Math
A 4D solid passing through
What a four-dimensional solid looks like from inside three-dimensional space: only its slice, and the slice changes as the solid moves through. A tesseract coming corner-first shows a tetrahedron that grows, loses its corners, becomes an octahedron and shrinks back; cell-first it shows one cube for the whole passage. The slices are built from the crossed cells, counted, measured, and their volume integrated over the passage is the 4-volume in every orientation - Cavalieri, one dimension up.
What happens
A flat creature watching a cube pass through its plane sees a square appear and vanish - or, corner-first, a point growing into a triangle, then a hexagon, then shrinking away. We are that creature one dimension up. A four-dimensional solid pushed through our space leaves a sequence of ordinary polyhedra, and the sequence depends on the direction: the tesseract cell-first is a cube that sits and goes; vertex-first it is a tetrahedron that truncates into an octahedron and back. Pick a solid and a direction, slide the hyperplane through, and count faces on the panel.
Equations used in the module
Equations the hyperplane w = c cuts every edge crossing it at t = w_a/(w_a − w_b) tesseract: 16 vertices (±1,±1,±1,±1), 32 edges, 8 cubes; 16-cell: 8 vertices, 24 edges, 16 tetrahedra cell-first: a cube; vertex-first: tetrahedron → truncated → octahedron → … every section is a closed polyhedron: V − E + F = 2
