What a loop of creases carries

A crease pattern’s interior vertices each have a rule. Kawasaki’s and Maekawa’s conditions, the big-little-big lemma and the rest say which mountain-and-valley letterings of a vertex’s own creases it admits, and a lettering of the whole sheet passes every local test exactly when it passes at every vertex. So the share of the sheet’s letterings that pass is a question about how the vertices’ rules combine, and the simplest guess is that they do not interact at all: multiply each vertex’s own share.

An array is priced by its vertices needed that guess to price arrays of slits before cutting, and tested it on five printed patterns by counting exactly. On four it held to the last digit. The Miura’s fifteen interior vertices each admit half their letterings and the sheet admits exactly $2^{-15}$; the square twist, the hexagon twist and the tapered corrugation behave the same way. On the Yoshimura it failed: the counted share was $2^{-24.059}$ against a product of $2^{-24.048}$, about three quarters of a per cent less. That essay ended by asking why, and offered a guess — the Yoshimura’s vertices meet round small triangles, three vertices pairwise joined, and perhaps a short loop is how one vertex’s verdict reaches another’s odds.

The guess is right, and it can be made exact.

A count is a product times a sum over loops #

Write a vertex’s rule as a function that is 1 on the letterings it admits and 0 on the rest, and expand it the way any function of a few binary letters can be expanded: as a sum of terms, one for each subset $S$ of its creases, each term a coefficient times $(-1)$ raised to the number of mountains among the creases of $S$. The coefficient on the empty set is the vertex’s own share. The coefficient on a single crease measures how much more often the rule admits that crease as a mountain than as a valley. The coefficient on a pair of creases, divided by the share, is the correlation between their letters over the letterings the vertex admits: $+1$ if the rule always gives them the same letter, $-1$ if always different, $0$ if knowing one says nothing about the other.

Multiply every vertex’s expansion together and average over all the sheet’s letterings. Almost every term averages to zero, because a crease that appears an odd number of times contributes a factor that is $+1$ half the time and $-1$ the other half. A crease belongs to the two vertices at its ends and no others, so a term survives only if every crease it names is named by both its vertices or by neither. Collect the survivors and the share comes out as

$$\text{share} = \prod_v \text{share}_v \times \sum_H \prod_v g_v(H_v),$$

where $H$ runs over sets of creases, $H_v$ is the part of $H$ at vertex $v$, and $g_v$ is the vertex’s coefficient divided by its share. The empty set contributes 1, which is the product guess. Everything else is a correction.

One more simplification makes the corrections pictures. A vertex rule that does not care which way up the paper is — and every rule here admits a lettering exactly when it admits the lettering with every mountain and valley swapped — has no coefficient on an odd number of creases. So a surviving $H$ meets every vertex in an even number of creases: it is a loop, or several loops, in the graph the creases make, weighted by the product of the correlations each vertex puts on the creases the loop uses there.

This is an identity rather than an estimate, and it was checked as one. The sum over loops, taken with the same column-by-column sweep that counts the letterings, reproduces the counted share divided by the product on all five printed patterns to twelve decimal places. Its shape is an old one: Van der Waerden’s 1941 expansion of the Ising model writes that model’s partition function as a sum over closed polygons on the lattice, and Chertkov and Chernyak’s loop series of 2006 corrects a message-passing estimate by a sum over loops in exactly this way. Here the lattice is the crease pattern and the coupling is the vertex rule.

A Miura vertex ties no pair of creases together #

The four patterns on which the product held all have vertices of degree four, and the figure shows what their rules look like in this light. A Miura vertex admits 8 of its 16 letterings, and they are exactly the letterings with an odd number of mountains — one mountain and three valleys, or three and one, which is Maekawa’s condition and nothing more. The big-little-big lemma is silent because the Miura’s two small sectors sit side by side and neither is strictly smallest; Kawasaki’s condition holds for every lettering because it is a condition on angles.

