The smallest sector decides — the big-little-big lemma generator
big-little-big 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
sectors: [50, 120, 130, 60]
sectors: [40, 120, 140, 60]
sectors: [18, 150, 162, 30]
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.
- the sectors close, satisfy Kawasaki, and contain a strictly smallest one — so the lemma has a case to forbid rather than a vacuous one ×1
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 region with no lettering
One turn angle at which a tessellation patch has no consistent lettering was found by sweeping a dial. Sweeping two dials finds nine patches with none, across three tilings, filling a corner of the parameter space — and never touching the square tiling, whose sectors have no sixty degrees to cross.
A tie is not a decision
The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.
How little the conditions decide
Local is not global is a statement about sufficiency: every vertex can pass and the sheet still fail. There is a sharper complaint available, and it is about strength. Fix one crease of a tessellation and propagate every condition the subject has to a fixed point: three creases out of a hundred and fifty-eight follow, and sixty-six vertices are still holding more than one answer.
Local is not global
Every vertex can satisfy every condition and the sheet still not fold. Deciding whether a whole crease pattern folds flat is NP-hard, which means no figure will settle it and no algorithm will scale.
The dial that decides nothing
Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.
The lengths are free
Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.
The order decides the count
Ask how many mountain-and-valley letterings a vertex admits and the answer looks as though it should depend on the angles. It does not. Three of the four conditions never see an angle at all, and the fourth asks only which sector is smallest — so the count is a function of a combinatorial arrangement, and a walk round the cycle that never looks at a vertex reproduces it exactly.
The smallest sector decides
Two assignments can satisfy both flat-folding theorems and only one of them folds. What separates them is a condition about the smallest angle, and it is the first rule in the subject that is not about counting.
The vertices nobody checks
The four conditions of flat folding are stated at an interior vertex, and the word interior carries the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, none of the four conditions says anything about them, and the condition that does decide them — a one-dimensional crease pattern, folded or refused exactly — was in hand all along.
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.
Where the lemma says nothing
The big-little-big lemma asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly. At a vertex whose two smallest sectors are equal the lemma has no opinion at all — and those are the vertices origami actually uses. The count of markings the conditions admit doubles, discontinuously, at exactly the angles everybody folds.
All the figures · The flat-folding field · The patterns a reader can fold