Origami & Folding
what one sheet can do
Concept

Quintic — where it appears

A polynomial equation of the fifth degree. It is the first degree beyond what one simultaneous fold settles, so it marks the boundary between the polygons a single fold constructs and the ones that need two folds made at the same instant.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

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.

construction · Multifold first pointsecond pointthe sixth axiom, at its full countthe first foldthe second foldthe third foldthree folds, each carrying one point onto one line and the other point onto the other — the most any single fold can offer

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.

construction · Multifold

Named alongside it

The objects these essays reach for when they reach for this one.

ConstructibilityCubicMultifoldEnumerationThe Huzita–Hatori axiomsOperation setTotient

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