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 107 times across 22 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

order-cost is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

`view: "ladder", family: "grid"`

`view: "restart"`

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

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

25 distinct letterings from 25 runs under the coin, and 1 from 40 under a constant order — every one of them verified against the conditions at every vertex and a folded sheet rebuilt from scratch ×4
32 patches over a stated grid of tiling and turn angle rather than five somebody chose: 26 have a consistent lettering and 6 have none at all ×4
9 patterns of one family, searched under a constant letter order: every one costs at most one node per panel, so the search reaches a lettering without turning round once ×4
a crease lies on between 1 and 10 of them, and that count is fixed by the drawing rather than by any lettering, because a lettering points an arc and cannot move it ×3
every one of these drawings starts cheap: at one period the cut cell is settled in well under a node per free letter, on all 4 tilings ×3
the cut sheet crosses on 1 of the 4 and not on the rest, so gluing is what brings the threshold within reach rather than what creates one ×3
the same search, the same patch and the same test at every node: 25 of 40 seeds finish inside 20,000 nodes under the coin, and every seed finishes in 80 under a constant letter order ×3
a witness costs 39 nodes under the standard variable order and 39 under the circuit order, so branching on the structure buys nothing when there is something to find ×2
so the two constant orders walk mirror trees, and on these 5 patches they cost the same node for node ×2
the cheapest cutoff is 100 nodes, at 563 in expectation, against 14746 for patience ×2
under ten branch orders the cut sheet's cost barely moves — a factor of 1.3 between its best and worst — while the sheet with no rim spans a factor of 357 ×2
a route that costs under a node per free letter is propagation; every sheet here starts at about a third of a node per letter at its smallest size ×1
a run that hit the budget is drawn as unfinished rather than as its budget, because the two are different facts ×1
and a constant letter order costs 80 nodes with nothing left to restart, because the randomness the strategy reseeds is the randomness that made the distribution ×1
and every one of them leaves that behaviour as the cell grows, at a size of its own — 4 on the square grid, 2 on the triangular grid — so the threshold is a property of the drawing rather than a size ×1
and every one of them leaves that behaviour as the cell grows, at a size of its own — 4 on the square grid, 3 on the honeycomb — so the threshold is a property of the drawing rather than a size ×1
and every one of them leaves that behaviour as the cell grows, at a size of its own — 4 on the square grid, 3 on the honeycomb, 2 on the rhombille tiling — so the threshold is a property of the drawing rather than a size ×1
and every one of them leaves that behaviour as the cell grows, at a size of its own — 4 on the square grid, 3 on the honeycomb, 2 on the triangular grid, 2 on the rhombille tiling — so the threshold is a property of the drawing rather than a size ×1
and the glued sheet's *best* order costs within a small factor of the cut sheet's, so most of what a single fixed order attributes to the gluing belongs to the route rather than to the sheet ×1
below the threshold the dearest of 8 orders costs at most 0.38 nodes per crease on every quiet sheet, which is propagation rather than search ×1
every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
on every cut sheet of the square, triangular, honeycomb and elongated tilings measured, the eight orders agree to within a factor of 1.3 ×1
so the variable order is worth four orders of magnitude on a negative and nothing here says which way round ×1
swapping every letter on the sheet maps an admissible lettering to an admissible one: Maekawa reads a difference, big-little-big reads a sameness, Kawasaki reads no letters, and every arc simply reverses ×1
the arc graph of the hexagonal patch has 130 arcs on 77 panels and a circuit rank of 54, with fundamental circuits from 4 to 16 arcs long ×1
the arc graph of the rhombille patch has 282 arcs on 157 panels and a circuit rank of 126, with fundamental circuits from 4 to 26 arcs long ×1
the arc graph of the square patch has 84 arcs on 49 panels and a circuit rank of 36, with fundamental circuits from 4 to 12 arcs long ×1
the cheapest of 8 orders never starts searching before the middle one, and crosses anywhere from the same period to two periods later: the square grid glued, middle at 4, cheapest at 5; the triangular grid glued, middle at 2, cheapest at 3; the honeycomb glued, middle at 2, cheapest at 4; the elongated triangular tiling glued, middle at 2, cheapest at beyond; the rhombille tiling glued, middle at 2, cheapest at 2; the rhombille tiling cut, middle at 2, cheapest at 3 ×1
the patches with none are not scattered — every one of them is at the shallow end of the turn, on 3 of the 4 tilings ×1
the rhombille's cut sheet does not: at 2 periods 67 to 13834 nodes, 1 of 8 orders out of budget; at 3 periods 7172 to 7172 nodes, 7 of 8 orders out of budget ×1
the same search, the same patch and the same test at every node: all 40 seeds finish inside 20,000 nodes under the coin, at 39–49 against the constant order's 39, so which letter is tried first costs the hexagonal patch nothing ×1
two patches with no lettering at all, and the two rules swap places between them: 511,999 against 63 on one, and 15 against 2,047 on the other ×1

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

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 the collection had spent so long measuring 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 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.

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.

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