by **Marek14** » Thu Aug 30, 2012 10:37 pm

Tetrahedral vertices in CRF polychora:

I used Mathematica to check on various tetrahedra for the purpose of CRF polychora:

Even though the list only shows tetrahedral verfs, it already points to several promising areas of research, I think.

They are in form of abcdef, where faces are abd, ace, bcf and def. Allowed faces are:

333, 334, 335, 344, 346, 348, 340, 355, 350, 366, 388, 300, 444, 445, 446, 447, 448, 449, 440, 450, 466, 468, 460, 555 and 566. I'll refer to them with the simplest polyhedron that has them, if there are several.

Seems that euclidean tilings tetrahedra appear here as well (for example 466664). Not sure how to weed them out.

333333: four tetrahedra (verf of pentachoron)

333334: two tetrahedra, two square pyramids

333335: two tetrahedra, two pentagonal pyramids

333344: tetrahedron, two square pyramids, triangular prism

333355: tetrahedron, two pentagonal pyramids, metabidiminished icosahedron

333444: three square pyramids, cube

333445: two square pyramids, pentagonal pyramid, pentagonal prism

334344: tetrahedron, three triangular prisms (verf of tetrahedral prism)

334346: tetrahedron, triangular prism, two triangular cupolas

334348: tetrahedron, triangular prism, two square cupolas

334340: tetrahedron, triangular prism, two pentagonal cupolas

334366: tetrahedron, two triangular cupolas, truncated tetrahedron

334388: tetrahedron, two square cupolas, truncated cube

334434: two square pyramids, two triangular prisms

334436: two square pyramids, two triangular cupolas

334438: two square pyramids, two square cupolas

334430: two square pyramids, two pentagonal cupolas

334444: square pyramid, two triangular prisms, cube

334446: square pyramid, triangular prism, triangular cupola, hexagonal prism

334448: square pyramid, triangular prism, square cupola, octagonal prism

334440: square pyramid, triangular prism, pentagonal cupola, decagonal prism

334466: square pyramid, two triangular cupolas, truncated octahedron

334530: square pyramid, pentagonal pyramid, pentagonal cupola, pentagonal rotunda

334544: pentagonal pyramid, two triangular prisms, pentagonal prism

335355: tetrahedron, three metabidiminished icosahedra

335350: tetrahedron, metabidiminished icosahedron, two pentagonal rotundas

335300: tetrahedron, two pentagonal rotundas, truncated tetrahedron

335450: square pyramid, metabidiminished icosahedron, pentagonal rotunda, diminished rhombicosidodecahedron

335535: two pentagonal pyramids, two metabidiminished icosahedra

335530: two pentagonal pyramids, two pentagonal rotundas

335555: pentagonal pyramid, two metabidiminished icosahedra, dodecahedron

336366: tetrahedron, three truncated tetrahedra (verf of truncated pentachoron)

336444: square pyramid, two triangular cupolas, cube

336446: square pyramid, triangular cupola, truncated tetrahedron, hexagonal prism

336466: square pyramid, two truncated tetrahedra, truncated octahedron

336544: pentagonal pyramid, two triangular cupolas, pentagonal prism

336566: pentagonal pyramid, two truncated tetrahedra, truncated icosahedron

338388: tetrahedron, three truncated cubes (verf of truncated tesseract)

338444: square pyramid, two square cupolas, cube

338448: square pyramid, square cupola, truncated cube, octagonal prism

330300: tetrahedron, three truncated dodecahedra (verf of truncated 120-cell)

330445: square pyramid, pentagonal cupola, pentagonal rotunda, pentagonal prism

330440: square pyramid, pentagonal cupola, truncated dodecahedron, decagonal prism

330450: square pyramid, pentagonal rotunda, truncated dodecahedron, diminished rhombicosidodecahedron

330540: pentagonal pyramid, pentagonal cupola, truncated dodecahedron, diminished rhombicosidodecahedron

330555: pentagonal pyramid, two pentagonal rotundas, dodecahedron

344443: four triangular prisms (verf of 3,3-duoprism)

