The Coxeter plane

With both tilts at 0 the projection plane is exactly the Coxeter plane: the tesseract's 16 vertices fall into two regular octagons, the coral zigzag of edges (the Petrie polygon) becomes the outer octagon, and the shadow lands back on itself at every 45° turn. Drag α or β and the plane tilts away, the regular polygons scatter, and the rotational symmetry of the vertices, measured in the corner, drops, usually to 2-fold, a half turn. Try the others: the cube gives a regular hexagon, the 5-cell a regular pentagon, the 24-cell two regular dodecagons, the 600-cell four rings of 30, and even the 120-cell's 600 vertices fall into groups of 30, with thirtyfold symmetry. Reflect once in each of its basic mirrors, in turn (three for the cube, four in four dimensions): that makes one move, and h moves bring every vertex back; in the Coxeter plane that move is a turn of exactly 360°/h. For the four-dimensional ones, drag both tilts to 90°: that is the other plane the move turns within, and the shadow is h-fold again, but looks different.