Origami & Folding
what one sheet can do
Series

Constructible polygons — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. compassfolding3242546276the first gap8496104111012413121461581681716186191820821122210232224825202612n across the top, φ(n) underneathcompass: φ(n) a power of two — Gauss, and Wantzel's proof that nothing else worksfolding: φ(n) with no factor above three, because one fold solves a cubic

    The heptagon a compass cannot reach

    Which regular polygons a tool can build is a condition on a single number. The compass needs it to be a power of two; a fold needs only that it has no factor above three — and seven is the first place the two answers differ.

    part 1 · construction
  2. pp − 1, factoredcompassfoldingthe regular p-gon32both tools52 · 2both tools72 · 3folding only112 · 5neither132 · 2 · 3folding only172 · 2 · 2 · 2both tools192 · 3 · 3folding only232 · 11neither292 · 2 · 7neither312 · 3 · 5neither372 · 2 · 3 · 3folding only11 is the first prime out of a fold's reach — 11 − 1 = 2 · 5the factor of five is the obstruction, and no arrangement of folds produces onea compass needs a power of two; a fold needs nothing above three

    The eleven-sided one nobody can fold

    Folding reaches the heptagon, which a compass cannot. It does not reach the hendecagon, and the obstruction is a single prime factor: ten has a five in it, a fold solves cubics, and no arrangement of cubics produces a five.

    part 2 · construction
  3. 050100150200250300350400020406080100120140160sides, up to npolygons reachablea fold — 155a compass — 40both sets computed by division to 400 · 4 Fermat primes and 13 Pierpont primes below it

    How many polygons a fold reaches

    The heptagon is what the extra axiom buys and it is one polygon. What it actually buys is a density: to a thousand sides a compass reaches fifty-two regular polygons and a fold reaches two hundred and seventy-five, and the ratio between them is still widening. The compass has five usable primes in the whole of arithmetic and may use each once; a fold keeps acquiring new ones and may repeat the factor of three as often as it likes.

    part 3 · construction
  4. how squarely the folds that fix a reference crossand what a crossing at that angle does to an error in the foldingthe four linear axioms565 references · worst 36.9°with the conic axiom16,890 references · worst 6.3°60° to 90°error × 1.264.6%54.4%45° to 60°error × 1.418.2%21.8%30° to 45°error × 2.017.2%17.1%15° to 30°error × 3.90.0%5.2%8° to 15°error × 7.20.0%1.2%under 8°error × 14.30.0%0.3%the linear axioms bottom out at 36.9° — a multiplier of 1.67 — and the conic axiom reaches 6.3°, a multiplier of 9.12 rounds on a square · a reference priced at 1 ⁄ sin of the widest angle its own folds make · shares, so the two sets are comparable

    What buys the reach costs the accuracy

    A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.

    part 4 · construction

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