Origami & Folding
what one sheet can do
Generator

How many pieces each printed pattern's letterings fall into

A generator in the flat-folding library, called 26 times across 4 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.

folding-pieces 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

How many pieces each printed pattern's letterings fall intoFor every crease pattern this site prints at true scale: the creases with an interior vertex at each end, and the number of mutually unreachable pieces that predicts. Two of the eight have none, and the Yoshimura has forty-eight.the bar is the creases with an interior vertex at each enda folder holding one of these patterns is in one piece of the count on the right, and cannot leave it107 of 862 moves survive across the shelf · 0 touch a buried creaseThe preliminary base0 buried · 1 piecesThe Miura fold22 buried · 4,194,304 piecesThe square twist4 buried · 16 piecesThe hexagon twist6 buried · 64 piecesThe Yoshimura pattern48 buried · 2.81 × 10^14 piecesFold and cut — the triangle0 buried · 1 piecesThe tapered corrugation27 buried · 1.34 × 10^8 piecesThe waterbomb tessellation42 buried · 4.39 × 10^12 pieces

view: "cut", pattern: "square-twist"

What one cut does to the piecesOne crease of a printed pattern cut — no paper removed, the two panels simply no longer joined — and the number of mutually unreachable pieces its letterings fall into, before and after. A cut is not local: it un-buries creases it was not made along.The square twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut16 pieces · 0 vertices releasedcutting a crease that reaches the edge4 pieces · 1 vertices releasedcutting a buried crease2 pieces · 2 vertices released

view: "cut", pattern: "hexagon-twist"

What one cut does to the piecesOne crease of a printed pattern cut — no paper removed, the two panels simply no longer joined — and the number of mutually unreachable pieces its letterings fall into, before and after. A cut is not local: it un-buries creases it was not made along.The hexagon twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut64 pieces · 0 vertices releasedcutting a crease that reaches the edge16 pieces · 1 vertices releasedcutting a buried crease8 pieces · 2 vertices released

view: "shelf", trials: 24

How many pieces each printed pattern's letterings fall intoFor every crease pattern this site prints at true scale: the creases with an interior vertex at each end, and the number of mutually unreachable pieces that predicts. Two of the eight have none, and the Yoshimura has forty-eight.the bar is the creases with an interior vertex at each enda folder holding one of these patterns is in one piece of the count on the right, and cannot leave it107 of 862 moves survive across the shelf · 0 touch a buried creaseThe preliminary base0 buried · 1 piecesThe Miura fold22 buried · 4,194,304 piecesThe square twist4 buried · 16 piecesThe hexagon twist6 buried · 64 piecesThe Yoshimura pattern48 buried · 2.81 × 10^14 piecesFold and cut — the triangle0 buried · 1 piecesThe tapered corrugation27 buried · 1.34 × 10^8 piecesThe waterbomb tessellation42 buried · 4.39 × 10^12 pieces

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 cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

The decision a crumple has taken

A sheet crumpled at random satisfies every condition in the subject, because it just folded. It also wrote itself a lettering — one of very many the pattern admits — and it is now in a piece of that space it cannot leave: at six folds a crumpled sheet carries thirty-nine buried creases, which is half a million million million pieces, and every change it admits stays inside one of them.

The pieces without the list

The letterings a pattern folds in fall into pieces no folder can cross, and the count was found by writing every lettering down — which stops at eighteen creases. The Miura has thirty-eight, the Yoshimura eighty-six, and the number of pieces can be read off the drawing without listing anything: four million and two hundred and eighty-one million million.

Thirty-two rules, thirty-two pieces

The waterbomb tessellation's surviving repeating rules fold to one object — same panels, same places, same areas. Put them in the space of letterings the patch admits and they occupy thirty-two different pieces of a quarter of a million, so no two of them can be reached from one another without unfolding the sheet.

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