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 regular 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 4D 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 projections
The vertex-first orthographic projection of the 24-cell has a rhombic dodecahedral envelope. The following image shows this projection, with hidden cells omitted.
The nearest vertex is at the center of the envelope where the 8 internal edges meet, shown here in yellow. It 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, as shown in the images below.
These are the ‘northern cells’.
Each of the 12 faces of the rhombic dodecahedron in the orthographic projection corresponds with an octahedral cell lying at the “equator” of the 24-cell. They are shown in the following images:
They appear flat because they are being viewed at from a 90° angle. On the other side of 24-cell, there another 6 cells arranged in cubic symmetry around the “south pole” vertex, mirroring the 6 northern. This makes a total of 24 cells.
The vertex-first perspective projection of the 24-cell has a similar structure, except that the 12 equatorial cells are hidden from view, and the faces of the rhombic dodecahedron aren't quite flat, making the shape of the projection envelope a tetrakis hexahedron. The following image shows this projection:
Cell-first projections
The cell-first parallel projection of the 24-cell has a cuboctahedral envelope, as shown in the following image:
The closest cell to the viewer lies in the center of the image, shown here in blue. Surrounding it are eight octahedra, shown in the following images:
These octahedra appear to be irregular, but this is just an artifact of the perspective projection. They are perfectly regular octahedra in 4D space.
At the vertices touching the central blue octahedron are another six octahedra, shown in the following images:
These cells lie on the limb of the 24-cell. They appear flattened into squares because they are being viewed at from a 90° angle.
Past these cells, on the far side of the 24-cell, lies another 9 cells mirroring the layout of the previous 9 cells. This makes a total of 24 octahedral cells.
The cell-first perspective projection of the 24-cell has a similar layout of cells, except that the 6 equatorial cells are out of view, and the faces of the cuboctahedral envelope are slightly curved into pyramids, making the shape of the envelope a tetrakis cuboctahedron. The following image shows this projection:
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.
