The four letterings a small map refuses
map-windows 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: "refused"
view: "forced"
view: "exact"
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.
- on the 3 by 3 map every one of 4 refusals is a circle exactly as long as the rim of the smallest rectangle it fits in ×6
- 6 by 6: 24.8% of 4000 letterings refused — 22.9% through a three-by-three window, 1.6% through a four-by-four one and no smaller, 0.30% only through a larger rectangle; the two window sizes alone predict 23.6% ×5
- the circle refusing the 3 by 3 lettering drawn crosses 8 creases, as many as there are panels round the rim of a 3 by 3 rectangle ×4
- a map 2 cells across has no refused lettering at any length — 2 by 2 none of 8 ×3
- a 4 by 3 map has 17 creases and 6 interior vertices, and exactly 2^11 = 2048 letterings pass every vertex ×1
- a sampled 10 by 10 lettering refused through a four-by-four window and no three-by-three one exists to draw ×1
- a sampled 6 by 6 lettering refused through a window exists to draw ×1
- a three-by-three map has 256 letterings that pass every vertex, and the loop test refuses exactly 4 of them ×1
- a three-by-three map refuses one lettering in 64, a four-by-four one in 512 with a circle that needs the whole of it, and a four-by-three none ×1
- at 20 a side, 0.22% of letterings pass and 0.07% are refused only through a rectangle larger than four by four; every other one is refused by a window three or four cells across ×1
- built cell by cell, the flat three-by-three and four-by-four maps give 4 of 256 and 2112 of 32768 letterings refused, 64 of them by no three-by-three window — the counts the crease-pattern construction gives ×1
- each of the four is unchanged by a quarter turn of the map: they are pinwheels ×1
- each shape's circle goes round eight of the map's nine interior vertices, leaving out one next to a corner ×1
- every one of them is a circle of twelve creases, as long as the ring round the map's rim, and every circle on the map is as long as its own rectangle's rim ×1
- every refusal on three by three, four by three and five by three is a refused three-by-three window, and four by four is the first map with refusals no window explains — 64 of 2112 ×1
- every two panels added to the circumference cut the share refused round the tube, by factors of 0.52 0.56 0.50 0.63; 0.58 0.53 0.61 0.55; 0.57 0.61 0.46 0.55 at heights 3, 5, 8 ×1
- from six panels long to eight the share refused round the tube moves by -1.5, -0.1, 0.1, -2.0 points at 4, 6, 8, 10 round, while the windows' share climbs by 10.1, 12.5, 13.9, 15.3 ×1
- of 156 sampled refusals that need a rectangle larger than four by four, 3 have a circle longer than the rim of its rectangle; the rest are as short as a circle spanning it can be ×1
- on every tube counted exactly, every refusal no three-by-three window explains is a circle round the tube — 32 of 64 on 2 by 3, 128 of 256 on 2 by 4, 1,856 of 4,096 on 4 by 3, 34,176 of 65,536 on 4 by 4, 64,568 of 262,144 on 6 by 3 — and the windows, counted across the seam, refuse at the flat map's rate ×1
- on maps three cells across, up to 6 long, every refusal is a three-by-three pinwheel: no longer circle closes ×1
- on the sampled maps most refusals are a refused window — 91% on 6 by 6, 90% on 8 by 8, 93% on 10 by 10 ×1
- on the tube 4 round and 3 long, the shortest circle in each drawn lettering winds exactly once round the tube: one 4 creases long in a single row, one 6 long across rows ×1
- on tubes 5 high the letterings no window refuses are almost all refused by a circle round the tube — 49.0% at 2 round, 54.9% at 4 round, 31.8% at 6 round, 16.9% at 8 round, 10.3% at 10 round, 5.6% at 12 round — and that share falls with the circumference while the windows' share grows with the area ×1
- one rectangle refuses on its own 3 by 3: one lettering in 64; 4 by 3: never; 4 by 4: one lettering in 512; 5 by 4: one lettering in 5461; 5 by 5: one lettering in 2857 ×1
- the 64 refusals of a four-by-four map that no three-by-three window explains are 4 shapes, each in sixteen versions, and not one is unchanged by any turn or mirror of the map: they are not larger pinwheels ×1
- the four-by-four map has letterings the loop test refuses with every window accepted ×1
- the lettering drawn passes every vertex, is refused as a whole, and has no refused window ×1
- the three-by-three windows of a tube, counted round the seam, refuse at the flat map's rate: 16.2% against 17.2% on 4 by 5, 25.9% against 24.7% on 6 by 5, 33.2% against 31.5% on 8 by 5, 39.3% against 37.7% on 10 by 5, 45.1% against 43.3% on 12 by 5, 44.6% against 43.3% on 6 by 8, 69.5% against 67.8% on 12 by 8 ×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 map refuses in small pieces
The one-pass refusal — a circle in the arcs a map's letters force — fires on four of a three-by-three map's 256 letterings and on none below it, and the question was how it grows. It grows by windows. Every lettering it refuses on a four-by-three or five-by-three map contains one of those four refused three-by-three patterns somewhere inside it, so the share refused is almost exactly one minus 63⁄64 to the power of the map's three-by-three windows. Four by four is the first map with a refusal no window explains, 64 of its 2,112. The rise is smooth, not sharp: a quarter at six by six, near half at eight, seven tenths at ten.
A tube refuses round itself
On a flat map the cheap refusal — a circle in the arcs a lettering forces — is almost entirely local: small windows of the map explain all but a sliver of it. Glue the map's left edge to its right and the windows still refuse at exactly the rate their number predicts, counted across the seam. But a new refusal appears that no window can contain, a circle that goes once round the tube. It refuses half of every lettering of a tube two panels round, over half of one four round, a quarter at six, and it falls by about half with every two panels of circumference while the tube's length hardly matters.
The long circles go round the middle
A four-by-four map has 64 letterings the one-pass refusal catches with no three-by-three pinwheel anywhere in them, and they are not larger pinwheels. They are four shapes, sixteen versions of each, none unchanged by any turn or mirror, and every one is a circle of twelve creases running round the middle of the map. Read the same way on larger maps, every refusal is a circle round some rectangle of panels, and it needs the whole of that rectangle: one in 64 three-by-three rectangles closes one, one in 512 four-by-fours, and rectangles three cells across close nothing a pinwheel does not. Two window sizes explain all but 0.07 per cent of a twenty-by-twenty map's letterings.
All the figures · The what it costs to know field · The patterns a reader can fold