530 acrochoron joined in pentagonal rotunda, or a blend of 334-050 acrochoron and 335-040 acrochoron joined in truncated dodecahedron.

3454-3443: Dodecahedron || rhombicosidodecahedron

3454-3404: Either a blend of 440-045 acrochoron and decagon || pentagonal cupola joined in decagonal prism, or a blend of diminished small rhombated 120-cell and 334-040 acrochoron joined in truncated dodecahedron.

3454-3663: The vertex can't be split because it would result in nonexistent 360 acrohedron. It looks like rhombicosidodecahedron with tetrahedra on triangular faces, triangular cupolas on square faces and truncated icosahedra on pentagonal faces.

3454-4334: hyperbolic solution.

3454-4444: standard prismatic solution.

3454-4406: hyperbolic solution.

3454-4554: Blend of 445-055 acrochoron and 344-450 acrochoron joined in diminished rhombicosidodecahedron.

3454-4664: hyperbolic solution.

3454-5445: Blend of 445-540 acrochoron and 340-545 acrochoron joined in diminished rhombicosidodecahedron. It looks like rhombicosidodecahedron with tridiminished icosahedra on triangular faces, pentagonal prisms on square faces and more pentagonal prisms on pentagonal faces.

3454-6336: The vertex can't be split because it would result in nonexistent 360 acrohedron. It looks like rhombicosidodecahedron with truncated tetrahedra on triangular faces, triangular cupolas on square faces and pentagonal pyramids on pentagonal faces.

3454-6446: Prismatotruncated 120-cell.

3454-6666: hyperbolic solution.

3454-8338: The vertex can't be split because it would result in nonexistent 380 acrohedron. It looks like rhombicosidodecahedron with truncated cubes on triangular faces, square cupolas on square faces and pentagonal pyramids on pentagonal faces.

3454-8448: hyperbolic solution.

3454-8668: hyperbolic solution.

3454-0330: Blend of 335-040 acrochoron and 340-030 acrochoron. It looks like rhombicosidodecahedron with truncated dodecahedra on triangular faces, pentagonal cupolas on square faces and pentagonal pyramids on pentagonal faces.

3454-0440: hyperbolic solution.

3454-0550: hyperbolic solution.

3454-0660: hyperbolic solution.
Marek14
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