A rule that says only “an odd number of mountains” correlates no pair of creases. Fix two of the four letters any way at all, and two of the four letterings of the other two creases complete an odd count. Fix three and exactly one completes it. Only when all four creases are named together does the rule say anything, and then it says everything: their product of signs is always the same. So the only non-zero correction a Miura vertex can contribute is on all four of its creases at once.

Now try to build a loop from such vertices. Every vertex the loop visits must give it all four creases, so the loop must continue along every crease at every vertex it reaches. It spreads through the whole connected pattern, and the pattern reaches the paper’s edge, where a crease ends at a vertex that rules nothing and has no correlation to offer. No loop can close, every correction is zero, and the product is the count. The square twist, the hexagon twist and the tapered corrugation have the same kind of vertex — the exact counts found eight of sixteen letterings at every interior vertex of all four patterns — and the same argument covers them.

An equilateral Yoshimura vertex ties every pair weakly #

An equilateral Yoshimura vertex has six creases with six sectors of sixty degrees. The big-little-big lemma is vacuous there — no sector is strictly smaller than both neighbours — so the vertex admits every lettering Maekawa allows: four mountains and two valleys, or two and four, fifteen of each, thirty of sixty-four.

Take any two of the six creases and count. Among the fifteen letterings with four mountains, the two creases are both mountains in six, both valleys in one and different in eight: alike seven times, different eight. The letterings with two mountains give the same split the other way round. So every pair of creases at the vertex is correlated by exactly $-1/15$ — a little more likely to differ than to agree, because a vertex with four of one letter and two of the other has more mixed pairs than matched ones. Nothing about which two creases they are enters the arithmetic, which is why the figure’s fifteen chords are all alike.

A correlation on every pair means a loop can turn any way at every vertex and still collect a non-zero weight, and the Yoshimura’s creases offer short loops everywhere: its horizontals and diagonals cut the sheet into small triangles.

The shortest loop is a triangle of three creases. Going round it collects a correlation of $-1/15$ at each corner, so each triangle contributes $(-1/15)^3 = -1/3375$ to the sum. The six-by-five Yoshimura has twenty-seven small triangles whose three corners are all interior vertices, and twenty-seven of them come to $-0.0080$. The triangles alone account for a shortfall of 0.80 per cent, against the 0.75 per cent the count found.

The rest is the larger loops, and they come in both signs.

A loop of four creases is two triangles sharing an edge with the shared edge left out — a small rhombus — and it collects four correlations, $(-1/15)^4 = +1/50625$. The pattern has thirty-three of them, which add back $0.00065$. Loops of five and six take a little away again; by loops of ten creases the running sum agrees with the count to the sixth decimal place, and the remaining terms, down to loops of thirty-six creases, change nothing visible. The Yoshimura’s three quarters of a per cent is twenty-seven triangles, corrected for their neighbours.

The guess the earlier essay made was the right one. A loop of three vertices is what lets one vertex’s verdict reach another’s odds: the letter on one crease of a triangle shifts the odds at both its ends, each of those shifts the odds on the triangle’s other two creases, and the effect comes back round to where it started. On the Miura the same loops exist — four vertices round every panel — but no vertex passes anything along them.

The same knife edge decides independence #

The Yoshimura’s correlations come from its vertices admitting everything Maekawa allows, and that depends on its proportion. A knife edge nine decimals wide found that the vertices admit thirty letterings when the rows are $\sqrt3$ half-columns tall or shorter and eight when they are taller, by any amount at all: above $\sqrt3$ the two sectors between the upper diagonals and the two between the lower ones become strictly smaller than their neighbours and the big-little-big lemma starts to bite, while below it the small sectors come in adjacent pairs and the lemma stays silent.

Read through the loop sum, that knife edge is also where the product becomes exact. At every proportion tried from 1.2 half-columns up to $\sqrt3$, the vertex admits the same thirty letterings, every pair is correlated by $-1/15$, and the counted share falls short of the product by exactly the same 0.7475 per cent. At every proportion tried above $\sqrt3$, from 1.75 to 2.4, the counted share is the product, exactly.

