Generator

The last free parameter — the universal molecule generator

A generator in the designing a base library, called 37 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.

universal-molecule 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: "alternating", show: "inset"`

`view: "alternating", show: "exact"`

`view: "alternating", show: "tight", tilt: 0.36`

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.

the molecule is built on a polygon of 4 corners ×3
sliding one corner of a pentagon along its edge changes the number of creases in its molecule 1 time — at 0.2872 ×2
among 1500 random convex pentagons, one crease shorter than 1%, 2%, 5%, 10% of the size occurs in 4.7%, 8.3%, 19.6%, 37.4%, and both creases that short — the neighbourhood of an incircle — in 0.00%, 0.00%, 0.60%, 3.67% ×1
among 1500 random convex polygons of each size, the share whose molecule has a crease under 1% of the polygon's size rises with the corners: 1.6% of 4-gons, 3.4% of 5-gons, 7.7% of 6-gons, 11.9% of 7-gons; under 5%, 7.7%, 19.7%, 34.4%, 51.3% ×1
as the quadrilateral shrinks, its sides change and the alternating sum of them stays 0.301 at every stage, until side 2 vanishes; the one interior crease is 0.151, which is 0.301 over twice the sine of half the 193.0° angle sum at that side ×1
at the event every crease of the molecule meets at one point, the centre of the circle touching all five sides ×1
at the flip the shortest crease has no length and the molecule's four lines — bottom, top, slant, left — touch one circle; either side it has the same 7 creases, and the short one separates a different pair of sides ×1
every corner of every inset polygon is exactly the requested distance from its own edge, to 1e-9 — the inset is solved rather than drawn ×1
five equal sides make both the regular pentagon, whose tangent lengths all give one radius, and a flexed convex pentagon whose five corners imply radii from 0.403 to 1.894 — so side lengths cannot confirm an incircle; they can refuse one, and refuse 71.1% of 1500 random pentagons ×1
in every one of 600 random quadrilaterals the interior crease is at least half the difference between the sums of opposite sides, and the median ratio is 0.511 — so a quadrilateral near to having an incircle is exactly one with a short crease ×1
near the event the crease that dies shrinks 1.19 of the polygon's side for each unit the corner moves, and the crease that is born grows only 0.52, so the shapes with a crease under 0.5 mm on a 100 mm polygon run 0.42 mm before the event and 0.96 mm after it ×1
of 1600 interior creases in 800 random pentagons, 1358 are also the crease of the quadrilateral their four side lines make, and every one of those is at least half that quadrilateral's Pitot difference (median ratio 0.521); the other 242 are not, and for them the bound fails, down to 0.010 ×1
on 300 4-gons, 300 6-gons, 300 8-gons, 300 10-gons, the interior creases add up to at least half the alternating sum of the sides every time — least ratios 1.000, 1.036, 1.121, 1.248 ×1
on 599 random convex quadrilaterals the interior crease equals the alternating sum over twice the sine of half the angle sum at the side that vanishes first, to 1.8e-14, and so is never below half the alternating sum ×1
on a 100 mm pentagon the shapes with a crease under 0.5 mm span 5.06 mm at h = 0.55, 2.61 mm at h = 0.6, 1.38 mm at h = 0.7, 1.67 mm at h = 0.8, 2.06 mm at h = 0.85, 2.88 mm at h = 0.9, 5.38 mm at h = 0.95 of the corner's travel, and at least 70% of each span lies after the event ×1
on a rectangle 1.15 wide the molecule has 7 creases at every position of the corner and changes only by two flips, at 0.2857 and 0.5150; the first is 1 − 1 ⁄ 2h, where four side lines touch the circle of radius ½ about (½, ½), and the right side is not one of them ×1
on every pentagon of the family the crease that is born grows more slowly than the dying one shrank — between 2.24 and 21.8 times more slowly ×1
on the square every change is a change of crease count; off it every change is a flip with the count kept, and the dying crease is not always the faster — its rate over the new one's runs from 0.75 to 1.41, against 2.28 on the square ×1
the change is a whole crease appearing, from 6 to 7, with nothing continuous about it ×1
the construction succeeds on exactly the convex polygons and refuses exactly the reflex ones — so the figure's sentence about its limit is tested rather than asserted ×1
the molecule changes its crease count at 0.2857, which is 1 − 1 ⁄ 2h for h = 0.7: the shape at which the pentagon has a circle touching all five sides ×1
the pentagon drawn round a circle has every crease meeting at the centre and no interior crease, while its five alternating sums, 0.67, 0.66, 0.70, 0.80, 0.81, are each twice the tangent length at the corner where the signs collide ×1
the polygon admits a shrinking, so the molecule has levels to draw ×1
with the top tilted by 0.36, the right side vanishes first; its two corners add to 195.5°, and the crease is 0.509 against half the alternating sum, 0.504 ×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.

Half the difference is a theorem

A quadrilateral's molecule has one crease that does not touch the outline, and on every quadrilateral measured it was at least half the difference between the sums of opposite sides. That was a measurement and has a proof. Shrinking the shape changes every side but not their alternating sum. The proof gives the crease exactly: the difference over twice the sine of half the two angles at the side that vanishes first. A projection argument extends the bound to every even polygon and explains why no odd polygon has one.

Six creases belong to the square

Every measurement of the pentagon whose molecule changes its mind was made on one family, and four of that family's sides lie on a square. A square has an incircle, so for the whole first stretch of the corner's travel four side lines touch one circle and the molecule carries a node where four creases meet: six creases, held degenerate for an interval. Widen the rectangle by a sixth and the molecule has seven creases at every position. What happens instead is two flips, where the short crease shrinks to nothing and regrows between another pair of sides — and a pentagon near an incircle is rarer than one near a flip by about the square.

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

The event is an incircle

The universal molecule changes its crease count at isolated shapes, and near one it carries a crease too short to fold. On the pentagon family where the change was found, the shape is exactly the one with a circle touching all five sides, and the short crease is born two to twenty times more slowly than it dies — so the unfoldable shapes lie mostly past the event, and the cleanest escape is to move onto it.

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

All the figures · The designing a base field · The patterns a reader can fold