Origami & Folding
what one sheet can do
Flat-folding

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.

Assumes The vertex the list does not have and Why the difference is two.

20 min read 8 figures The pattern is the objectFlat is rare

Ink does not say what it meant. Two segments drawn across one another and four segments drawn out from a point make the same mark on paper, and the difference between them is not visible: it is a fact about which lines the person drawing thought they were drawing. The first cannot fold and the second is the commonest vertex in the subject.

The reason the first cannot fold is worth having in full, because it is not one reason. Of the four conditions this collection asks at every interior vertex, three of them can be satisfied by a crossing. Only one refuses it every time, and which of the other three lets it through depends on the angle and on the letters.

What a crossing is, read as a vertexTwo creases drawn across one another, and the vertex the drawing has there. Its four sectors come in two equal pairs, so Kawasaki's two alternating sums are equal only when the lines are square to one another; and its four spokes belong to two creases, so the mountains and valleys can never differ by the two Maekawa's theorem asks for.the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge
Fig. 1 Two creases crossing at twenty-three degrees, read as the vertex the drawing has there. Its four sectors come in two equal pairs, so the two alternating sums Kawasaki compares are 45.8° and 314.2°; and its four spokes belong to two creases, so the letters can only be four and none, two and two, or none and four.

The first condition never notices

Developability asks whether the sectors around an interior vertex sum to a full turn — whether the paper there was cut or stretched. At a crossing they always do, because two straight lines through a point divide the plane into four sectors and four sectors around a point are a full turn by construction. Across every angle and every lettering swept here, thirty-two of thirty-two crossings are developable.

That is not a weakness of the condition. Developability is about whether the paper exists, and the paper at a crossing exists perfectly well: it is a flat sheet with two lines drawn on it. What has gone wrong is not in the material.

The second condition depends on the angle

Kawasaki’s condition asks the alternating sum of the sectors to vanish — equivalently, that the odd-numbered sectors and the even-numbered ones each come to a half turn. The condition is an equality, which is why a pattern satisfying it is a coincidence rather than a typical drawing.

At a crossing the four sectors are a, 180° − a, a, 180° − a. The two alternating sums are therefore 2a and 360° − 2a, and they are equal exactly when a is a right angle. So the condition reads, at a crossing, as a single sentence about the drawing: two creases may cross only if they cross squarely.

Swept over eight angles and four letterings each, four of the thirty-two satisfy it, and they are the four right-angled ones. At 23°, the two sums are 45.8° and 314.2° — not a near miss, a gross disagreement, and a near miss would be no better since the condition is an equation and not a tolerance.

Every lettering of a crossing, and none of them foldsA crossing at eight angles, with every way of lettering it. The bar is how many letterings satisfy both Kawasaki's condition and Maekawa's; the note says how many satisfy each on its own. Lettered as two ink lines the bar is empty everywhere, and lettering the spokes independently fills it only at the right angle — where the drawing is four creases meeting rather than two crossing.the bar is the letterings that satisfy both conditionseach ink line carries one letter along its whole length11°0 of 40 pass Kawasaki · 0 pass Maekawa23°0 of 40 pass Kawasaki · 0 pass Maekawa34°0 of 40 pass Kawasaki · 0 pass Maekawa46°0 of 40 pass Kawasaki · 0 pass Maekawa90°0 of 44 pass Kawasaki · 0 pass Maekawa109°0 of 40 pass Kawasaki · 0 pass Maekawa126°0 of 40 pass Kawasaki · 0 pass Maekawa149°0 of 40 pass Kawasaki · 0 pass MaekawaMaekawa refuses a crossing at every angle; Kawasaki refuses every angle but the right one
Fig. 2 Every way of lettering a crossing, at eight angles. The bar is how many letterings satisfy both Kawasaki’s condition and Maekawa’s; the note counts each separately. Kawasaki fills only at the right angle, Maekawa nowhere, and the bar is therefore empty everywhere.

The third condition depends on the letters

The big-little-big lemma is the condition that reads a sector against its neighbours: a sector strictly smaller than both of them must be bounded by two creases of opposite letter, or the panel it carries has nowhere to go.

At a crossing with a less than a right angle, the two sectors of size a are each strictly smaller than both neighbours, and each of them is bounded by one crease from each line. So the lemma reads as a sentence about the letters: the two lines must carry different letters. Where they do, it is satisfied — sixteen of sixteen. Where they carry the same letter, it fails — fourteen of sixteen, the exceptions being the two right-angled cases, where all four sectors are equal and no sector is strictly smallest, so the lemma has nothing to say.

