Sector angles — where it appears
Named by 22 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
Which vertices are the random ones
Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.
Walking between two foldings
The letterings a vertex folds in are always counted and never navigated. Counting says a generic degree-six vertex has eight of them; navigating says that changing any two creases turns any one into any other, and that changing two neighbouring creases does not — and that the vertices which come apart are the ones with no coincidences in them, which is the opposite of what every other measurement here would suggest.
The other grid
Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.
The cost is in the coincidences
How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.
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.
A leaf ends its pattern
A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.
Where a sector crosses sixty
Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.
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.
The dial and the tiling that is not alike
Four of the five tilings a twist tessellation can be built on behave identically under every dial the construction has. The fifth has two kinds of vertex, and everything about it is different: it is the only one whose search has a tail, the only one whose shallow patches take minutes to draw, and the only one where a distance has to be solved rather than assumed.
One step per panel is a table size
Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.
Six creases and the same straight line
The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.
A knife edge nine decimals wide
Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.
The most decided vertex here
Sixteen ways to letter four creases; Maekawa allows eight; the big-little-big lemma allows four. A twist polygon's corner is one of the few vertices in this collection where the second cut applies, so it keeps four labellings where a grid, a leaf, a Miura and a crumple all keep eight — and the family the collection long called difficult turns out to be the one whose conditions decide the most.
The designer's grid is the dearest thing here
Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.
A count is not a length
Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.
Parity is not enough
A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.
An alternating sum of angles
Kawasaki's condition says the sectors round a vertex alternate to a straight angle. A glued band has no vertices and obeys a condition of exactly the same shape: the crease angles have to alternate to a multiple of a straight angle. Two different quantities, two different sheets, one arithmetic — and in both cases what is being said is that a product of reflections came back the right way.
How rare a band that folds is
Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.
Named alongside it
The objects these essays reach for when they reach for this one.
The big-little-big lemmaInterior vertexAssignmentSearch costCrimpingFlat-foldabilityGenericityTwistBox pleatingClosureConstraint propagationCorrugation