Origami & Folding
what one sheet can do
Series

Origami numbers — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the degree of the equation, and what it is made ofnumberdegreemade ofwhere it comes from½11a fold in half√222^1the diagonal of the squareφ22^1the silver rectangle's cousin∛233^1doubling the cube2 cos(2π/7)33^1the regular heptagon∜242^2a square root of a square root∛2 · √262^1 · 3^1a product of two of them2^(1/5)5not twos and threesa fifth root2 cos(2π/11)5not twos and threesthe regular hendecagonchecked by exhaustion: no number here satisfies a rational equation of lower degree with coefficients up to 6

    The numbers a fold reaches

    Folding solves cubics, which is one fact about one fold. The reason the subject has a theory rather than a bag of tricks is a second fact about all of them: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots. Constructions can therefore be built out of constructions — and no tower of them ever arrives at a fifth root.

    part 1 · construction
  2. degree is how far out the number is; height is what the construction costsnumberdegreesteps in the towerwhat the steps are½10 stepsa fold in half√221 stepthe diagonal of the squareφ21 stepthe silver rectangle's cousin∛231 stepdoubling the cube2 cos(2π/7)31 stepthe regular heptagon∜242 stepsa square root of a square root∛2 · √262 stepsa product of two of them2^(1/5)5no tower reaches ita fifth root2 cos(2π/11)5no tower reaches itthe regular hendecagon2^(1/8)83 stepsthree square roots2^(1/9)92 stepstwo cube roots2^(1/12)123 stepstwo squares and a cube2^(1/16)164 stepsfour square roots1 pairs invert: 2^(1/9) is of degree 9 and 2 steps, 2^(1/8) of degree 8 and 3a degree 2^a · 3^b is a steps of square root and b of cube root, so the height is a + b and nothing else

    Reachable is not cheap

    The closure is what makes folding a theory rather than a bag of tricks: constructions can be built out of constructions. What that also means is that constructions have lengths and the lengths compose, so every reachable number has a height as well as a degree — the number of extension steps the shortest tower to it must take. The two orderings disagree, and a ninth root is a shorter tower than an eighth.

    part 2 · construction
  3. what the plane specifies, and what the paper carries2 rounds from a bare sheet, every proportion at the same rulesheeton the paperlost to the edgesquare1 × 1.000565 references72%1,440 off the paperA-series1 × 1.414114,927 references47%102,162 off the paper3 : 21 × 1.500135,281 references48%123,872 off the paperdouble square1 × 2.00044,673 references59%65,294 off the paper2 rounds of a1, a2, a3, a4 · a crossing is a reference only where the paper is · square loses the largest share, at 72%

    The field has no edge

    Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.

    part 3 · construction
  4. every degree a fold reaches up to 200, as a square root count against a cube root counta point at (a, b) is the degree 2 to the a times 3 to the b, and a tower to it takes a + b steps0123456701234square rootscube roots13927812618541624123610882472164814432966419212825 of the first 200 degrees, and they are the lattice points under a line of slope minus log 2 over log 3the pale points are the degrees a compass reaches as well

    Twos and threes run out

    A fold reaches a number exactly when the degree of its equation is a product of twos and threes, which sounds like a large set because it is infinite and because it is so much larger than the compass's. Counted, the reachable degrees are the lattice points under a straight line, so there are about half a log-squared of them: twenty of the first hundred, a hundred and forty-two of the first million. The share falls from a fifth to one part in seven thousand, and the factor by which folding beats the compass rises at every decade without ever settling.

    part 4 · construction
  5. how many extension steps the shortest tower to each polygon takesa polygon of n sides needs the degree of two cosine of a turn over n, which is Euler's totient halved3 sides0degree 1 · square roots only, so a compass reaches it4 sides0degree 1 · square roots only, so a compass reaches it5 sides1degree 2 · square roots only, so a compass reaches it6 sides0degree 1 · square roots only, so a compass reaches it7 sides1degree 3 · 1 cube root8 sides1degree 2 · square roots only, so a compass reaches it9 sides1degree 3 · 1 cube root10 sides1degree 2 · square roots only, so a compass reaches it12 sides1degree 2 · square roots only, so a compass reaches it13 sides2degree 6 · 1 square root and 1 cube root14 sides1degree 3 · 1 cube root15 sides2degree 4 · square roots only, so a compass reaches it16 sides2degree 4 · square roots only, so a compass reaches it17 sides3degree 8 · square roots only, so a compass reaches it18 sides1degree 3 · 1 cube root19 sides2degree 9 · 2 cube roots20 sides2degree 4 · square roots only, so a compass reaches it21 sides2degree 6 · 1 square root and 1 cube root24 sides2degree 4 · square roots only, so a compass reaches it26 sides2degree 6 · 1 square root and 1 cube root27 sides2degree 9 · 2 cube roots28 sides2degree 6 · 1 square root and 1 cube root30 sides2degree 4 · square roots only, so a compass reaches it32 sides3degree 8 · square roots only, so a compass reaches it34 sides3degree 8 · square roots only, so a compass reaches it35 sides3degree 12 · 2 square roots and 1 cube root36 sides2degree 6 · 1 square root and 1 cube root37 sides3degree 18 · 1 square root and 2 cube roots38 sides2degree 9 · 2 cube roots39 sides3degree 12 · 2 square roots and 1 cube root40 sides3degree 8 · square roots only, so a compass reaches itthe pale bars are the polygons a compass reaches, and they are not the cheap ones — 4 of the one-step polygons need a cube root

    Gauss's polygon is the expensive one

    Which regular polygons a fold reaches is a condition on the factorisation of Euler's totient, and every polygon that passes it also has a height — the number of extension steps the shortest tower to it takes. Read that column instead of the verdict and the field inverts: the heptagon, which no compass reaches, costs one step; the seventeen-sided polygon that made Gauss famous costs three, the most on the list; and the polygons a compass finds easy are the ones a folder pays most for.

    part 5 · construction
  6. what each axiom set specifies, and what of it the sheet carrieson a unit square, from its four corners and four edgesaxiomshow manyroundsfold lineson the paperoff itlosta line through two points116500.0%a line through two points126500.0%and a point onto a point218900.0%and a point onto a point22321336432.5%and the other two linear ones411291257.1%and the other two linear ones42925651,44071.8%and a point onto a line through a point51283310475.9%and a point onto a line through a point52———past the capthe conic axiom loses three quarters of its crossings at one round, and cannot be run a second

    The axiom that reaches furthest wastes most

    The field of origami numbers is defined on an unbounded plane and a folder has a square. Counted axiom set by axiom set on the same sheet, the share of crossings that land off the paper rises with every axiom added: nothing at all from the first two, fifty-seven per cent from the four linear ones at a single round, and seventy-six per cent from the conic axiom at a single round — more, in one round, than the linear four lose in two. The instrument that reaches furthest into the field delivers the smallest share of what it specifies.

    part 6 · construction

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