The smallest sector decidesTwo assignments of the same four creases. Both satisfy Kawasaki and Maekawa. The left one folds; the right one does not, because the strictly smallest sector has the same assignment on both sides and the paper either side of it has nowhere to go.MVMM40°foldsopposite across the small sectorMMVM40°does not foldthe same on both sidesboth satisfy Kawasaki and Maekawa — the angles and the counts are identical
Fig. 3 The lemma on a vertex that can satisfy it: the smallest sector is strictly smaller than both its neighbours, so the two creases bounding it must carry opposite letters. At a crossing the same reading applies twice over — both small sectors are bounded by one crease of each line — which turns the lemma into a statement about whether the two lines are lettered alike.

That is worth pausing on, because it is the one place a crossing is a reasonable object. Two lines of different letters crossing obliquely satisfy three of the four conditions the subject checks, and satisfies them not marginally but exactly. Whatever refuses it is going to be the fourth.

The fourth condition refuses every crossing

Maekawa’s theorem says the mountains and valleys at a flat-foldable interior vertex differ by exactly two. The reason is a winding argument rather than a fact about paper: the folded cross-section turns through a full circle, and the turns are the creases.

A crossing cannot produce a difference of two, and the argument is a count rather than a construction. A drawn line is one crease and carries one letter along its length. The four spokes at a crossing are therefore two creases counted twice — two of one letter and two of the other, or four of one, or four of the other. The differences available are four, nought and four. Two is not among them.

That holds at every angle, including the right angle Kawasaki allows, and under every lettering, including the different-letter ones the lemma allows. Of the thirty-two crossings swept here, none satisfies Maekawa’s condition.

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
Fig. 4 Why the difference has to be two, drawn as the cross-section it comes from: three creases one way and one the other, and the walk closes. A crossing lettered as two lines gives two and two, or four and none — walks that turn and come back rather than closing, which is the same failure the count reports and which the figure refuses to draw.

The survivors are a different drawing

Something does fold with four creases at a point and two of them collinear, and the enumeration finds it as soon as the four spokes are allowed to carry letters independently rather than inheriting them from two lines.

Of the hundred and twenty-eight spoke letterings swept, eight fold. Every one of them is at the right angle — the only angle Kawasaki allows — and every one of them changes letter across the point: the spoke going left and the spoke going right are lettered differently, and so are the spoke going up and the spoke going down.

Lettering the four spokes instead, and what survivesA crossing at eight angles, with every way of lettering it. The bar is how many letterings satisfy both Kawasaki's condition and Maekawa's; the note says how many satisfy each on its own. Lettered as two ink lines the bar is empty everywhere, and lettering the spokes independently fills it only at the right angle — where the drawing is four creases meeting rather than two crossing.the bar is the letterings that satisfy both conditionseach of the four spokes is lettered on its own11°0 of 160 pass Kawasaki · 8 pass Maekawa23°0 of 160 pass Kawasaki · 8 pass Maekawa34°0 of 160 pass Kawasaki · 8 pass Maekawa46°0 of 160 pass Kawasaki · 8 pass Maekawa90°8 of 1616 pass Kawasaki · 8 pass Maekawa109°0 of 160 pass Kawasaki · 8 pass Maekawa126°0 of 160 pass Kawasaki · 8 pass Maekawa149°0 of 160 pass Kawasaki · 8 pass MaekawaMaekawa refuses a crossing at every angle; Kawasaki refuses every angle but the right one
Fig. 5 The same sweep with the four spokes lettered independently. The bar fills only at the right angle, and the letterings that fill it are exactly the ones whose letters change across the point — which is four creases meeting there, not two lines drawn over one another.

A crease whose letter changes partway along it is not one crease. It is two, and the point where they meet is a vertex — the vertex the drawing appeared to have all along, and which the ink never distinguished from a crossing. The centre of the preliminary base is that vertex eight times over: four straight lines through the middle of a square, and the assignment that works is five of one letter and three of the other, which no reading of it as four lines could produce.

So the sentence the subject never quite states is: a crossing folds only when it is not a crossing. Where the ink admits both readings, only one of them is a pattern, and the pattern is the one in which the letters change at the point.

The size of the failure

The four conditions return a bit each. There is a fifth reading that returns a number, and on a crossing the number is large.

Placing a folded state is a composition of reflections: step across a crease and the panel beyond carries the near panel’s motion followed by reflection in that crease. Go all the way round a vertex and the composition has to come back to where it started, or the paper cannot be put down. What comes back is a rotation by twice the alternating sum of the sectors — zero exactly when Kawasaki’s condition holds, which is why a vertex that satisfies it contributes nothing and a vertex that does not contributes everything.

