# Category 1: Regular Polychora

There are 16 regular polychora where 6 are convex and 10 are starry. Compare this to 3 regular polytera, and 9 regular polyhedra. So here in four dimensions, there seems to be an abundance of regular figures. The regular convex polychora have been known since the 1800s. Except for tho, all of these are orientable. Following is a list of the 16 regular figures along with tho, a faceting of hex.

## Convex Regulars

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".

## Star Regulars

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.

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.

## This Little Polychoron Gets a Free Ride

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 and Conjugates

Below shows how the 16 regulars pair up as duals and as conjugates.

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.

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