Six creases belong to the square

Two essays on the universal molecule rest on one family of pentagons. The skeleton changes its mind slid one corner of the pentagon $[0,0]\ [1,0]\ [1,1]\ [t,1]\ [0,h]$ along its top edge and found the molecule filling it holding six creases, jumping, and holding seven. The event is an incircle located the jump exactly at $t = 1 - 1/(2h)$, recognised it as the one shape with a circle touching all five sides, and measured the crease born there growing two to twenty times more slowly than the crease that died.

Both essays were right about the family. The family is not a typical pentagon, and the reason is in its coordinates: four of its five sides lie on the unit square. A square has an incircle — the circle of radius one half about the centre touches all four of its sides — so every member of the family has four side lines that touch one circle, whatever $t$ is. That is a coincidence a random pentagon has with probability zero, and it turns out to be responsible for the six creases, the jump and the lopsided rates, all three.

Widen the square by a sixth #

Put the family’s four fixed sides on a rectangle $w$ wide instead — the pentagon $[0,0]\ [w,0]\ [w,1]\ [t,1]\ [0,h]$ — and slide the corner the same way. At $w = 1.15$ with a left side 0.7 high, the molecule has seven creases at every position of the corner. Halving on the crease count finds nothing to halve on, because the count never changes. On the dial, every width from 1.05 to 1.4 does the same.

Something still happens, and the picture shows what. The shortest interior crease falls to zero at two isolated positions, 0.2857 and 0.5150 of the way along, and rises again after each. At each of those shapes the two branch points at the ends of the short crease merge into one, where four creases meet; either side, there are two branch points and seven creases, but the short crease joins them across a different pair of sides. This is a flip: the molecule’s combinatorics change and its crease count does not.

The first flip does not move with the width, and that gives the game away. It sits at $t = 2/7$ on every rectangle, which is $1 - 1/(2h)$ at $h = 0.7$, the event the incircle essay found. The four side lines meeting at it are the bottom, the top, the slanted side and the left — and the right side is not one of them. The circle of radius one half about $(\tfrac12, \tfrac12)$ touches the bottom, top and left of every rectangle one unit high, and it touches the slanted side exactly at $t = 1 - 1/(2h)$. On the square it touches the right side as well, and only then does the flip become an incircle.

What six creases actually were #

Back on the square, the same bookkeeping explains the stretch of six. A generic pentagon’s molecule has three branch points, each where three of the shrinking outline’s edges meet, and seven creases. The square family before the event has two branch points, and one of them is exactly at $(\tfrac12, \tfrac12)$, where four creases meet: the bottom, the right, the top and the left sides all shrink onto the square’s centre at the same instant, because the square’s four sides are all half a unit from it.

So the six-crease molecule is not a different kind of molecule that holds for a while and then gives way. It is a flip held open for a whole interval, a degenerate molecule the family cannot leave because the square keeps four of its lines tangent to one circle for every value of $t$. The “jump” to seven at the event is the moment the slanted side reaches the same circle, all five lines are tangent, and the degeneracy finally resolves into a generic molecule.

That also explains the numbers the incircle essay measured and found puzzling. On the square the crease that dies at the event shrinks 1.19 of the height per unit of the corner’s travel, and the crease that is born grows 0.52: a ratio of 2.28, and more than twenty on other left-side heights. Off the square, the dying crease is the same crease at the same rate, 1.19, because its four lines do not include the right side; the born crease grows at 0.92, and the ratio is 1.30.

The table carries the rest. At the second flip, the one that exists only off the square, the rates are the other way round — 0.77 dying against 0.89 born at $w = 1.15$, and a ratio between 0.75 and 0.86 on every rectangle measured. “Born slowly, every time” was a statement about the square. On a generic pentagon a flip is roughly symmetric, the born crease sometimes faster and sometimes slower, and the band of unfoldably short creases round it sits roughly evenly either side rather than seventy per cent past it.

The span, off the square #

The incircle essay’s practical result was a width in millimetres: on a pentagon a hundred millimetres across, how far the corner can move while the molecule carries a crease shorter than half a millimetre, the scale below which hand folding cannot place a crease. Its answer depended on the two rates, because the span is the distance the dying crease takes to fall from half a millimetre to nothing plus the distance the born one takes to grow back.

