Above are the vertex figures of the regular polytera in cross sections.
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.
Below are fields of sections of the regular polytera.
Above are the vertex figures of the facetings of tac rendered in cross sections.
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. Now 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.
Below are fields of sections of the tac facetings.
Above are the vertex figures of the hin regiment rendered in cross sections.
7. Hin - demipenteract, also known as the hemipenteract. The symbol is oo8x. 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.
Below are fields of sections of the hin regiment.
The following are self duals: hix.
The following are dual pairs: pent-tac.
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