Origami & Folding
what one sheet can do
Generator

Which polygons need more than one fold at a time

A generator in the axioms and construction library, called 9 times across 3 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

multifold-reach is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

Which polygons need more than one fold at a timeFor every regular polygon up to twenty-four sides: the totient of its side count, that number's prime factors, and which tool reaches it. A compass needs the factors to be twos, a single fold allows threes as well, and two folds at once allow fives.nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

upTo: 34

Which polygons need more than one fold at a timeFor every regular polygon up to twenty-four sides: the totient of its side count, that number's prime factors, and which tool reaches it. A compass needs the factors to be twos, a single fold allows threes as well, and two folds at once allow fives.nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 225202 · 2 · 526122 · 2 · 327182 · 3 · 328122 · 2 · 329282 · 2 · 73082 · 2 · 231302 · 3 · 532162 · 2 · 2 · 233202 · 2 · 534162 · 2 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

upTo: 28

Which polygons need more than one fold at a timeFor every regular polygon up to twenty-four sides: the totient of its side count, that number's prime factors, and which tool reaches it. A compass needs the factors to be twos, a single fold allows threes as well, and two folds at once allow fives.nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 225202 · 2 · 526122 · 2 · 327182 · 3 · 328122 · 2 · 3the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

upTo: 24

Which polygons need more than one fold at a timeFor every regular polygon up to twenty-four sides: the totient of its side count, that number's prime factors, and which tool reaches it. A compass needs the factors to be twos, a single fold allows threes as well, and two folds at once allow fives.nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

A crease that does not exist yet

Simultaneous folding is usually described as a problem of dexterity — several coincidences to be achieved in the same instant. The reference graph says otherwise: of the hundred and five two-fold operations, twenty-eight need no simultaneity and forty-nine can be done in an order, leaving twenty-eight whose folds each name the other. Those are not hard to hold. They are hard to know, and a loop that guesses and re-solves finds them at eight per cent of the error a pass.

Counting operations is not counting power

The catalogue of simultaneous-fold operations runs from seven to nearly ten million between one fold and five. What a construction can reach does not: each fold admits at most three lines, because two parabolas have three proper common tangents and not four, so m folds admit at most three to the m — and the largest polynomial degree they actually settle is smaller again, at twice m plus one. Three counts of the same subject, growing at three speeds.

Two creases at once

The seven axioms describe what one fold can do, and the restriction to one fold is a rule somebody imposed rather than a property of paper. Allow two creases to be made simultaneously and the reachable degree rises — and the hendecagon nobody could fold becomes foldable.

Every generator · The axioms and construction field · The patterns a reader can fold