Imagine the hyperbolic 24-cell sitting in the upper half-space model of hyperbolic 4-space. We can assume that one of its ideal vertices is the point at infinity. The small balls in slide 1 shows the intersections of the 0-skeleton with the flat 3-space at infinity. Notice slide 1 shows 23 vertices. Vertex 24 is the point at infinity, so it isn't shown. Now imagine projecting the 1-skeleton of the 24-cell down vertically in the upper half-space to the 3-space at infinity. The geodesic edges project to straight lines in Euclidean 3-space. Slide 1 already shows 24 of these edges. Slide 2 through 7 add progressively more of the projected edges to the picture. Slide 7 shows the complete projection of the 1-skeleton of the 24-cell vertically down to the flat 3-space at infinity. One could count that there are 112 edges in slide 7. There are, in fact, 120 edges in the 120-cell. The remain 8 edges where vertical in the upper half-space, and have projected to the eight red balls of slide 7.
Slides 8 through 29 show parts of the projected 2-skeleton. Each triangle of the 2-skeleton projects vertically down to an honest Euclidean triangle at infinity. The slides progress around the 24-cell showing projections of the 24 octahedra forming the walls of the 24-cell. Note that 12 of the 24-cell's 2-cells are vertical in upper half-space, and project down to edges in the pictures. These edges form the outer cube. The 24-cell has 120 2-cells, and thus the pictures show only 118 of them.
These pictures were made by Peter Storm in 2007 using Mathematica and POV-Ray. The slideshow is written in JavaScript.