Generator

The oldest book cuts the paper — the senbazuru cuts generator

A generator in the who found it, and when library, called 42 times across 7 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.

senbazuru-cuts 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

`view: "perfect", show: "cuts"`

`view: "perfect", show: "loops", at: 10`

`view: "perfect", show: "cases", at: 8`

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.

slit length per crane rises from 1.00 to 1.67 sides as the piece grows — the rule is broken harder, not less ×6
the 1 cut sets the search finds on 8 by 8 are all different ×5
every perfect tree the colour search finds is checked join by join on the 8 by 8 grid — every crane held, one piece, no merge wasted ×4
these 21 joins keep every crane of the 8 by 8 grid held and the whole piece in one, checked join by join, with a slack of 0 ×4
the 8 by 8 tree leaves out 4 points of its colour, a third of its 12 loops ×3
at 6 by 6, 99.64% of the 33412811 failing subsets fail by leaving a crane hanging from nothing ×2
even at 5 by 5 the piece needs 10 of its 16 joins, and there are only 50 ways to spend that few ×2
its 21 joins are linked by 20 shared cranes, one fewer than the joins, which is what a tree is ×2
the 8 by 8 grid has a perfect tree to draw ×2
the cut search finds exactly one perfect tree on 8 by 8 ×2
the stray join has 5 neighbouring joins and every one of them is cut, so the four cranes it holds are joined to nothing else — while every crane on the sheet still keeps a join ×2
a second look — no kept join with all its neighbouring joins cut — never refuses a cutting that holds, and it raises the share of what passes that holds at every size: 65.6% to 91.3%, 64.6% to 81.0%, 54.3% to 76.8% ×1
and it is the only perfect tree on 16 by 16, and it is four copies of the 8 by 8 one with a join at the middle ×1
and of the subsets that pass the local test, the share that hold together falls with the size — 65.6%, 64.6% ×1
and of the subsets that pass the local test, the share that hold together falls with the size — 65.6%, 64.6%, 54.3% ×1
and the steps do not grow with the joins each size adds, which rise from 15 to 21 — so the losses scale with the border, not with the sheet ×1
at 6 by 6, 76,331 of the 119,004 sound cuttings that come apart leave a single join standing alone as a piece of its own ×1
cranes minus corner joins is exactly 2n − 1 at every size, which is the lattice identity the arrangement rests on ×1
every connected-crane arrangement in the table requires the sheet to be slit — none is reachable from an uncut square ×1
every fewest arrangement the count reads back is checked directly — every crane held, one piece, and the number of joins it claims ×1
every odd grid is refused before any search, by the corner colours or by the floor ×1
every subset is tried: 16, 512, 65,536 of them, the largest in well under a second ×1
every subset is tried: 16, 512, 65,536, 33,554,432 of them, the largest in well under a second ×1
every subset that holds the piece together also leaves no crane hanging, so the local test never refuses a cutting that works ×1
four perfect 8 by 8 trees and one join at the middle hold all 256 cranes with 85 joins, which is the floor ⌈(n² − 1)⁄3⌉ exactly, every join merging four separate pieces ×1
from eight by eight on, each added row and column takes a nearly constant 0.071 off the log of the share — the steps run 0.069, 0.073, 0.070, 0.072 ×1
from five by five on, the share of sound cuttings that hold falls at every size — 64.6%, 54.3%, 51.1%, 46.7%, 43.6%, 40.5%, 37.7%, 35.1% ×1
of the grids whose counting floor is a whole number — 4, 5, 7, 8, 10, 11 — only 4 and 8 meet it with no merge to spare, eight by eight in exactly one way, and ten by ten needs one join more than its floor ×1
on 10 by 10 the tree's colour has 41 points and its links close 24 loops; a perfect tree keeps 33 of the points, so it cuts 8, a third of the loops ×1
on a three by three every one of the four joins is load-bearing — exactly one of the 16 subsets leaves the piece connected, and it is the full set ×1
on a three by three the only subset that leaves no crane hanging is the full set, and it is also the only one that holds together — the local test is the whole census there ×1
on every grid exhausted, more than 99 per cent of the sound cuttings that come apart do so through pieces touching the border — 430 of 430 at 5 by 5, 118,379 of 119,004 at 6 by 6 ×1
on the 5-grid the four points the corner cranes force are of two colours, and on the 4-grid they are of one ×1
on the even grids the fewest joins that hold the piece is exactly the counting floor ⌈(n² − 1)⁄3⌉, and it is reached in one way only — 5 at 4 by 4 ×1
on the even grids the fewest joins that hold the piece is exactly the counting floor ⌈(n² − 1)⁄3⌉, and it is reached in one way only — 5 at 4 by 4, 12 at 6 by 6 ×1
the census is an exhaustion over every subset of joins with a union-find on each, so a connected arrangement is found rather than argued for ×1
the column-by-column count agrees with trying every subset at four, five and six by six — 21 of 32, 785 of 1,215, 141,621 of 260,625 ×1
the cut search on 10 by 10 ends in 7 cases, not one of them a perfect tree ×1
the cut search on 8 by 8 ends in 2 cases, 1 of them a perfect tree ×1
the doubled tree on 32 by 32 holds every crane in one piece with no merge wasted ×1
the grids with a perfect tree among those searched are 2, 4, 8, 16, every one a power of two ×1
the rim forces a smaller share of the tree as the grid grows, so the middle is where a perfect tree is decided ×1
the share of admissible cuttings falls at every size — 6.3%, 4.1% ×1
the share of admissible cuttings falls at every size — 6.3%, 4.1%, 1.2% ×1
the six-by-six count agrees with the exhaustion's 141,621 arrangements that hold together ×1

