Origami & Folding
what one sheet can do
Generator

Why the difference is two — the maekawa reason generator

A generator in the flat-folding library, called 11 times across 7 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.

maekawa-reason 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

Why the difference is twoThe cross-section of a flat-folded vertex is a closed path that turns through exactly one full circle. Every mountain turns it one way by half a turn and every valley the other, so the counts must differ by exactly two — which is Maekawa's theorem, and it is a statement about winding rather than about paper.MVMwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

assignment: "MVMM"

Why the difference is twoThe cross-section of a flat-folded vertex is a closed path that turns through exactly one full circle. Every mountain turns it one way by half a turn and every valley the other, so the counts must differ by exactly two — which is Maekawa's theorem, and it is a statement about winding rather than about paper.MVMwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

assignment: "MMVM"

Why the difference is twoThe cross-section of a flat-folded vertex is a closed path that turns through exactly one full circle. Every mountain turns it one way by half a turn and every valley the other, so the counts must differ by exactly two — which is Maekawa's theorem, and it is a statement about winding rather than about paper.MMVwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

assignment: "MVVV"

Why the difference is twoThe cross-section of a flat-folded vertex is a closed path that turns through exactly one full circle. Every mountain turns it one way by half a turn and every valley the other, so the counts must differ by exactly two — which is Maekawa's theorem, and it is a statement about winding rather than about paper.MVVwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

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 ring and a line

A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

A sheet that routes itself

DNA origami folds one long strand into a shape by holding it against itself with a few hundred short ones. There is no sheet and no crease — what has to be designed is a route — and the first thing that can go wrong is a counting argument crease patterns already know under another name.

The loop a vertex cannot close

A crease pattern's letters can contradict themselves, and the contradiction is never local. Enumerate every mountain-valley labelling of a single interior vertex at degree four, six and eight — a hundred and fifty pass every condition the subject has — and not one of them sends its panels round in a circle. The one labelling that would is refused by Maekawa, alone: Kawasaki holds on it and so does the big-little-big lemma.

The sheet has two sides

Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.

Thickness has a sign

Swap every mountain for a valley and back again. Kawasaki does not notice, Maekawa gets the same condition the other way round, the lemma still asks the two creases to differ, and the layers come out mirrored. Every theorem on this site is blind to which side of the paper it is looking at — and a hinge in a panel with depth is not. The fold closes one way and jams at nothing at all the other.

Two creases that cross

A crossing is four creases at a point, so the four conditions of the subject apply to it — and three of them can be satisfied. It is developable at every angle, it satisfies the big-little-big lemma whenever its two lines carry different letters, and it satisfies Kawasaki's condition when the lines meet squarely. Maekawa's refuses it always, at every angle and under every lettering, because a crossing's four spokes belong to two creases and can only be four and none, two and two, or none and four.

Why the difference is two

Maekawa's theorem says mountains and valleys differ by exactly two at every flat-foldable vertex. The constant is not empirical — it is a full turn, and the theorem is about winding rather than about paper.

Every generator · The flat-folding field · The patterns a reader can fold