1. Hix - hexateron, also known as the 5-simplex. The symbol is oooox. This polyteron is the simplest one, it has 6 pens for it's "terons" and has 6 vertices. Its verf is a pen. This is the 5-D version of the tetrahedron. I usually refer to it's symmetry group as "hixic".
2. Pent - penteract, 5-cube, or decateron. This is the 5-D cube. The symbol is oooo'x. This polyteron is the "block" (or measure) polytope of dimension 5, it has 10 tesseracts for it's terons and 32 verts. Its verf is a pen. I usually call its symmetry group "pentic".
3. Tac - triacontaditeron, or better known as pentacross. The symbol is xooo'o, it can also be symbolized as xo8o. This polychoron is the 5-D version of an oct, it has 32 pens for it's terons and 10 verts. Its verf is a hex. It also has not one but three other regiment members - hehad, phap, and nophap. Hex has pentic symmetry.
4. Hehad - (HE had) hexadecahemidecateron, can also be called the hemipentacross - it is the 5-D version of thah. It has 16 pens and 5 hexes as terons and 10 verts. Its verf is a tho. It has "hinnic" or demipenteractic symmetry. For a while I assumed that this was the only faceting of tac, but I was wrong! - I found the next two polytera a few years ago (actually I knew of their shapes, just didn't realize they were uniform).
5. Phap - (FAP) pentachoric hemiantiprism. It has 12 pens and 5 thoes, with 10 vertices. Its verf is a faceting of hex with tet pyramid symmetry. Phap has pentachoron antiprism symmetry. To help see what this thing looks like, first consider the dyad prism (which turns out to be a square - it has 1 top dyad, 2 side dyads, and 1 bottom dyad - notice the pattern 1 2 1. No consider the triangle antiprism - 1 top triangle, 3 mid-top ones, 3 mid-bottom ones, and 1 bottom one - pattern 1 3 3 1. Tetrahedron antiprism has a 1 4 6 4 1 pattern - look familiar? - the Pentachoron antiprism has a 1 5 10 10 5 1 pattern of pentachora from top to bottom, phap contains the pentachora at the "1" and "5" locations.
6. Nophap - (NO fap) spinopentachoric hemiantiprism. It has 20 pens and 5 thoes, with 10 vertices. Its verf is a faceting of hex with tet pyramid symmetry. It has pentachoron antiprism symmetry. Nophap has the pentachora at the "10" locations.
7. Hin - demipenteract, also known as the hemipenteract. The symbol is ooo6. This polychoron is semiregular. Its verf is a rap. It has 10 hexes and 16 pens as terons, it also has 16 vertices - it is formed by alternating the vertices of a penteract. It has hinnic symmetry and there are 6 members in this regiment, all of which have hinnic symmetry and 16 vertices.
8. Dah - dis16. It has 16 pens and 16 raps. Its verf is a firp.
9. Han - 16penteract. It has 16 raps and 10 hexes. Its verf is a pinnip.
10. Radah - (RAY dah) retrodis16. It has 16 pens and 16 firps. Its verf is a faceting of firp.
11. Rinah - (RI nah) retrointercepted 16. It has 10 hexes and 16 firps. Its verf looks like pinnip, this polyteron is a copycat of han.
12. Hit - hexadecateron. It has 16 pinnips and is a noble polyteron (like vertices and like facets). Its verf is a spiky faceting of firp.
Duals
The following are self duals: hix.
The following are dual pairs: pent-tac.
Conjugates
The following are self conjugates: hix, pent, tac, hehad, phap, nophap, hin, dah, han, radah, rinah, and hit.
The following are conjugate pairs: none.
Category D --- back to Polyteron Page --- Category 2