344444: two triangular prisms, two cubes (verf of 3,4-duoprism)

344445: two triangular prisms, two pentagonal prisms (verf of 3,5-duoprism)

344446: two triangular prisms, two hexagonal prisms (verf of 3,6-duoprism)

344447: two triangular prisms, two heptagonal prisms (verf of 3,7-duoprism)

344448: two triangular prisms, two octagonal prisms (verf of 3,8-duoprism)

344449: two triangular prisms, two enneagonal prisms (verf of 3,9-duoprism)

344440: two triangular prisms, two decagonal prisms (verf of 3,10-duoprism)

34444n: two triangular prisms, two n-gonal prisms (verf of 3,n-duoprism)

344463: two triangular prisms, two triangular cupolas

344464: triangular prism, triangular cupola, cube, hexagonal prism

344466: triangular prism, triangular cupola, hexagonal prism, truncated octahedron

344468: triangular prism, triangular cupola, octagonal prism, truncated cuboctahedron

344460: triangular prism, triangular cupola, decagonal prism, truncated icosidodecahedron

344483: two triangular prisms, two square cupolas

344484: triangular prism, square cupola, cube, octagonal prism

344486: triangular prism, square cupola, hexagonal prism, truncated cuboctahedron

344404: triangular prism, pentagonal cupola, cube, decagonal prism

344405: triangular prism, pentagonal cupola, pentagonal prism, diminished rhombicosidodecahedron

344663: triangular prism, two triangular cupolas, truncated tetrahedron

344664: two triangular cupolas, cube, truncated octahedron

344665: two triangular cupolas, pentagonal prism, truncated icosahedron

344684: triangular cupola, square cupola, cube, truncated cuboctahedron

344883: triangular prism, two square cupolas, truncated cube

345454: triangular prism, metabidiminished icosahedron, two pentagonal prisms

345450: triangular prism, metabidiminished icosahedron, two diminished rhombicosidodecahedra

345404: triangular prism, pentagonal rotunda, pentagonal prism, decagonal prism

345054: pentagonal cupola, metabidiminished icosahedron, pentagonal prism, diminished rhombicosidodecahedron

346444: triangular prism, triangular cupola, cube, hexagonal prism

346446: triangular prism, triangular cupola, hexagonal prism, truncated octahedron

346448: triangular prism, triangular cupola, octagonal prism, truncated cuboctahedron

346440: triangular prism, triangular cupola, decagonal prism, truncated icosidodecahedron

346464: triangular prism, truncated tetrahedron, two hexagonal prisms (verf of truncated tetrahedral prism)

346466: triangular prism, truncated tetrahedron, two truncated octahedra (verf of great rhombated pentachoron)

346468: triangular prism, truncated tetrahedron, two truncated cuboctahedra (verf of great rhombated tesseract)

346460: triangular prism, truncated tetrahedron, two truncated icosidodecahedra (verf of great rhombated 120-cell)

346644: two triangular cupolas, two hexagonal prisms

346646: two triangular cupolas, two truncated octahedra

346648: two triangular cupolas, two truncated cuboctahedra

346640: two triangular cupolas, two truncated icosidodecahedra

346663: two triangular cupolas, two truncated tetrahedra

346664: triangular cupola, truncated tetrahedron, hexagonal prism, truncated octahedron

346844: triangular cupola, square cupola, hexagonal prism, octagonal prism

346846: triangular cupola, square cupola, truncated octahedron, truncated cuboctahedron

346864: square cupola, truncated tetrahedron, hexagonal prism, truncated cuboctahedron

346044: triangular cupola, pentagonal cupola, hexagonal prism, decagonal prism

346046: triangular cupola, pentagonal cupola, truncated octahedron, truncated icosidodecahedron

346064: pentagonal cupola, truncated tetrahedron, hexagonal prism, truncated icosidodecahedron

348444: triangular prism, square cupola, cube, octagonal prism

348446: triangular prism, square cupola, hexagonal prism, truncated cuboctahedron

