The 24-Cell is regular figure which exists only in 4-space. That is, it has no analogous figures in any of the lower or higher spaces! This figure is bounded by 24 octahedral cells. It contains 96 regular triangular faces (each shared by 2 cells) and 96 edges (each shared by 3 cells) and 24 vertices (each shared by 6 cells). The image below displays a view of the 24-cell. The 2 smaller figures show the 2 projections of the octahedral cells found in the large figure. There are 12 of each projection.
It is interesting that each of the 2-D octahedral cell projections contain 2 regular triangular faces and a base which projects to a square. Actually all the bounding cells are regular in 4-space, but they are distorted when projected into 3-space or 2 space.
24-Cell Construction
There are several ways to construct a 24-cell. One of the easiest to visualize is to start with a Hypercube and inscribe an octahedron within each of the cube cells that enclose it. Looking at the figure to the left, the veritices of the green otahedron touch the centers of each face of the blue cube. If you draw a similar octahedra within all 8 cube cells and remove the original hypercube, a 24-cell is formed. 8 of the octahedral cells of the 24 cell are the ones that you drew and the remaining 16 are formed by the way the original 8 are joined.

Below is the resulting figure. It is somewhat enlarged approximately 2X.

This is the same figure as shown above, but it has been rotated in 4-space and the resulting 2-space projection is different. There are 8 of each octahedral projections in this view. Notice that there is an image of a Hypercube in this 2-space projection as well.
HyperspaceCross Polytope120-Cell
last revised: 9/28/02