Slicing a polytope

Drag the slider and our space sweeps through the polytope from one end to the other; the picture shows the slice at that moment. The small picture on the left looks at it edge on: a gold line crosses the polytope's shadow, and that line is our space; the edges that cross it are gold too, and every vertex of the slice comes from one of them. When the tesseract comes through vertex first, at first contact the slice is a single point; farther in, a regular tetrahedron grows, its four corners are cut off, and halfway through the slice is a regular octahedron. Cell first, it arrives as a whole cube and stays a cube all the way. Whenever the slice is a solid, V − E + F is 2. In the corner the volumes of the slices, each times its thickness, are added up; once the sweep is over the total is the same whichever way it went, and it is the polytope's volume in four dimensions. Send the 120-cell through cell first, and a quarter of the way in, the slice is a football.