Origami & Folding
what one sheet can do
Generator

States the motion never reaches

A generator in the rigid folding library, called 6 times across 2 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

reachable-states is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

States the motion never reachesFor five degree-four vertices: how many mountain-and-valley assignments satisfy every local flat-folding condition, and how many of those a continuous rigid motion actually arrives at. The two numbers are computed by machinery that shares no code, and where they differ there is a folded state that exists and cannot be got to without bending a panel.60° / 90°all 4 reached30° / 120°all 4 reached45° / 45°2 of 8 reached50° / 70°all 4 reached80° / 55°all 4 reachedsectorseach square is one assignment the theorems allowfilled — a rigid motion arrives there · open — a flat state with no path to itthe gap opens where two sectors are equal, and nowhere else on this listbig-little-big has nothing to forbid there — the linkage still does

cases: [60,90, 30,120, 45,45, 50,70, 80,55]

States the motion never reachesFor five degree-four vertices: how many mountain-and-valley assignments satisfy every local flat-folding condition, and how many of those a continuous rigid motion actually arrives at. The two numbers are computed by machinery that shares no code, and where they differ there is a folded state that exists and cannot be got to without bending a panel.60° / 90°all 4 reached30° / 120°all 4 reached45° / 45°2 of 8 reached50° / 70°all 4 reached80° / 55°all 4 reachedsectorseach square is one assignment the theorems allowfilled — a rigid motion arrives there · open — a flat state with no path to itthe gap opens where two sectors are equal, and nowhere else on this listbig-little-big has nothing to forbid there — the linkage still does

cases: [60,90, 45,45]

States the motion never reachesFor five degree-four vertices: how many mountain-and-valley assignments satisfy every local flat-folding condition, and how many of those a continuous rigid motion actually arrives at. The two numbers are computed by machinery that shares no code, and where they differ there is a folded state that exists and cannot be got to without bending a panel.60° / 90°all 4 reached45° / 45°2 of 8 reachedsectorseach square is one assignment the theorems allowfilled — a rigid motion arrives there · open — a flat state with no path to itthe gap opens where two sectors are equal, and nowhere else on this listbig-little-big has nothing to forbid there — the linkage still does

cases: [30,120, 80,55, 45,45]

States the motion never reachesFor five degree-four vertices: how many mountain-and-valley assignments satisfy every local flat-folding condition, and how many of those a continuous rigid motion actually arrives at. The two numbers are computed by machinery that shares no code, and where they differ there is a folded state that exists and cannot be got to without bending a panel.30° / 120°all 4 reached80° / 55°all 4 reached45° / 45°2 of 8 reachedsectorseach square is one assignment the theorems allowfilled — a rigid motion arrives there · open — a flat state with no path to itthe gap opens where two sectors are equal, and nowhere else on this listbig-little-big has nothing to forbid there — the linkage still does

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A state no motion reaches

Flat-foldability asks whether a folded state exists. Rigid-foldability asks whether there is a path to it. The two sets are different, and the difference can be counted on a single vertex.

Opening with nothing to pull

A leaf is not opened by a hand, a hinge or a motor. It opens because it keeps growing — which puts a condition on the pattern that no folder ever has to satisfy, because a person can always push.

All the figures · The rigid folding field · The patterns a reader can fold