Category 2: The Truncates

tip tat thex titho tico thi tex tiffix tighi taghi tigaghi tigfix taggix tissidtixhi tidtidohi tiggidtixhi quitit quit sishi quit gishi quit gashi quit gogishi


There are 21 polychoron truncates where 6 are convex, 6 are truncated stars, 3 are ditrigonary, 5 are quasitruncates, and then there's titho. Fourteen regiments are represented amongst the truncates. All of these are orientable except for titho. The ten 3-D truncates are the following: tut, tic, toe, tid, ti, tigid, tiggy, quith, quit sissid, and quit gissid. Following is a list of the 21 four dimensional truncates.
Truncates

Convex Truncates

Verfs of the Convex Truncates

18. Tip - truncated pentachoron. The symbol is ooxx. Its cells are 5 tets and 5 tuts. It has 20 verts (vertices). Its verf (vertex figure) is a trigonal pyramid. It has 10 pieces with two types, LOC = 4. This is the 4-D version of tut. It has pennic symmetry.

19. Tat - truncated tesseract. The symbol is oox'x. Its cells are 8 tics, and 16 tets. It has 64 verts. Its verf is also a trigonal pyramid. It has 24 pieces with two types and has tessic symmetry. LOC = 4. This is the 4-D version of tic.

20. Thex - (THEKS) truncated hexadecachoron. The symbol is xxo'o, it can also be symbolized as o9x. This polychoron is the 4-D version of tico, it has 16 tuts and 8 octs for it's cells and 48 verts. Its verf is a square pyramid. It also has 24 pieces with two types and has tessic symmetry. LOC = 4. It also has another regiment member - titho.

21. Tico - (TI co) truncated icositetrachoron, it also acts as the great rhombated hex. The symbol can be oo'xx, xxx'o, or xx$x. It has 24 cubes and 24 toes as cells, it also has 192 verts. Its verf is a trigonal pyramid. It has 48 pieces with two types, LOC = 4. It has icoic symmetry.

22. Thi - truncated hecatonicosachoron. The symbol is oox^x. This polychoron is the 4-D tid, it has 2400 verts and has 120 tids and 600 tets as cells. Its verf is a trigonal pyramid. It has 720 pieces in two types, LOC = 4. It has hyic symmetry.

23. Tex - truncated hexacosichoron. The symbol is xxo^o. This polychoron is the 4-D version of ti, its cells are 120 ikes and 600 tuts. It has 1440 verts with a pentagonal pyramid verf. It has 720 pieces in two types, LOC = 4. Tex is hyic and has one other regiment member - tiffix.


Star Truncates

Verfs of the Star Truncates

All of these have hyic symmetry, Tigaghi has 2400 verts, the others have 1440 verts. Regiments are as follows: tex(above)-tiffix, tighi-taghi, tigaghi, tigfix-taggix.

24. Tiffix - (TIF fix) truncated fix. The symbol is xx^o*o. Its cells are 120 gads and 120 tis. Its verf is a star pyramid. It has 2520 pieces with two types, LOC = 13.

25. Tighi - (TIG hi) truncated gohi. The symbol is o^o*x^x. Its cells are 120 sissids and 120 tigids. Its verf is a long narrow pentagon pyramid. Tighi has 3720 pieces with two types, LOC = 15.

26. Taghi - (TAG hi) truncated gahi. The symbol is o*ox^x. Its cells are 120 gikes and 120 tids. Its verf is a star pyramid. It has 21720 pieces in 4 types, LOC = 71.

27. Tigaghi - (tih GAG hi) truncated gaghi. The symbol is oo*x^x. Its cells are 120 gissids and 120 tigids. Its verf is a trigonal pyramid. Tigaghi has 10920 pieces in three types, LOC = 42.

28. Tigfix - (TIG fix) truncated gofix. The symbol is xx*o^o. It has 120 sissids and 120 tiggies. Its verf is a pentagon pyramid. It has 15960 pieces in four types, LOC = 48.

29. Taggix - (TAG gix) truncated gax. The symbol is xxo*o. Its cells are 120 gikes and 600 tuts. Its verf is a star pyramid. This is one of the more wilder looking truncates with 80760 pieces in 9 types, LOC = 269 - highest LOC in this category.


