# Q.E.D.

58 Visualisation Take one of the Platonic solids, choose a point outside but very close to the center of one of the faces and from this point project the skeleton of the solid onto the face. The resulting 2d image contains most of the information about the solid. By performing the analogous operation for the regular 4d polytopes, we arrive at the following 3d images. Note that the first three images above generalise to those below. The projections of the 120 and 600cell are too complex to be reproduced here. Construction Constructing the nd simplex, orthoplex and cube is easy, and we can make the other regular polytopes from these standard ones. On page 41 we constructed the icosahedron from the octahedron and a dodecahedron can be inscribed as a dual in the icosahedron below, right such that the midpoints of the faces of the second are the vertices of the first. This gives all the Platonic solids. The midpoints of the faces of a 4d cube are the vertices of a 24cell. By dissecting its octahedral cells as described on page 41, we get 96 points and 24 icosahedra. For each such icosahedron there is a point in 4d at the same distance from all its vertices as the icosahedral edge length. Then the 24 points corresponding to the icosahedra plus the 96 points are the vertices of a 600cell. Finally, a 120cell can be inscribed as a dual into the 600cell the midpoints of the cells of the second are the vertices of the first.