348484: triangular prism, truncated cube, two octagonal prisms (verf of truncated cubic prism)

348486: triangular prism, truncated cube, two truncated cuboctahedra (verf of great rhombated 24-cell)

348644: triangular cupola, square cupola, hexagonal prism, octagonal prism

348646: triangular cupola, square cupola, truncated octahedron, truncated cuboctahedron

348684: triangular cupola, truncated cube, octagonal prism, truncated cuboctahedron

348846: two square cupolas, two truncated cuboctahedra

348883: two square cupolas, two truncated cubes

348046: square cupola, pentagonal cupola, truncated cuboctahedron, truncated icosidodecahedron

340444: triangular prism, pentagonal cupola, cube, decagonal prism

340445: triangular prism, pentagonal cupola, pentagonal prism, diminished rhombicosidodecahedron

340454: triangular prism, pentagonal rotunda, pentagonal prism, decagonal prism

340404: triangular prism, truncated dodecahedron, two decagonal prisms (verf of truncated dodecahedral prism)

340405: triangular prism, truncated dodecahedron, two diminished rhombicosidodecahedra

340644: triangular cupola, pentagonal cupola, hexagonal prism, decagonal prism

340646: triangular cupola, pentagonal cupola, truncated octahedron, truncated icosidodecahedron

340656: triangular cupola, pentagonal rotunda, truncated icosidodecahedron, truncated icosahedron

340604: triangular cupola, truncated dodecahedron, decagonal prism, truncated icosidodecahedron

340846: square cupola, pentagonal cupola, truncated cuboctahedron, truncated icosidodecahedron

340046: two pentagonal cupolas, two truncated icosidodecahedra

340003: two pentagonal cupolas, two truncated dodecahedra

355553: four metabidiminished icosahedra

355555: two metabidiminished icosahedra, two dodecahedra

355503: two metabidiminished icosahedra, two pentagonal rotundas

355003: metabidiminished icosahedron, two pentagonal rotundas, truncated dodecahedron

356566: metabidiminished icosahedron, truncated tetrahedron, two truncated icosahedra

350054: two pentagonal rotundas, two diminished rhombicosidodecahedra

350003: two pentagonal rotundas, two truncated dodecahedra

366663: four truncated tetrahedra (verf of decachoron)

366664: two truncated tetrahedra, two truncated octahedra (verf of bitruncated tesseract)

366665: two truncated tetrahedra, two truncated icosahedra (verf of bitruncated 120-cell)

368684: truncated tetrahedron, truncated cube, two truncated cuboctahedra [mysterious!]

388883: four truncated cubes (verf of 48-cell)

444444: four cubes (verf of tesseract)

444445: two cubes, two pentagonal prisms (verf of 4,5-duoprism)

444446: two cubes, two hexagonal prisms (verf of 4,6-duoprism)

444447: two cubes, two heptagonal prisms (verf of 4,7-duoprism)

444448: two cubes, two octagonal prisms (verf of 4,8-duoprism)

444449: two cubes, two enneagonal prisms (verf of 4,9-duoprism)

444440: two cubes, two decagonal prisms (verf of 4,10-duoprism)

44444n: two cubes, two n-gonal prisms (verf of 4,n-duoprism)

444450: cube, pentagonal prism, decagonal prism, diminished rhombicosidodecahedron

444466: cube, two hexagonal prisms, truncated octahedron (verf of truncated octahedral prism)

444468: cube, hexagonal prism, octagonal prism, truncated cuboctahedron (verf of truncated cuboctahedral prism)

444460: cube, hexagonal prism, decagonal prism, truncated icosidodecahedron (verf of truncated icosidodecahedral prism)

444555: three pentagonal prisms, dodecahedron (verf of dodecahedral prism)

444566: pentagonal prism, two hexagonal prisms, truncated icosahedron (verf of truncated icosahedral prism)

445544: four pentagonal prisms (verf of 5,5-duoprism)

