Regular polychora are extensions of regular polyhedra to the 4D world. Each shape is surrounded by regular 3D shapes, each of which gather around a vertex in a uniform manner.
Compared to 5 Platonic solids in 3D, there are six regular 4D shapes, making it more diverse than in 3D.
From now on, since it's tiring to write the full names of shapes, I will use short names coined by mathematician Jonathan Bowers.
These are pronounced according to the rules of phonics, just like ordinary Eng words.
This article features 3 of six regular 4D shapes: 5-cell, 8-cell, 16-cell; the smaller ones.
This is made of these elements: Cells: 5 Tet's. Faces: 10 triangles. Edges: 10. Vertices: 5.
This is the simplest one, in terms of number of vertices. It is the only one of the six with odd numbers of vertices and cells.
This is a pyramid based on Tet. Four of five points form a Tet, and the fifth point is "above" it, connecting to all existing points.
In 4D space, some of the coordinate values must be irrational number. To construct it easily, choose unit points in 5D space: \((1,0,0,0,0)\) and its permutations.
Elements: 8 cubes. 24 squares. 32 edges. 16 vertices.
Commonly known as tesseract. The 4D hypercube, also called measure polytopes. Natural extension of the square and cube.
Just like the square and cube, we can think of this as the origin of all 4D shapes.
This is a prism based on the cube. Two cubes are in different realms, and all corresponding points are connected, making 6 additional cubes.
Among hypercubes in all dimensions, the Tes is the only one that the circumradius - distance between the center and any vertex - is equal to the edge length. This is because the square root of the dimensions, which is the diagonal length of the hypercube, is the integer 2 in 4D.
Elements: 16 Tet's. 32 triangles. 24 edges. 8 vertices.
The 4D cross polytope, also called orthoplex. Extension of Oct in 3D. The Oct has 6 vertices, 4 of which form a square; other two are above and below the square. Thus, the Oct is two square pyramids attached at their bases.
Similarly, in 4D, the Hex is formed by adding 2 vertices "above and below" the Oct. Two octahedral pyramids attach on the bases to form a Hex.
Additionally, just like the Oct is a triangular antiprism, the Hex is a tetrahedral antiprism.
Among orthoplexes in all dimensions above 2D, the Hex is the only one that is a space-filling shape.
The dual of a 4D shape is a shape whose vertices and cells are switched, and edges and faces are switched.
Dual relationships of regulars are described below:
Since the regulars are the shapes with the highest symmetry, the dual relationships are most powerful. When we have trouble analyzing regulars, looking at their dual helps us.