The reason is in the third drawing of the vertex figure. A taller vertex admits eight letterings, and they are eight because the lemma ties three pairs of creases completely: the two upper diagonals, which bound a strictly smallest sector, always differ; the two lower diagonals always differ; and with those fixed, Maekawa forces the two horizontals to agree. Each of the three pairs then has one free letter, two to the third is eight, and every other pair of creases is uncorrelated — knowing an upper diagonal’s letter says nothing about a horizontal’s.

So a loop through these vertices can only turn where a vertex locks a pair. Entering a vertex along a horizontal crease, it must leave along the other horizontal, and it runs straight along the row to the paper’s edge. Entering along an upper diagonal, it must leave along the other upper diagonal, which carries it up to the vertex above, where it arrives along a lower diagonal and must leave along that vertex’s other lower diagonal. It zigzags between two rows, sideways, until it reaches the edge. No path through locked pairs ever closes, any union of such paths still ends at the rim, and with no closed loop there is no correction. The taller Yoshimura’s vertices decide independently, and the product over vertices is a count.

That is the surprising part of the result, and it inverts what the earlier essay’s shelf seemed to say. Degree six was the obvious suspect — six creases and the same straight line had already found the Yoshimura’s degree changing a constant where a different arithmetic was expected — but the Yoshimura did not fail the product because it has degree-six vertices or because it is triangulated. It failed because it is drawn at the one proportion — and below it — where its vertex rule is weakest. A rule that admits nearly half of all letterings passes a little information along every pair of creases; a rule that admits an eighth locks a few pairs completely and leaves everything else free. Strong, sparse constraints were independent here; weak, dense ones were not.

The shortfall is a price per triangle #

If triangles are the leading loops, the shortfall should grow with the number of triangles, and it does — nearly in proportion. Across eight sizes, from a four-by-three Yoshimura with five triangles to an eight-by-six with fifty-two, the shortfall is between 4.00 and 4.07 ten-thousandths of a power of two per triangle. One triangle on its own would cost $-\log_2(1 - 1/3375)$, about 4.28 ten-thousandths; the slightly smaller price per triangle on a real sheet is the rhombi paying a little back, the same pattern the loop sum showed term by term. A ten-by-seven Yoshimura, counted the same way outside the figure, has eighty-five triangles and falls short by $2^{-0.0339}$, which is 3.99 ten-thousandths each.

So the product’s error compounds with the sheet’s area. It is three quarters of a per cent on the printed six-by-five pattern, over two per cent on ten by seven, and on a sheet of a thousand triangles it would be a factor of about $2^{0.4}$, the product claiming a third more letterings than exist — a linear extrapolation, not a count, but the eight sizes give no sign of the price per triangle drifting.

What this does to the ledger’s promise #

The point of the product, in an array is priced by its vertices, was practical: price an array of slits before cutting by reading each vertex’s sectors and the cuts that will reach it. That essay concluded that the promise held on patterns whose vertices are independent “and the shelf does not say in advance which those are”. The loop sum says in advance. Read the pair correlations each vertex’s rule puts on its creases; if no closed loop of the pattern collects a non-zero product of them, the product of vertex shares is the count. For a pattern of Maekawa-only degree-four vertices that is automatic, and for a Yoshimura it is a matter of which side of $\sqrt3$ it was drawn on.

The test that essay proposed — slit one crease of each triangle and watch the ratio return to one — turns out not to be a test of loops at all on the equilateral Yoshimura. A slit there releases the vertices at its ends completely: a vertex with one slit becomes a strip of five creases at equal sixty-degree intervals, and a strip of equal segments folds in every lettering, as a slit tip keeps one condition found that a released vertex keeps only what its strip’s own lemma imposes. Slit all eighteen buried horizontal creases of the six-by-five sheet and the ratio does return to one, but only because every vertex a slit reaches stops ruling anything; there is nothing left to be correlated. The proportion is the clean experiment, because it removes the loops’ weights while leaving every vertex ruled.

