# Polyhedron Category 1: Regulars

The regular polyhedra have congruent faces, congruent edges, and congruent vertices. Also all of their faces are also regular. The first five are convex and have been known since the ancient times, they are called the Platonic solids. The next four are the regular star polyhedra, they are known as the Kepler-Poinsot polyhedra. All of these are orientable. These polyhedra show up quite a bit as cells of uniform polychora. The polychoron category most similar to this one is category 1. 1. Tet - (TET) tetrahedron. Symbol is oox, also s s s. Faces are 4 triangles. It has 4 vertices and its verf is a triangle.  2. Cube - (KYOOB) cube, or hexahedron. Symbol is oo'x, also ox x, and x x x. Faces are 6 squares. It has 8 vertices and its verf is a triangle. This is the most famous polyhedron of all.

3. Oct - (OCT) octahedron. Symbol is xo'o, also oxo, ss s. Faces are 8 triangles. It has 6 vertices and its verf is a square.  4. Doe - (DOE) dodecahedron. Symbol is oo^x. Faces are 12 pentagons. It has 20 vertices and its verf is a triangle.

5. Ike - (IKE) icosahedron. Symbol is xo^o, also ss'o and sss. Faces are 20 triangles. It has 12 vertices and its verf is a pentagon.  6. Gad - (GAD) great dodecahedron. Symbol is o*o^x. Faces are 12 pentagons. It has 12 vertices and its verf is a star, it is in the ike regiment.

7. Sissid - (SIS sid) small stellated dodecahedron. Symbol is x*o^o. Faces are 12 stars. It has 12 vertices and its verf is a pentagon.  8. Gike - (GIKE) great icosahedron. Symbol is xo*o, also ss"o and ss,s. Faces are 20 triangles. It has 12 vertices and its verf is a star, it is in the sissid regiment.

9. Gissid - (GIS sid) great stellated dodecahedron. Symbol is oo*x. Faces are 12 stars. It has 20 vertices and its verf is a triangle.

Dual Pairs: cube-oct, doe-ike, gad-sissid, gike-gissid.

Self Dual: tet.

Conjugate Pairs: doe-gissid, ike-gike, gad-sissid.

Self Conjugates: tet, cube, oct.