On the square at a left side 0.7 high, the dying crease covers its last half-millimetre in 0.42 mm of the corner’s travel and the born crease needs 0.96 mm to grow one: a span of 1.38 mm, seventy per cent of it past the event. That was the reason for the incircle essay’s rule that the cleanest escape is to move onto the event rather than away from it.

Off the square the same arithmetic on the first flip gives 0.42 mm before and 0.54 mm after — a span of 0.96 mm, 57 per cent of it past the flip. At the second flip, where the born crease is the faster, the span is 1.21 mm and only 46 per cent of it lies past. So the band of unfoldable shapes around a generic flip is narrower than the square’s by about a third and sits roughly evenly either side, and the rule about which way to escape turns into a much weaker one: the flip itself is still the nearest shape with no short crease for about half of the band, but there is no longer a direction that is systematically better than the other.

That is worth stating plainly because the square’s numbers were the only ones anybody had measured. The shortest crease is not a crease measured how often a printed pattern carries creases too short to place; a designer reading the incircle essay would have budgeted a band of up to five millimetres and a strong preference for one side of it, both of which were the square talking.

An incircle is two flips at once #

The general statement is short. A pentagon’s molecule flips wherever four of its consecutive side lines touch one circle — one condition on the shape, so the flips of a sliding family are isolated points. It has an incircle where all five touch — two conditions, so a sliding family generically never meets one at all. A flip is codimension one and an incircle is codimension two, and the incircle is exactly the place where two flips coincide: both interior creases vanish together and the molecule’s three branch points collapse to one.

The census of random pentagons shows how different those two neighbourhoods are.

One crease shorter than a hundredth of the pentagon’s size occurs in 4.7 per cent of fifteen hundred random pentagons, and shorter than a twentieth in 19.6 per cent — a different draw from the one the incircle essay’s census made, which found 3.4 and 19.7. Both creases that short, which is what nearness to an incircle means for a pentagon, occurs in none at a hundredth or a fiftieth, in 0.6 per cent at a twentieth and in 3.7 per cent at a tenth. The first share grows in proportion to the limit and the second as its square, which is what one condition against two predicts.

So the short creases a designer actually meets in pentagonal molecules are almost never near an incircle. In tree-method terms a pentagon is the space a packing leaves between five flaps’ circles, each flap having cost a circle on the sheet, and nothing in a packing makes all five of the leftover polygon’s sides touch one circle except the designer arranging it; packing is the hard part because those leftovers are not chosen directly. They are near flips — near a shape where some four of the side lines nearly share a circle and the fifth is nowhere near it — and the incircle essay’s advice to “go all the way to the event” becomes: go to the flip, where four creases meet at one point and the short crease is gone. That shape exists in every family that crosses a flip, which is most of them.

The ruler check a pentagon does have #

The incircle essay’s most useful result was a check a designer can make with a ruler. For a quadrilateral, Pitot’s theorem says an incircle exists exactly when the two pairs of opposite sides have equal sums, and the molecule’s one interior crease was never shorter than half the difference. It asked for the pentagon analogue.

The flip is what gives it. A pentagon’s interior crease sits between two branch points, and between them it touches four of the pentagon’s side lines — every line but one. Drop that side, extend its two neighbours until they meet, and the four lines make a quadrilateral, and the crease is often that quadrilateral’s own interior crease. When it is, the quadrilateral bound applies to it directly.

Of 1,600 interior creases in 800 random pentagons, 1,358 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, with a median of 0.52 — the quadrilateral result carried over exactly. The other 242 are creases the quadrilateral would draw differently, because in the quadrilateral a different pair of edges shrinks away first; for those the bound fails, down to a hundredth of the difference.

So the pentagon’s ruler check is five Pitot checks, not one: for each side, drop it, extend its neighbours, and compare the sums of the resulting quadrilateral’s opposite sides. A flip is a zero of one of the five, and a crease is protected by its own quadrilateral’s difference whenever that quadrilateral shrinks the same way the pentagon does. A pentagon that cannot be filled at all is the other case, the molecule that does not exist, and it is not a near-flip: a reflex corner is not a matter of four lines nearly sharing a circle. What the check cannot do without the skeleton is say which of the five belong to the molecule’s two creases.

What five sides can say #

The incircle essay hoped for a single number from the sides alone. For a pentagon there is a reason that cannot work, and it is about odd numbers.

