The Regular Polychora

The Platonic Solids

In 3D, we have 5 regular polyhedra, the polyhedra made solely of congruent regular polygons. They are:

1. 

   The tetrahedron, made of 4 triangles, with 3 meeting at each corner.

2. 

   The cube, made of 6 squares, with 3 meeting at each corner.

3. 

   The octahedron, made of 8 triangles, with 4 meeting at each corner.

4. 

   The dodecahedron, made of 12 pentagons, with 3 meeting at each corner.

5. 

   The icosahedron, made of 20 triangles, with 5 meeting at each corner.

These are also known as the Platonic solids, and are the only regular polyhedra possible in 3D.

The Regular Polychora

In 4D, we have the regular polychora, also known as the 4D polytopes, which are built from Platonic solids folded into 4D space. Just as 3D polyhedra are bounded by polygonal faces, 4D polychora are bounded by polyhedral cells. It turns out that there are only 6 possible regular polychora in 4D. They are:

1. The pentatope, also known as the 5-cell, made of 5 tetrahedra, joined 3 to an edge.

2. The 4D hypercube, the 4D equivalent of a 3D cube, also known as the 8-cell. It is bounded by 8 cubes, joined 3 to an edge.

3. The 16-cell, the 4D equivalent of the octahedron. It is bounded by 16 regular tetrahedra joined 4 to an edge.

4. The 24-cell, bounded by 24 octahedra. A unique object with no exact analog in any other dimension.

5. The 120-cell, bounded by 120 regular dodecahedra.

6. The 600-cell, bounded by 600 tetrahedra joined 5 to an edge.