The 24-Cell
The 24-cell is a very unique object with special properties that is matched by no other polytope in any dimension.
It is the only self-dual polytope that isn't also a simplex, besides regular polygons in 2D. A self-dual polytope is one where if you interchange its facets and its vertices, you get the same polytope. In all other dimensions above 2D, the only self-dual polytopes are the n-simplices: the tetrahedron, 5-cell, etc.. (All regular polygons in 2D are self-dual, so this property is not distinguishing in 2D.)
It is the only regular space-tiling polytope that isn't a hypercube. This means that you can stack 24-cells together to tile a 4D region of space with no gaps in between. The hypercubes have this property: squares tile 2D space, cubes tile 3D space, and 8-cells tile 4D space. In 3D, the rhombic dodecahedron also has this property, but it is not a regular polytope.
Its closest equivalents in 3D are the rhombic dodecahedron and the cuboctahedron. However, unlike the rhombic dodecahedron which has faces that are not regular polygons, and the cuboctahedron which has two different types of regular polygons for its faces, the cells of the 24-cell are all identical, regular octahedra.
Vertex-first projection
The vertex-first orthographic projection of the 24-cell has a rhombic dodecahedral envelope. The vertex is at the center of the envelope, and may be thought of as the ‘north pole’ of the 24-cell. There are 6 octahedra (slightly flattened by the projection) packed around it in a cubic symmetry. These are the ‘northern cells’. Each of the 12 faces of the rhombic dodecahedron corresponds with an octahedral cell viewed at from a 90-degree angle, hence they appear flat. These are the ‘equitorial cells’. On the other ’side‘ of these equitorial cells are another 6 cells arranged in cubic symmetry around the ‘south pole’ vertex. These 6 cells would be hidden from view if a 4D viewer were looking at the 24-cell from this viewpoint. This makes a total of 24 cells.
The vertex-first perspective projection of the 24-cell has the same structure as described for the orthographic projection, except that due to foreshortening, its equitorial cells are not completely flat and so its envelope is a tetrakis hexahedron. The following diagram shows this perspective projection, with hidden cells omitted.
The blue edges lie inside the envelope, and meet at the “north pole” vertex.
Cell-first projection
The cell-first parallel projection of the 24-cell has a cuboctahedral envelope. The closest cell to the viewer lies in the center of this cuboctahedron, and its vertices touch the centers of the square faces of the cuboctahedron. Each triangular face of the central octahedron is in dual orientation with the triangular face of the cuboctahedron in the same octant. These pairs of triangles are actually opposite faces of octahedral cells, which appear in projection to be distorted to exactly fill the space between the central octahedron and the cuboctahedral envelope. There are 8 such pairs.
Furthermore, each square face of the cuboctahedron actually corresponds with an octahedral cell viewed at from a perpendicular angle, hence they appear as flat squares in projection. These 6 squares connect the outer half of the 24-cell containing the central octahedron and the 8 surrounding cells with another central octahedron and its 8 surrounding cells, which lie on the other side of the 24-cell in the 4th direction. Hence, there are 9 + 6 + 9 = 24 cells in total. These other 9 cells are hidden from view to a 4D viewer looking at the 24-cell from this viewpoint.
The cell-first perspective projection of the 24-cell has an envelope in the shape of a modified cuboctahedron, where the square faces are replaced by shallow pyramids. The layout of cells within the envelope are similar to that in the parallel projection. The following diagram shows this perspective projection, with hidden cells omitted:
The central cell is highlighted in blue. This is the cell closest to the 4D viewer.
Construction
One possible way to construct the 24-cell is by a process analogous to the construction of the rhombic dodecahedron in 3D. That is, by cutting the 4D hypercube into 8 cubical pyramids and attaching them to the faces of a second 4D hypercube. This construction is easiest to understand if one compares the cell-first projection of the hypercube and the vertex-first projections of the 24-cell.
Another construction is via the rectification of the 16-cell. Rectification is to cut off the vertices of a polytope such that the cutting hyperplane lies on the midpoints of the edges meeting at the vertex. This construction is easiest to understand if one compares the vertex-first projection of the 16-cell and the cell-first projection of the 24-cell.