A polygon with an incircle has, at each corner, two tangent lengths equal to each other: the distances from the corner to where the circle touches its two sides. So each side is the sum of two tangent lengths, and for a quadrilateral the four equations for four sides can be solved only when the alternating sum of the sides is zero — which is Pitot’s condition. For a pentagon, five equations in five tangent lengths always have exactly one solution, whatever the sides are. The sides alone never rule an incircle in, because a set of tangent lengths is always available; whether a circle actually touches all five depends on the corners’ angles agreeing with those lengths, and the angles are not in the sides.

The one thing the sides can do is refuse. If the unique solution has a negative tangent length, no incircle exists whatever the angles are, and among fifteen hundred random pentagons 71 per cent are refused that way by their sides alone. The other 29 per cent pass a test that means nothing on its own: the regular pentagon and a flexed pentagon with identical sides both pass it, and only one of them has an incircle. Every odd number of sides behaves this way, and every even number above four needs the alternating sum to vanish and the angles to agree as well.

What the picture cannot show #

Whether a flip folds. At a flip four creases meet at a branch point instead of three, and with the hinge creases running in from the outline that point carries more creases than any generic branch point. The lettering of such a molecule is not checked here, and it is the same open question the incircle essay left about its single-point molecule, now asked of a far more common shape.

Whether the molecule is the skeleton. Every molecule drawn is the polygon’s straight skeleton, which is what Robert Lang’s universal molecule is when no path between two non-adjacent corners becomes active during the shrink. None of the polygons here came from a tree, so none was checked for an active path, and a real design’s molecule may split where these do not.

And a rectangle is still a special shape. Widening the square breaks the one coincidence that made the six creases, but a rectangle has four right angles and two pairs of parallel sides, and the families drawn are one-parameter slices through a five-dimensional space of pentagons. The census is what speaks for pentagons in general, and it says flips are common and incircles rare.

The idealisations underneath #

The paper is of zero thickness and the creases are lines. The molecule is the straight skeleton, computed by shrinking the outline edge by edge. Flips are found by halving on the molecule’s combinatorics — which pairs of sides each interior crease separates — rather than on its crease count, since off the square the count never moves; rates are measured a five-hundredth of the height either side of each flip. The random pentagons are hulls of seven random points in a square, the same population the census in the incircle essay used, and share its bias toward some shapes over others.

How the claims were checked #

The square’s six-to-seven change and the rectangles’ flips are found the same way, by halving, and the first flip on every rectangle is required to fall at $1 - 1/(2h)$ to a hundred-thousandth, while the crease count is required to be the same at every sampled position of the corner off the square and to change exactly once on it.

At each flip the molecule is required to have four creases meeting at one point — or a crease of no length — with the same crease count either side and the short crease separating different pairs of sides before and after.

The quadrilateral bound is required on every shared crease, at least half its quadrilateral’s Pitot difference, and the unshared creases are required to include some below it, so the figure cannot quietly claim a bound for creases it does not hold for.

The flexed pentagon is required to keep all five sides equal to the regular one’s, to be convex, and to imply radii spread by more than a twentieth, while the regular one’s five implied radii agree to nine figures.

Still open: a flip in a design #

The measurement this sets up is the one both earlier essays pointed at from the other side. A population of molecules from real tree-method layouts would say how often their pentagons sit near flips, and whether the four-line coincidences that make flips are more common in designed layouts than in random ones — which they may well be, since a designer’s packings are full of equal flaps and aligned edges, and the last free parameter is exactly the freedom that moves a molecule toward or away from one. From packing to crease pattern is where such a population would come from.

The lettering of a flip is the other. A tree cannot argue found molecule letterings free of contradictions on seven outlines because a molecule’s panels form a tree; a four-crease branch point changes the tree locally but not its having no loops, so the prediction is that a flip letters as freely as its neighbours do. That is a finite check on the three molecules drawn above.

Sideways from here, the square’s degeneracy is a warning about test cases that belongs beside the corner that splits the shrink, whose split event is also measured on hand-drawn outlines. A family chosen for easy coordinates tends to be chosen with right angles and equal lengths in it, and those are exactly the coincidences that make a generic event degenerate.

The habit worth carrying is about the families measurements are made on. Before generalising from a sweep, ask what the family’s easy coordinates have made coincide. Four sides on a unit square looked like the simplest possible pentagon, and it was: simple enough that four of its lines always shared a circle, which turned a flip into six creases, a jump and an incircle.