Origami & Folding
what one sheet can do
Series

The axioms — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

    One fold at a time, and there are exactly seven of them

    A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.

    part 1 · construction
  2. hh/263°42.0°21.0°The foldone fold, made so that the cornerreaches the lower crease at the sameinstant as the point above it reachesthe ray — two conditions, one creaseWhat it produces63° divided into21.00° and 42.00°a third of 63° is 21.00°— measured off the fold, not drawnA cubic, so no compass reaches it.mountainvalley

    Folding beats the compass, by exactly one degree

    Straightedge and compass solve quadratics. A single fold solves cubics. That one-step difference settles two problems Greek geometry could not, and leaves a third exactly as impossible as it was.

    part 2 · construction
  3. axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

    Why the list stops at seven

    The seven axioms are not seven useful folds somebody collected. They are every fold there is, and the proof is an exercise in counting degrees of freedom that takes about a minute.

    part 3 · construction
  4. startend8x³ + 4x² − 4x − 1legs 1.000, 0.500, -0.500, -0.125each turn a right angle3 real rootsx = -0.900969x = -0.222521x = 0.623490each ray lands on the end to 1.1e-16the launch angle's negative tangentis the root — which is the folda right-angled bounce off two linesat once is exactly what one fold does

    Where the cubic comes from

    Folding solves cubics, and the usual explanation stops at the sixth axiom. The reason is older and better: a right-angled bounce along a path of coefficients is a root-finder, and one fold is exactly such a bounce.

    part 4 · construction
  5. distinct fold lines this axiom specifies and no other doesaxiomafter 4 pointsafter 9 pointsafter 565 pointsA1 — through two points08121054A2 — one point onto another08142649A3 — one line onto another4564994A4 — perpendicular through a point001661distinct lines in all1292274300the four operations name 38 folds at the first round and draw 12 lines with them

    What each axiom is worth

    The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.

    part 5 · construction
  6. the bar is how many folds the alignment names, averaged over the trialsa1 — the fold through two points1.000.0% none · 0.0% twoa2 — one point onto another1.000.0% none · 0.0% twoa3 — one line onto another2.000.0% none · 100.0% twoa4 — a perpendicular through a point1.000.0% none · 0.0% twoa5 — a point onto a line, through a point1.5223.8% none · 76.2% twoan alignment with no fold is not a failed construction; it is an alignment the paper cannot make

    An axiom may name no fold

    The seven operations are stated about points and lines in a plane, and a plane has no edges. On a square, three of them always name exactly one fold and always land it on the paper; placing a line on a line names two, and 13.3% of the folds it specifies are creases the sheet never reaches; and placing a point on a line through a second point names two folds, one, or — 23.8% of the time — none at all.

    part 6 · construction
  7. the axiom is satisfied by both, and says nothing about whichthe two are square to one another, alwaysthe thin lines are the two the axiom is given; the two heavy ones are its answers

    The axiom that names two folds

    Bring one line onto another and the operation is satisfied by either of two folds, always exactly perpendicular to one another, creasing the paper in completely different places. Over four thousand random line pairs on a square, both folds land on the sheet two thousand nine hundred and sixty-five times, and the statement of the axiom does not say which one is meant. The fifth axiom is worse: its two answers are at any angle at all, from half a degree apart to square.

    part 7 · construction
  8. axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

    Which of the seven survive

    The seven axioms are the complete list of ways one fold can be specified by aligning marked things. Every one of them names points and lines on a sheet, three of them quietly assume that a line has two sides, and on a closed sheet a line need not — so the list is complete for a disc and shorter for anything else.

    part 8 · construction

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