What the expansion cannot show #

Folding. Everything here is about letterings that pass local tests, which local is not global separated from folding at the start, and how little the conditions decide measured how far passing every local test is from folding flat. The loop sum makes one local count exact; it says nothing about which of those letterings have a layer order.

Patterns with strictly smallest sectors at degree four. Every degree-four vertex tested here admits all eight Maekawa letterings. A generic degree-four vertex with one strictly smallest sector admits four, and its lemma locks the pair of creases round that sector; whether locked pairs on a general quadrilateral mesh can close a loop — whether such a pattern is independent the way the tall Yoshimura is — was not measured.

Why the larger loops’ signs alternate the way they do. The series is exact and converges fast on every pattern tried, but nothing here bounds the terms in general; a pattern with stronger correlations and many short loops could have a sum that converges slowly or to something far from one.

Rules on letters, not paper #

The vertices are idealised in the usual way: creases are lines meeting at points, paper has no thickness, and a vertex’s rule is the set of letterings its local conditions admit. The Yoshimura is the pattern Yoshimura described in 1955 as the buckling mode of a crushed cylinder — one of the patterns nobody designed — drawn flat with its rows $\sqrt3$ half-columns tall unless stated, at six columns by five rows; the Miura, the twists and the corrugation are the printed patterns of the earlier counts. The correlations are averages over the letterings each vertex admits, every lettering weighted equally, which is the weighting a count makes and not one any physical folding process is known to follow.

Two sums that must agree #

The loop sum is computed by a sweep that shares its structure with the counting sweep but none of its arithmetic: one adds up letterings, the other multiplies Fourier coefficients. They are required to agree to twelve decimal places on every pattern drawn, the triangles’ term is required to be exactly their number times $(-1/15)^3$, and no loop of one or two creases may contribute. The vertex figure requires the Miura vertex to correlate no pair and only all four creases, the equilateral Yoshimura vertex to correlate all fifteen pairs by exactly $-1/15$, and the tall vertex to lock exactly three pairs. The proportion figure requires the same shortfall at every proportion up to $\sqrt3$ and an exact product at every one above, and the size figure requires the price per triangle to lie between nine-tenths of one triangle’s cost and that cost.

Still open: the general degree-four mesh #

The cleanest next question is a generic quadrilateral mesh. Its vertices are degree four like the Miura’s, but each has a strictly smallest sector, so each locks one pair of creases and leaves the rest — which pairs depends on the angles, vertex by vertex. Whether locked pairs can close a loop on such a mesh is a question about the mesh’s geometry that the loop sum turns into a path-following exercise, and the answer would say whether independence is typical of flat-foldable degree-four patterns or a property of the regular ones.

The second is a pattern whose loops add rather than subtract. The Yoshimura’s triangles all carry the same negative sign; a pattern whose shortest correlated loops have an even number of corners would carry positive terms first, and its product would undercount rather than overcount. The printed waterbomb tessellation, counted the same way outside the figures, is not that pattern: its degree-six vertices correlate their pairs by $-1/15$ like the Yoshimura’s, but each is fenced off from the next by Maekawa-only degree-four vertices, which a loop can only pass by taking all four of their creases. Nothing shorter than six creases contributes, and the counted share is 0.99992 of the product — the same sign, a hundred times smaller. A pattern with correlated vertices round squares is still to be found.

Sideways, one cut removes one arc and a cut is not local both found that a single cut’s effect spreads past the vertices it touches. The loop sum is a way to say how far: a cut changes the vertex rules at its ends, and the change reaches exactly as far as the loops through those vertices carry non-zero weight.

The habit worth carrying is about independence assumptions. When a product of local probabilities is nearly right, expand the error before explaining it. Here the expansion was exact, its leading term was a picture — small triangles on a sheet — and it pointed straight at the proportion where the error vanishes.