Origami & Folding
what one sheet can do
Generator

The coin's forty answers and the constant's one, on the rhombille patch

A generator in the what it costs to know library, called 124 times across 25 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

order-cost 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

The coin's forty answers and the constant's one, on the rhombille patchNode counts for 40 runs of one lettering search on one crease pattern of 157 panels and 282 creases, ranked. Under the search's own random choice of which letter to try first the cost runs from 86 to 15872 with 15 runs unfinished; under a constant choice every run costs 80.the dot is one run's cost, ranked; the rule is the constant order1001e+31e+4nodes visited40 seeds, ranked by cost80 nodes, every seed15 unfinished at 20,000same pattern, same conditions, same test at every node — the only difference is which letter is tried first

view: "ladder", family: "grid"

One node per panel: the orthogonal grid a box-pleated base is drawn onNodes visited against panels, for 9 crease patterns of one family searched under a constant letter order. Every point lies on or under the diagonal, which is a search that never backtracks.each point is one pattern: panels across, nodes up00100100200200one node per panelnodes visitedpanels2 by 2 to 16 by 16, and not one backtrack anywhere in the family

view: "restart"

A strategy against the absence of the problem it solvesThe expected total cost of cutting a lettering search off after a given number of nodes and starting again with a new seed, against the cost of not randomising the search at all. The curve is a correct answer about a distribution the search itself produced.the curve is stop-and-restart; the rule is a constant letter order1001e+31e+41001e+31e+4563 at a cutoff of 10080 nodes, deterministic, nothing to restartexpected nodes in totalcutoff, in nodes

view: "mirror", kinds: [square, elongated, hexagonal, triangular, rhombille]

Trying mountain first and trying valley first cost the sameNode counts for the same lettering search run twice on each of 5 crease patterns, once trying a mountain at every choice and once trying a valley. Every point lies on the diagonal, which is what a symmetry of the problem looks like when it is measured rather than assumed.each point is one patch, searched twice002020404060608080square · 26elongated · 32hexagonal · 39triangular · 39rhombille · 80nodes, mountain firstnodes, valley firstthe dashed line is y = x, and nothing has been fitted to anything

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.

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 corrugation never backtracks

As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.

A crumple has no tail

The least structured crease pattern this collection can produce is a sheet folded at random and flattened. Its consistent letterings get rarer as it deepens — thirty-four of forty down to eleven — and finding one costs one step per panel from beginning to end, with no wrong guess anywhere. Disorder and difficulty turn out to be unrelated quantities.

A population nobody chose

Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon measured for so long belongs to the one tiling the grid leaves out.

A proof in no nodes at all

A parity refuses a sheet before any search begins. It costs one addition, it is certain, and it says nothing about why — while a search that exhausts on the same sheet costs thousands of nodes and produces a proof of the same fact. Two proofs of one thing, and the cheap one is available only where somebody has noticed the invariant.

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 search with nothing to reorder

One search on a crease pattern costs eighty steps or fifteen thousand depending on the order it takes its decisions in. The other search on the same crease pattern costs 1,188,571 steps whatever order it is given — twelve permutations of the panels, twelve identical counts. The difference between them is one line of code that neither has and one has.

Each drawing has its own threshold

Gluing a cell's edges was measured once, at one size, and found to cost three orders of magnitude — which cannot tell a threshold from a slope, nor say whether a cut sheet has one further out. Swept from one period to five on four tilings, every sheet starts at about a third of a node per free letter and every drawing leaves that behaviour at a size of its own: four periods on the square grid, three on the honeycomb, two on the triangular grid and two on the rhombille, where even the cut sheet crosses.

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.

One witness or forty

Taking the randomness out of a search made it three orders of magnitude cheaper in the worst case and cost it thirty-nine of its forty answers. The compromise everybody reaches for — randomise only the choices that cannot matter — recovers four of the forty on two patches and none on the other three, because the diversity was never where it looked.

Restarting what cannot be restarted

Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.

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.

The cheapest route crosses later

A search for a consistent lettering has a threshold: below it the letters propagate and the cost is a third of a node per crease, above it the search backtracks and the cost explodes. The threshold was measured with one branch order. Measured with eight, the cheapest route never starts searching before the typical one, and on most sheets it starts a period or two later — so part of every threshold on the record belongs to the route. And the one cut sheet past its threshold, the rhombille's, spreads across nearly three orders of magnitude of cost, which moves the spread off the gluing and onto the threshold.