At a crossing that rotation is 4a − 360°, doubled. Two creases crossing at 60° turn the paper through 240° on the way round; at 40° it is 400°; at 23°, 536°; at 11.5°, 628°. Only the right angle gives nought.

What a crossing is, read as a vertexTwo creases drawn across one another, and the vertex the drawing has there. Its four sectors come in two equal pairs, so Kawasaki's two alternating sums are equal only when the lines are square to one another; and its four spokes belong to two creases, so the mountains and valleys can never differ by the two Maekawa's theorem asks for.the vertex nobody listedthe two lines meet at 90.0°sectors 90.0° 90.0° 90.0° 90.0°alternating sums 180.0° and 180.0°Kawasaki holds — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge
Fig. 6 The one crossing Kawasaki allows: two creases meeting squarely, four equal sectors, both alternating sums a half turn. It still fails Maekawa as long as each line keeps one letter, and its four sectors are all equal, so the big-little-big lemma has nothing to say about it either.

Two things follow. The failure has a location — the vertex the ink has and the list does not — and it has a size, which is a rotation in degrees rather than a verdict. That is what makes a crossing legible in a patch of a hundred and twenty-five panels: the panels can be walked, the disagreement between two routes to the same panel measured, and the measurement traced back to a single point.

Three lines through a point do fold

The refusal is about two lines, and it does not survive adding a third. That is the surprise in this, and it comes out of the same two lines of arithmetic.

Take n straight lines through one point, each carrying one letter. The 2n sectors repeat after half a turn — the sector opposite any sector is equal to it — so Kawasaki’s alternating sum pairs each sector against its own opposite. When n is odd those pairs come with opposite signs and the sum vanishes identically: three lines through a point satisfy Kawasaki at any angles, without a condition on them at all. When n is even the halves reinforce instead, and the condition becomes a genuine equation the angles have to solve.

Maekawa behaves the same way for the same reason. With n lines the letters come in pairs, so the difference between mountains and valleys is twice the difference between the lines of one letter and the lines of the other. Two is twice one, so a difference of two needs the lines to split as k and k ± 1 — which is impossible for two lines and easy for three.

Swept at generic angles: of the eight ways of lettering three lines through a point, all eight satisfy Kawasaki, six satisfy Maekawa, all eight satisfy the big-little-big lemma, and six of the eight fold. Of the four ways of lettering two lines, none does. Of the sixteen ways of lettering four lines at generic angles, none satisfies Kawasaki — the even case has returned, and the preliminary base escapes it only because its sectors are all equal, which is exactly the special solution of that equation.

Where the sheet fails to close, markedA tessellation patch with a ring drawn round every vertex whose reflections do not compose back to the identity. Composing the reflections around a vertex returns a turn of twice the amount Kawasaki's condition is out by there, so a vertex that satisfies it contributes nothing — and the vertices that do turn are exactly the places two creases were drawn across one another.the rings are the vertices whose reflections do not compose to the identityread as ink54 vertices in the drawing42 compose to the identity12 do not, and turn instead240.0° to 240.0° of turn12 places two creases cross
Fig. 7 The same question asked of a whole patch rather than of one crossing: which of its vertices fail to compose back to the identity. Every failure is a crossing and every crossing is a failure, which is the two-crease case repeated forty-five times.

So the offending object is narrower than creases crossing. It is two creases crossing, and it is the parity of the number of lines through the point that decides — which is Maekawa’s theorem being a statement about parity all the way down.

The parity, with the counts written out

The three-line case is not an exception to be remembered alongside the two-line one. Both are instances of a single statement about the number of lines through the point, and the statement is worth having with its arithmetic attached because it settles every case at once.

Take nn straight lines through a point, each carrying one letter along its whole length. The mountains and valleys then come in pairs, so if kk of the lines are mountains the difference between the counts is 2k−2(n−k)2k - 2(n-k), which is 2(2k−n)2(2k - n). Maekawa wants that to be two, so 2k−n2k - n must be one — which needs nn odd, and then kk must be (n±1)/2(n \pm 1)/2.

At an even number of lines Maekawa is unsatisfiable, whatever the angles are. Two lines is the smallest case and the one this essay is about; four lines is the next, and it is why the centre of the preliminary base cannot be read as four lines each with a letter, however square its sectors are. An even-line vertex must have a letter change somewhere along one of its lines, or it is not a vertex at all.

At an odd number it is not merely satisfiable but common, and the count is a pair of binomial coefficients. The letterings satisfying Maekawa number (n(n−1)/2)+(n(n+1)/2)\binom{n}{(n-1)/2} + \binom{n}{(n+1)/2}: six of eight at three lines, twenty of thirty-two at five, seventy of a hundred and twenty-eight at seven. Kawasaki is free at every odd nn for the reason the sectors repeat, so those counts are very nearly the counts of what folds.