445540: two pentagonal prisms, two diminished rhombicosidodecahedra

445644: two pentagonal prisms, two hexagonal prisms (verf of 5,6-duoprism)

445640: pentagonal prism, hexagonal prism, diminished rhombicosidodecahedron, truncated icosidodecahedron

445744: two pentagonal prisms, two heptagonal prisms (verf of 5,7-duoprism)

445844: two pentagonal prisms, two octagonal prisms (verf of 5,8-duoprism)

445944: two pentagonal prisms, two enneagonal prisms (verf of 5,9-duoprism)

445044: two pentagonal prisms, two decagonal prisms (verf of 5,10-duoprism)

445n44: two pentagonal prisms, two n-gonal prisms (verf of 5,n-duoprism)

446466: cube, three truncated octahedra (verf of truncated 24-cell)

446468: cube, truncated octahedron, two truncated cuboctahedra

446566: pentagonal prism, two truncated octahedra, truncated icosahedron (verf of great rhombated 600-cell)

446644: four hexagonal prisms (verf of 6,6-duoprism)

446646: two hexagonal prisms, two truncated octahedra (verf of great prismated decachoron)

446648: two hexagonal prisms, two truncated cuboctahedra (verf of great prismated 48-cell)

446744: two hexagonal prisms, two heptagonal prisms (verf of 6,7-duoprism)

446844: two hexagonal prisms, two octagonal prisms (verf of 6,8-duoprism)

446846: hexagonal prism, octagonal prism, truncated octahedron, truncated cuboctahedron (verf of great prismated tesseract)

446944: two hexagonal prisms, two enneagonal prisms (verf of 6,9-duoprism)

446044: two hexagonal prisms, two decagonal prisms (verf of 6,10-duoprism)

446046: hexagonal prism, decagonal prism, truncated octahedron, truncated icosidodecahedron (verf of great prismated 120-cell)

446n44: two hexagonal prisms, two n-gonal prisms (verf of 6,n-duoprism)

447744: four heptagonal prisms (verf of 7,7-duoprism)

447844: two heptagonal prisms, two octagonal prisms (verf of 7,8-duoprism)

447944: two heptagonal prisms, two enneagonal prisms (verf of 7,9-duoprism)

447044: two heptagonal prisms, two decagonal prisms (verf of 7,10-duoprism)

447n44: two heptagonal prisms, two n-gonal prisms (verf of 7,n-duoprism)

448844: four octagonal prisms (verf of 8,8-duoprism)

448846: two octagonal prisms, two truncated cuboctahedra

448944: two octagonal prisms, two enneagonal prisms (verf of 8,9-duoprism)

448044: two octagonal prisms, two decagonal prisms (verf of 8,10-duoprism)

448n44: two octagonal prisms, two n-gonal prisms (verf of 8,n-duoprism)

449944: four enneagonal prisms (verf of 9,9-duoprism)

449044: two enneagonal prisms, two decagonal prisms (verf of 9,10-duoprism)

449n44: two enneagonal prisms, two n-gonal prisms (verf of 9,n-duoprism)

440044: four decagonal prisms (verf of 10,10-duoprism)

440n44: two decagonal prisms, two n-gonal prisms (verf of 10,n-duoprism)

44nn44: four n-gonal prisms (verf of n,n-gonal duoprism)

44mn44: two m-gonal prisms, two n-gonal prisms (verf of m,n-gonal duoprism)

466664: four truncated octahedra

555555: four dodecahedra (verf of 120-cell)

Tetrahedra that are too large or impossible:

333555, 334300, 334468, 334460, 334533, 334540, 334566, 335533, 338544, 330444, 330544, 344403, 344406, 344604, 344003, 346843, 346043, 348843, 348043, 348044, 340446, 340406, 340844, 340043, 340044, 340053, 350053, 360604, 300003, 446460, 446640, 448566, 448046, 440566, 440046, 450054, 450064, 466665, 466684, 466604, 468864, 468064, 460064, 556566, 566665

Tetrahedra that come out planar:

334433, 346643, 348844, 340045