The cost of asking the wrong sheet

A test written for a sheet with an edge, run on a sheet without one, does not fail. It exhausts — proving, at three, thirty-five and three thousand four hundred and fifty-five nodes, that no lettering exists — and the letterings it proved impossible fold, on the collection's own machinery, at every size they were tried at.

The cure was named first

A heavy-tailed search runtime, the arithmetic for cutting it off and restarting, and the reason restarts work at all were established in the study of search between 1993 and 1998. This collection imported all three, and inherited with them the phenomenon they answer — which is that randomising a search's choices is what makes the tail.

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.

The difficulty was in the coin

One tessellation patch, one search, one test at every node — and a cost that runs from eighty-six steps to fifteen thousand depending on nothing but the starting seed. The heavy tail is real, it was measured carefully, and it was made by a single line of the search that nobody had thought of as a choice at all.

The edge is what makes it hard

Grids, crumples, leaves, corrugations and fold-and-cut patterns all give up a consistent lettering at one step per panel with no wrong guess anywhere. The one family that does not is a tessellation clipped to a square, and what separates it from the others is not disorder, not size and not irregularity. It is having a rim.

The motion has no letters to choose

A flat-folding search picks a letter for every crease and can pick badly. A rigid folding does not pick anything: the fold angles are real numbers, determined by the panels through equations that have a solution or do not. Replacing a discrete choice with a continuous solve removes every ordering question at once, and introduces a failure of its own.

The order that is its own mirror

Trying a mountain first and trying a valley first are two different searches, and on a hundred and forty-two crease patterns they cost the same number of steps — not on average, not nearly, but identically, pattern for pattern. The reason is a symmetry of every condition the subject has, and it is four lines long.

The order that proves nothing exists

Twelve crease patterns with no consistent lettering at all. Proving it takes fifteen steps under one rule and half a million under another — and on three of the twelve the two rules swap places, so neither is the good one. The cost of a negative is two to the power of how many free choices sit above the contradiction.

The proportion everyone draws is the hard one

The Yoshimura is drawn equilateral, and every cost and count of it has been measured there. Moved a quarter of a step along its own height dial, a search for a stacking that took 268,650 steps takes 808, and a sheet with 9,790 folded states has 92. The equilateral pattern is the most expensive member of its family on every sheet measured, because exact coincidence removes the conditions that a panel lying across a crease would impose: from 1,024 conditions to none. A drawing chosen for being easy to construct is the one instance where almost nothing constrains the layers, and a population of one chosen drawing had been reporting that freedom as the difficulty of the pattern.

The route built from the gluing

A search for a consistent lettering of a glued sheet costs a third of a node a crease on its cheap routes and thousands on its dear ones, and an account was offered of which routes are cheap: the ones that take first the creases whose choice propagates through the gluing. That account can be turned into a route rather than tested on samples, and there are two ways to read it. Built the first way, a route finishes nearly twice as many searches past the threshold as a drawn one does, and on the rhombille torus — where none of eight drawn routes finishes inside twenty thousand nodes — six of eight built routes do, the cheapest in 1,183. So the rhombille's difficulty belongs to the route. But which reading works changes with the tiling, and on the elongated tiling, where every drawn route finishes, the built ones lose searches.

The route, not the sheet

Every cost measured for a glued sheet has been one number from one branch order, and a backtracking search's cost belongs to the pair. Asked under eight orders instead of one, a cut cell's cost barely moves — 42 to 55 nodes — while the torus over the same drawing runs from 69 to 24,636, with one order giving up entirely. The glued sheet's best order costs less than twice the cut sheet's, so most of what a single order charged to the gluing belongs to the route through it.

The torus keeps its tail

Reading why a branch failed removes the heavy tail from the search for a lettering of a cut sheet: on the rhombille drawing at two periods five of eight searches finish becomes eight of eight, and glued along one pair of edges none of eight becomes eight of eight. Glued along both, it stays at one of eight, and it stays there when every refused loop is also remembered. Two accounts of why are refuted on the way: every refusal on every sheet is a loop, so tilings do not differ by how they fail, and a torus's refusals blame no more of the stack than a cut sheet's. What separates them is the length of the proof. Undo twenty decisions from a stopped run and the proof that the letters left are already lost takes 33 nodes on a cut sheet and about 2,400 on a torus, so a jump on a torus has nothing short to read.

Which choice the cost lives in

A backtracking search takes two decisions at every step — which thing to decide, and what to decide about it. The literature is almost entirely about the first. On these crease patterns the whole of the cost was in the second, and the structural improvement everybody reaches for first makes matters worse on fifty-two patterns out of eighty-seven.

All the figures · The what it costs to know field · The patterns a reader can fold