Which way the odd case runs as it grows

The share, though, does not stay at three quarters. Two binomial coefficients from the middle of a row divided by the whole row is a quantity that shrinks as the row lengthens, roughly as one over the square root of the number of lines: three quarters at three, five eighths at five, a little over a half at seven, and falling slowly after that.

So an odd-line vertex is never refused outright and it is steadily less permissive as more lines are drawn through the point. That is Maekawa doing at a concurrent vertex what it does everywhere else — accepting a little under half of what it is offered and getting no stricter — with the parity deciding whether the little under half is available at all.

The two behaviours together make a rule that fits in a sentence and covers every drawing of concurrent lines the subject contains. An even number of lines through a point folds only if some line changes letter at the point; an odd number folds at any angles, for about half its letterings. A crossing is the first case, four lines through the centre of a square is the second instance of the same case, and the preliminary base’s eight spokes are what a drawing looks like once the letter change has been admitted.

Nobody draws one on purpose

It is worth asking why an object this badly behaved is not a familiar hazard, and the answer is that a hand does not produce it.

A crease pattern drawn by a folder comes from a sequence: fold, unfold, fold again, and the creases are where the paper was turned. Paper turned about a line leaves a line, and two such lines meet at a point that was on both folds — which is a vertex, and which the folder will draw as one because they watched it happen. The same is true of every pattern published as mathematics: the Miura is a grid of parallelograms whose creases are its edges, the Yoshimura is what a crushed cylinder does, and neither has anywhere for a crossing to come from.

Across the eight patterns printed here at true scale, the count of crossings is nought. Across every pattern this collection draws from a rule rather than from a tiling patch, it is also nought. The crossings that exist here were made by a construction, and specifically by one deciding where to stop — which is the general case: an object nobody draws by hand is exactly the object that appears when drawing is done by rule.

The vertices a crease list does not haveEvery crease pattern here, read twice: once as the list of vertices and edges it is built from, and once as the ink on the page. The bar is how many vertices the second reading has to invent, which is how many places two creases cross with nothing recorded there.the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly
Fig. 8 Everything this collection draws, read as ink rather than as a list. The printed patterns need no invented vertex at all; the bars belong to patches cut out of tessellations, where a rule decided where the paper stopped.

What this makes a crossing worth

A proof of failure that costs one comparison, and the comparison is not about folding.

Testing a pair of drawn segments for a crossing is arithmetic on their endpoints: no letters, no angles, no folded state, no panels. Where it says yes, the pattern has no flat folded state, and the reason can be given as a theorem rather than as a search — Maekawa’s, applied to a vertex nobody wrote down.

That is unusually strong for something so cheap. Most of what can be said quickly about a crease pattern is necessary and not sufficient, and most of what is sufficient cannot be computed at any size worth drawing. A crossing is on the other side of both: it decides, and it decides in the direction that matters, since a no is the answer a checker can give honestly.

Where the argument stops

It is about a crossing, not about a pattern. A drawing with no crossing has passed one condition of many, and the conditions at every vertex together still do not decide the sheet.

It assumes a drawn line is one crease. That is the convention every crease pattern in this collection follows and every published one the collection has read, but it is a convention. A notation in which a line may change letter partway along it without marking the point would make the ink ambiguous in the other direction — and the mark that resolves it, a dot at the vertex, is exactly what the subdivision adds.

The sweep is a sample, and the argument is not. Eight angles and every lettering is a demonstration rather than a proof; what carries the claim is the algebra above it, which is two lines of arithmetic about a and a count of four objects. The sweep is there because an assertion that has never rejected anything proves nothing, and it does reject: four of its thirty-two pass one condition, eighteen pass another, and none passes all four.

And it assumes the two lines are straight. A curved crease is sampled into segments before any of this is asked, so a crossing found between two samples is partly a fact about the sampling — the same qualification every measurement on a curve here carries, and the reason the sampling is stated with the number.

Where this goes next

the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly The vertex the list does not have Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other. 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 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. 102°60°78°120°one smallest sectorthe lemma constrains one pair4 foldable assignmentsof the 16 markings120°60°60°120°two smallest sectors equalthe lemma constrains nothing8 foldable assignmentsof the 16 markings 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.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 11 that link here.

The objects this essay names

Each one links to every other essay that touches it.

The big-little-big lemmaCrease assignmentCrease patternCrossingFlat-foldabilityInterior vertexNecessary conditionSector angles