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 I'm too lazy to write the full names of shapes, I will use short names coined by mathematician Jonathan Bowers. Tet: tetrahedron; Oct: octahedron; Doe: dodecahedron; Ike: icosahedron; and so on. These are pronounced according to the rules of phonics.
Pen = Regular pentachoron
This is the simplest 4D shape in terms of number of vertices. Four points form a Tet, and the fifth point is "above" it, connecting to all existing points & making a total of 5 Tet's. Thus, it's a tetrahedral pyramid. It is the only one of the six with odd numbers of vertices and cells.
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.
Tes = Tesseract / Regular octachoron
A 4D hypercube, also called measure polytopes. Natural extension of square & cube. Composed of eight cubes. We can obtain it by vertically moving and duplicating a cube and connecting the corresponding vertices.
Among hypercubes in all dimensions, the Tes is the only one that the 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.
Hex = Regular hexadecachoron
The 4D cross polytope, also called orthoplex. Analogous to the Oct in 3D. First, we tend to view the Oct as two square pyramids joined at their bases. Similarly, two octahedral pyramids attach on the bases to form a Hex. Second, just like the Oct is a triangular antiprism, the Hex is a tetrahedral antiprism.
If we bring the center of this figure to the origin of the coordinate axes, two vertices are on each of the four axes, equidistant from the origin.
Among orthoplexes in all dimensions above 2D, the Hex is the only one that is a space-filling shape.
Ico = Regular icositetrachoron
This is the most special of the regular polychora, with no counterpart in regulars in 3D. What this corresponds to is the non-regular shape, rhombic dodecahedron (Rad). Similar to the procedure for understanding the Hex, it is necessary to clearly understand the Rad first. Since many of you have never heard of this name, let's start with the idea of its formation. It begins with a cube.
A cube has eight vertices and six faces. Grab the centers of these six faces and stretch them outward to form four-sided pyramids. Continue until the two adjacent triangles are co-planar. Then, a rhombus is formed at each of the 12 ridges of the original cube. The Rad has two types of vertices: the 8 vertices of the original cube (3-shot) and the 6 vertices "pulled" from the center of the faces (4-shot). Edges of the original cube are now diagonals of rhombi and are no longer boundaries of the figure.
What happens if we perform the same operation in 4D? The Tes has 16 vertices and 8 cells. We imagine that we grab the centers of these 8 cubic cells and stretch them outward to form a cubic pyramid with each cell. When two adjacent square pyramids are stretched until they are corealmic, the two inverted square pyramids form an octahedron.
Since a Tes has 24 (two-dimensional) faces, a total of 24 Oct's can be formed where the faces were. And we can prove geometrically that the Oct's are in fact regular! All the faces of the original Tes disappear, leaving hollow square structures within the Oct's. There are two types of vertices: the 16 vertices of the original Tes (4+4=8-shot) + the 8 vertices that have been "pulled out" from the center of the cells (8-shot).
Therefore, the higher-dimensional version of the Rad successfully "promotes" to a regular polychoron! What a miracle! I understand this phenomenon this way: since the diagonal of the Tes is an integer 2, the lengths of diagonal edges and the original axial edges are equal.
Just as the Rad has a dihedral angle of 120° and is a 3D space-filling shape, the Ico has a dichoral angle of 120° and fills 4D space.
Hi = Regular hecatonicosachoron
Made up of 120 Doe's. Three Doe's gather on each side.
Beginners of 4D geometry tend to think of the final regular in 4D, 600-cell, as the most fascinating & daunting. However, in terms of numbers of vertices, 600-cell has only 120, but this 120-cell has 600. Therefore, I may say that 120-cell is the hardest to comprehend.
It has the most vertices among the regular polychora, 600. Not only it's the largest, it "contains" all other regulars in 4D. That is, if we choose & connect these vertices correctly, we can make all five other types. It even contains 5 Ex's, which is in the next section, and 120 Pen's, which is not contained in any other regular 4D shape. In 3D, a Doe contains cubes and Tet's, but the "capacity" of the Hi is much higher than that of the Doe. Perhaps because of this property, the entire universe is said to be in the shape of the Hi.
Ten Doe's are on the equator, forming a ring which looks like a necklace in a projection. Thus, the dichoral angle is equivalent to the interior angle of a regular decagon, 144°.
If we consider the whole as a glome (hypersphere), 120 Doe's are arranged in the following order: 1 at the North Pole, 12 at the Arctic Circle, 20 at mid-latitudes, 12 at the Tropic of Cancer, 30 at the equator, and so on. In fact, the "equator" has the different meaning from in the previous section. Here is a clue to understanding the structure of the glome: I will tell you more about it in another article.
Ex = Regular hexacosichoron
It has the most cells, 600 Tet's. Five Tet's around a single edge make the angle close to 360°, so it takes this many to cover the center. At each vertex, 12 lines and 20 Tet's form an Ike.
We may obtain this shape by wedgeing 120 Ike's together instead of 600 Tet's. This is a type of star polychoron. It is similar to the relationship between the Ike and the great dodecahedron (Gad).
There is a way to construct an Ex based on an Ico. First, separate each edge of Oct cells into golden sections, which are then rearranged into an Ike. Then, attach an Ike pyramid to each Ike to form a total of 480 Tet's. Finally, fill the gaps with 120 Tet's.
From the icosahedron, we can remove at most three pentagonal pyramids, resulting in a polyhedron consisting of regular triangles and pentagons. Similarly, by removing Ike-pyramids from an Ex, a polychoron consisting of Tet's and Ike's can be formed. There are more than 300 million different kinds of them.
Entire list
The Pen and the Ico are duals of themselves, while the Tes and the Hex as well as the Hi and the Ex are duals of each other. The Schläfli symbols are reversed for each pair.
Surprisingly, four out of six have integers as dichoral angles.
Symbol: Schläfli symbol. Angle: dichoral angle. C,V,E,F: # of cells, vertices, edges, faces. Shot: # of lines touch a vertex. Lobed: # of faces touch a vertex. Hung: # of cells touch a vertex. Piece: # of faces touch an edge. Chambered: # of cells touch an edge. Fold: # of cells touch a face (always 2 in 4D).