Titho

Verf of Titho

Like tho, titho is a one-of-a-kind.

30. Titho - (TIH tho) truncated tho. Its cells are 8 thahs, 8 tuts, and 4 toes. It has 48 verts with a bow-tie pyramid verf. It belongs to the thex regiment and has 72 pieces with 3 types, LOC = 17.


Ditrigonary Truncates

Verfs of the Ditrigonary Truncates

These three have hyic symmetry and belong to the same regiment. They have 2400 vertices and are the truncates of 3 members of the sishi regiment.

31. Tissidtixhi - (TIS sid TIK she) truncated sidtixhi. The symbol is x(x*oo). Its cells are 120 sidtids, 600 tuts, and 120 tiggies. Its verf is a ditrigon pyramid. It has 15960 pieces in 5 types, LOC = 66.

32. Tidtidohi - (tid TID uh hee) truncated dittady. The symbol is x(x*'oo^). Its cells are 120 ditdids, 120 tiggies, and 120 tis. Its verf is a tripod pyramid. It has 67800 pieces in 10 types, LOC = 186. It ties with quit gogishi for the most piece types in category 2.

33. Tiggidtixhi - (TIJ jid TIK she) truncated gidtixhi. The symbol is x(x^o,o). Its cells are 120 gidtids, 600 tuts, and 120 tis. Its verf is a tripod pyramid. It has 57720 pieces in 9 types, LOC = 199.


Quasitruncates

Verfs of the Quasitruncates

These are the 4-D versions of quith, quit sissid, and quit gissid. All of these have hyic symmetry except for quitit which has tessic symmetry.

34. Quitit - (QUID it) quasitruncated tesseract. The symbol is oox"x. It has 64 verts with a trigonal pyramid verf. Its cells are 8 quiths and 16 tets. It is the 4-D version of quith. It looks like a crowned tesseract in the same way a quith looks like a crowned cube. It has 120 pieces with 3 types, LOC = 25.

35. Quit Sishi - (quit SISH she) quasitruncated sishi. The symbol is oo^x*'x. It has 2400 verts with a trigonal pyramid verf. Its cells are 120 quit sissids and 120 does. It is the 4-D version of quit sissid. It looks like a crowned hi in the same way a quit sissid looks like a crowned doe. It has 3960 pieces in 3 types, LOC = 25.

36. Quit Gishi - (quit GISH she) quasitruncated gishi. The symbol is o^ox*'x. It has 1440 verts with a pentagonal pyramid verf. Its cells are 120 quit gissids and 120 ikes. It has 35040 pieces in 7 types, LOC = 120. Quit gishi and quit gashi form a regiment.

37. Quit Gashi - (quit GASH she) quasitruncated gashi. The symbol is o*o^x*'x. It has 1440 verts with a star pyramid verf. Its cells are 120 quit sissids and 120 gads. It is quite wild looking with 51840 pieces in 7 types, LOC = 148.

38. Quit Gogishi - (QUIT go GISH she) quasitruncated gogishi. The symbol is oox*'x. It has 2400 verts with a trigonal pyramid verf. Its cells are 120 quit gissids and 600 tets. It is the 4-D version of quit gissid, although it doesn't look it. It has 103,200 pieces in 10 types, LOC = 252. It has the highest piece count in this category - it also ties with tidtidohi with largest number of piece types in this category.


Conjugates and Regiments

Below shows how the truncates pair up as conjugates and group into regiments.

Conjugates

The following are self conjugates: tip, thex, titho, tico, and tidtidohi.

The following are conjugate pairs: tat-quitit, thi-quit gogishi, tex-taggix, tiffix-tigfix, tighi-quit gashi, taghi-quit gishi, tigaghi-quit sishi, tissidtixhi-tiggidtixhi.

Regiments

The following are singletons: tip, tat, tico, thi, tigaghi, quitit, quit sishi, and quit gogishi.

The following are grouped into regiments: thex-titho, tex-tiffix, tighi-taghi, tigfix-taggix, tissidtixhi-tidtidohi-tiggidtixhi, and quit gishi-quit gashi.

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