Matpic
mathematics, in pictures
Ladder

Scissors congruence — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

    Equal area is enough, and equal volume is not

    Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

    rung 1 · computation
  2. A dissection that never comes apart. The three pieces of the triangle-to-rectangle dissection drawn at 4 moments of the swing. Each top piece turns about a pin at the end of the slice it stands on, and the pieces stay joined throughout.

    A dissection that never comes apart

    The plane theorem lets the pieces be picked up and put down anywhere. Require instead that they stay joined at their corners and swing, and the theorem survives — which was open for a century and is a much stronger statement about the same cuts.

    rung 2 · computation
  3. The quantity a cut cannot change and a turn can. 4 polygons, each with the spikes of its translation invariant drawn round a dial: the length of the edges facing each direction, less the length of those facing the opposite way. It vanishes everywhere for 3 of them.

    Slid, but never turned

    Every construction on this ladder turns its pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.

    rung 3 · computation
  4. What the chain costs on a 6-gon: 39 pieces. A regular 6-gon fanned into 4 triangles, each with the three cuts that turn it into a rectangle, beside the running count of the pieces the whole chain produces — 39 of them.

    Finitely many, and nobody says how many

    The theorem promises a dissection exists and the proof produces one. Running the proof on a hexagon produces thirty-nine pieces, ingenuity produces five, and there is no method for proving that five cannot be four.

    rung 4 · computation

All ladders