1. Pen - pentachoron, also known as the 5-cell or 4-simplex. The symbol is ooox. This polychoron is the simplest one, it has 5 tets for it's cells and has 5 verts (vertices). Its verf (vertex figure) is also a tet. It obviously has 5 pieces in one type, LOC = 1. I usually refer to it's symmetry group as "pennic".
2. Tes - tesseract, best known as the hypercube (which is really generic for cubes of all dimensions). The symbol is ooo'x. This polychoron is the most famous one, it has 8 cubes for it's cells and 16 verts. Its verf is a tet. It has 8 pieces in one type, LOC = 1. The tes is actually the square of a square. I usually call its symmetry group "tessic".
3. Hex - hexadecachoron, also known as the 16-cell. The symbol is xoo'o, it can also be symbolized as o8x. This polychoron is the 4-D version of an oct, it has 16 tets for it's cells and 8 verts. Its verf is an oct. It has 16 pieces in one type, LOC = 1. The hex is actually the diamond of the diamond. It also has another regiment member - tho. Hex has tessic symmetry.
4. Ico - (I co) icositetrachoron, also known as the 24-cell or the octaplex. The symbol is oo'ox, oxo'o, or o9o. This polychoron is somewhat like a 4-D co and acts like a rectified hex, it has 24 octs for it's cells and 24 verts. Its verf is a cube. It has 24 pieces in one type, LOC = 1. I usually call its symmetry group "icoic". Ico also forms a regiment of 14 figures plus one compound.
5. Hi - hecatonicosachoron, also known as the 120-cell, twelvtychoron, or the dodecaplex. The symbol is ooo^x. This polychoron is the 4-D doe, it has 600 verts and 120 does for it's cells. Its verf is a tet. It has 120 pieces in one type, LOC = 1. I call it's symmetry group "hyic".
6. Ex - hexacosichoron, also known as the 600-cell or the tetraplex. The symbol is xoo^o. This polychoron is the 4-D version of ike, it has 600 tets for it's cells and has 120 verts. Its verf is an ike. It has 600 pieces in one type, LOC = 1. Ex forms a regiment of 5 figures, 4 are regular, one is the small swirlprism, there's also the additional exotic-celled case "sachi". It can be colored isogonally under several other symmetries - ixitic, 3-swirldoic, 3-skidoic, 2-skidoic, 1-skidoic, 5-spyritic, and 5-skitettic.
All of these have hyic symmetry, Gogishi has 600 verts, the others have 120 verts.
7. Fix - faceted 600choron, also known as the icosahedral 120choron, it could also be called the icosaplex. The symbol is xo^o*o. This polychoron is the 4-D version of gad, its cells are 120 ikes. Its verf is a gad. It has 2400 pieces in one type which are grouped as 600 tet-shaped dimples of 4 pieces, LOC = 4.
8. Gohi - (GO hi) great 120choron. The symbol is o^o*o^x. It has 120 gads for its cells. Its verf is a sissid. Gohi has 3600 pieces in one type which are grouped as 1200 triangle shaped dimples of three pieces. LOC = 6.
9. Gahi - (GAY hi) grand 120choron. The symbol is o*oo^x. It has 120 does for its cells. Its verf is a gike. Fix, gohi, and gahi belong to the ex regiment. It has 21600 pieces with 3 types. LOC = 44.
10. Sishi - (SIS she) small stellated 120choron. The symbol is oo^o*x. It has 120 sissids and its verf is a doe. It is the 4-D version of sissid. Sishi has 1440 pieces of one type shaped like pentagon pyramids, LOC = 4. Sishi looks like a hi with blunt points on its facets. Sishi leads a regiment that contains gaghi, several swirlprisms, and all of the category 17 polychora.
11. Gaghi - (GAG hi) great grand 120choron. The symbol is oo*o^x. It has 120 gads and its verf is a gissid. It has 10800 pieces with 2 types, LOC = 33. Gaghi also has 1200 spiky edges.
12. Gishi - (GISH shi) great stellated 120choron. The symbol is x*oo^o. It has 120 gissids and its verf is an ike. It has 2400 pieces with one type, LOC = 8. The gishi regiment contains gishi, gashi, gofix, and gax as well as the swirlprism gisp and the exotic-celled gachi.
13. Gashi - (GASH she) grand stellated 120choron. The symbol is o*o^o*x. It has 120 sissids and its verf is a gad. It has 7200 pieces of one type, LOC = 16.
14. Gofix - (GO fix) great facet 600choron, also called great icosahedral 120choron. The symbol is xo*o^o. It has 120 gikes and its verf is a sissid. It has 7200 pieces of one type, LOC = 12. Hop David has painted a picture that has a gofix section in it.
15. Gax - (GAKS) grand 600choron. The symbol is xoo*o. It has 600 tets and its verf is a gike. It is the 4-D version of gike. It has 36000 pieces of 4 types, LOC = 118 - it is the most complex of this category. It can be colored isogonally under several other symmetries - ixitic, 3-swirldoic, 3-skidoic, 2-skidoic, 1-skidoic, 5-spyritic, and 5-skitettic.
16. Gogishi - (go GISH she) great grand stellated 120choron. The symbol is ooo*x. It has 120 gissids and its verf is a tet. Gogishi is very spiky, for it is the 4-D version of gissid. It has 9600 pieces of two types, LOC = 30.
17. Tho - (THO) tesseractihemioctachoron. It belongs to the hex regiment, and therefore has 8 verts. Its cells are 8 tets and 4 octs (which go through the center). It's verf (shown above) is a thah, which is what tho is a 4-D version of. Tho doesn't seem to fit any category, so it gets a free ride with hex in this category. It has 40 pieces of two types, LOC = 5. Tho is non-orientable.
Duals
The following are self duals: pen, ico, gohi, and gashi.
The following are dual pairs: tes-hex, hi-ex, fix-sishi, gahi-gishi, gaghi-gofix, gax-gogishi.
Conjugates
The following are self conjugates: pen, tes, hex, tho, and ico.
The following are conjugate pairs: hi-gogishi, ex-gax, fix-gofix, gohi-gashi, gahi-gishi, sishi-gaghi.
Category B --- back to Polychoron Page --- Category 2