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 perfect tree is the joins it leaves out

On the colour a perfect tree lives on, the lattice points and the cranes they share form a graph full of loops, and a perfect tree is that graph with some points cut out. Euler's count fixes how many: every cut removes three loops' worth, and the loops always number exactly three times the cuts a perfect tree needs — so a set of that many cuts, no two sharing a crane, leaves a perfect tree exactly when what remains is one piece. Searched that way, ten by ten fails in seven cases that fit on one page, fourteen in 761, and sixteen's one tree is four copies of eight's diamond of cuts with a cross of cuts along the middle.

Nearly every cutting fails at one crane

Six by six connected cranes have twenty-five joins and thirty-three million ways to keep some of them, and an exhaustion over all of them takes a fifth of a second. Of the 33,412,811 that fail, 99.64 per cent fail at a single crane — one left holding none of the joins at its corners — which a maker can check by looking at each crane in turn. The arrangements that pass that check hold together less often as the grid grows: all of them at three by three, 54 per cent at six by six.

No odd grid holds a perfect tree

A perfect tree of joins holds a slit grid of cranes with every join merging four pieces that were separate. Two joins side by side share two cranes and close a loop, so a perfect tree never has two — and so all its joins lie on one colour of the checkerboard the lattice points make. The four corner cranes each touch one lattice point, which must be a join; on a grid with an odd side those four points are two colours. So no odd grid holds a perfect tree, and seventeen by seventeen, the size the count was pointing at, is settled without counting. On one colour the search is small enough to finish: sixteen by sixteen has exactly one perfect tree, the doubled one, and twenty by twenty has none.

The border is where the cranes come apart

Counted one join at a time rather than one subset at a time, the slit grid of connected cranes runs to twelve by twelve, where there are 2¹²¹ ways to keep some of the joins. Among the cuttings that hold every crane by something, the share that also hold together keeps falling — 54.3 per cent at six by six, 35.1 at twelve — and from eight by eight on it falls by the same factor at every size. A constant factor is the signature of the border: at six by six, 99.5 per cent of the sound cuttings that come apart do so through a stray piece touching the outermost ring of joins.

The oldest book cuts the paper

The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.

The prediction held at eight and broke at ten

A join in a slit grid of cranes merges at most four pieces, so n² cranes need at least ⌈(n² − 1)⁄3⌉ joins, and exhaustion found four and six by six meeting that floor in exactly one way. The guess was that eight by eight would too, with twenty-one joins. Counted a join at a time, it does — twenty-one, one way. Ten by ten does not: it needs thirty-four against a floor of thirty-three, and has 7,076 ways to spend them. The grids that meet the floor with no merge to spare are four and eight by eight among every size to fourteen, and sixteen by sixteen by construction, because a perfect tree of joins on a grid twice as wide is four perfect trees and one join in the middle. The even grids were never the pattern; the doublings are.

Which cranes can stay joined

The 1797 book slits a square into a grid and leaves the cranes attached at the interior lattice points, each of which holds four of them at once. Of the sixteen ways to choose which of a three-by-three's four points to leave joined, exactly one leaves the piece in a single object — and it is the one that uses all four. The cutting is very nearly forced rather than chosen.

All the figures · The who found it, and when field · The patterns a reader can fold