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A polychoron, also known as a 4-polytope, is a four-dimensional shape. There are six regular convex 4-polytopes, the 5-Cell, 8-Cell, 16-Cell, 24-Cell, 120-Cell, and 600-Cell. You can learn more about them here: https://en.wikipedia.org/wiki/Regular_4-polytope
4-polytopes cannot be seen in three-dimensional space due to their extra dimension, but there are several techniques that can be used to “project” 4-polytopes into three-dimensional space. These studies use the cell-centered stereographic projection method, chosen for the interesting aesthetic appearance.
I've posted my three favourite here, the 24-Cell, 120-Cell, and 600-Cell. In each study, select cells have been removed to allow maximal viewing of the various cells, including the center cell. In the 120-cell and 600-cell, the outer cells have been removed to allow better visualization of the inner cells, and because their relative size is so large compared to the remaining cells.
While these three-dimensional shapes have curved faces and edges of various lengths, it is interesting to imagine that in four-dimensional space, the faces are flat, and the edges are straight and identical in length.
The 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube.
The 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.
The 600-cell is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell.
