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.

**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.

* 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.

* 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.

* 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.

* 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**

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.

Category 1 --- back to Polychoron Page --- Category 3