The event is an incircle

The skeleton changes its mind slid one corner of a pentagon along its top edge and counted the creases in the molecule that fills it. The count sat at six, jumped, and sat at seven, and the jump was located by sampling at 0.2872 of the way along. The essay closed on the question that matters to anybody holding paper: the event is a single shape, but a shape a hair away from it has a molecule with a very short crease in it, and a very short crease is a fold nobody can make. How wide is the neighbourhood, in units a folder would recognise?

Answering it turns up three things. The event is not an accident of the sampling; it is the one shape of the family that has an inscribed circle, and the jump sits at exactly two sevenths. The short crease is born much more slowly than it dies, so the neighbourhood is lopsided and most of it lies past the event. And for polygons with more corners than five, a molecule sitting in such a neighbourhood is common rather than rare.

Where the jump is, exactly #

The molecule here is the polygon’s straight skeleton, the tree traced by the corners of the polygon as every side moves inward at the same speed. Each point where the tree branches is a point at the same distance from three or more sides at once: the centre of a circle touching those sides. A generic pentagon’s skeleton has three such points, each touching three sides, joined by two interior creases.

The slid pentagon has corners at $(0,0)$, $(1,0)$, $(1,1)$, $(t,1)$ and $(0,h)$. The circle of radius one half centred at $(\tfrac12, \tfrac12)$ touches its bottom, right and top sides, and it touches the left side at $(0, \tfrac12)$, which is on that side whenever $h$ exceeds a half. It touches the fifth, slanted side only for one position of the sliding corner, and solving for it gives

$$t^{*} = 1 - \frac{1}{2h}.$$

At that shape the pentagon has a circle touching all five sides, every inward-moving side reaches the centre at the same instant, and every crease of the molecule meets at one point. For a left side 0.7 high the formula gives two sevenths, 0.2857. The earlier essay’s 0.2872 was the first sample past it on a grid of two hundred steps; the event is exact and the sampling was not.

That identifies the event with something a designer can test with a ruler rather than a skeleton algorithm. A polygon is at an event of this kind exactly when it has an incircle, and near one exactly when it nearly does. For a quadrilateral that is a classical condition: Pitot’s theorem says a convex quadrilateral has an inscribed circle exactly when the sums of its opposite sides are equal. For a pentagon there is no such clean test on the sides alone, but the geometry is the same — the molecule’s interior creases measure how far the polygon is from having a circle that touches every side.

The crease that is too short #

The shortest crease is not a crease sets the scale this needs. Aligning two points by eye to better than half a millimetre is hard, and a crease shorter than the error in placing its own ends has a direction nothing the folder does can determine. So take half a millimetre as the shortest crease a folder can make on purpose, and ask for which shapes of the slid pentagon the molecule has a crease shorter than that.

The shortest crease is a straight line in the corner’s position on each side of the event, and the two lines have different slopes. With the left side 0.7 high, the dying crease shrinks by 1.19 of the polygon’s side for each unit the corner moves; once past the event the new crease grows by only 0.52. On a pentagon a hundred millimetres across, the molecule carries a crease shorter than half a millimetre for 0.42 mm of the corner’s travel before the event and 0.96 mm after it, 1.38 mm in all.

That is a small neighbourhood on this pentagon, and it is not small everywhere. Move the dial and the event slides along the edge and the slopes change with it; the neighbourhood is narrowest near a left side 0.7 high and widens on either side.

Born slowly, every time #

The asymmetry is not a quirk of one pentagon. Across the whole family, the crease that dies always dies faster than the new one is born.

The dying crease’s rate rises gently from about 0.85 at a left side 0.55 high to 1.41 near the top of the range. The new crease’s rate rises to about half a side per unit near 0.7 and then falls away toward nothing at both ends, so the ratio between them runs from 2.2 at its best to 21.8 at a left side 0.975 high. A molecule that has just passed an event keeps its short crease for between two and twenty times longer than one about to reach it.

The two creases are not the same object, and there is no reason for their rates to match. Before the event, the short crease joins the centre of a circle touching the bottom, left and slanted sides to the centre of one touching the right, top and slanted sides; after it, the two branch points that separate touch different triples of sides. How fast a branch point moves as the polygon changes is set by the angles between the sides it touches, so each crease’s rate is a fact about its own triples, and the family happens to give the new crease the slower pair every time. Which angles decide the ratio, and whether some other family of polygons reverses it, is not worked out here; the measurement is that across this whole family it never reverses.

The span in millimetres #

Putting the slopes and the folder’s scale together gives the neighbourhood’s width for any pentagon of the family and any shortest foldable crease.

At half a millimetre the span is 1.38 mm at its narrowest and grows to 5.06 mm for a left side 0.55 high and 5.38 mm at 0.95. The spans scale in proportion to the shortest crease a folder allows, so a folder who needs two millimetres to crease cleanly faces spans four times wider — over twenty millimetres at the ends of the family, a fifth of the polygon’s side. And the share of each span lying past the event never falls below seventy per cent.

The span in millimetres does not depend on the polygon’s size at all: it is the shortest foldable crease divided by each of the two rates, and the rates are pure numbers. What changes with size is the span as a share of the polygon. A molecule forty millimetres across, a common size inside a base folded from a square of twenty-five centimetres, has the same 1.38 mm span at its best, which is two and a half times as large a share of its side as on the hundred-millimetre polygon. Small molecules are where the unfoldable neighbourhood takes the largest bite out of the design space.

A quadrilateral’s crease is a ruler reading #

The quadrilateral makes the incircle connection exact enough to use. A convex quadrilateral’s molecule has at most one crease that does not touch the outline, and by Pitot’s theorem the quadrilateral has an incircle — and that crease vanishes — exactly when $a + c = b + d$ for its sides taken in order.

Measured on six hundred random quadrilaterals, the interior crease is never shorter than half the difference $\lvert a + c - b - d \rvert$, and its median is 0.51 of that difference. So a quadrilateral whose opposite sides’ sums differ by more than twice the shortest foldable crease cannot have a crease that short in its molecule, and one whose sums nearly agree nearly always does. That is a check a designer can make with the side lengths of a layout and no skeleton at all. The lower bound held on every quadrilateral drawn and is not proved here; the upper side of the cloud spreads out, because the crease can be much longer than half the difference when the quadrilateral’s angles are unequal.

How often a molecule sits near an event #

A family of pentagons slid along an edge is a line through the space of shapes, and a real design is a point somewhere else in it. The question for a designer is how likely any given molecule is to be near an event, and that needs a population.

Drawn as the convex hulls of random points in a square, fifteen hundred of each size, the polygons whose molecules carry a crease shorter than one per cent of their own size are 1.6 per cent of quadrilaterals, 3.4 per cent of pentagons, 7.7 per cent of hexagons and 11.9 per cent of heptagons. At five per cent the shares are 7.7, 19.7, 34.4 and 51.3. Half of all random heptagons carry a crease shorter than a twentieth of their size.

The growth with corners follows from the counting. A polygon with $k$ corners has $k - 3$ interior creases in a generic molecule, so more corners means more creases that could be short, and each is short near its own kind of event. For small limits the shares grow roughly in proportion to the limit, which is what a crease length that falls linearly to zero at an event predicts: the fraction of shapes within a distance $\varepsilon$ of any event is proportional to $\varepsilon$.

The escape is to go all the way #

The lopsided span suggests a rule for a design that finds itself near an event, and it is not “move away”.

A molecule sitting at the event has no short crease. All of its creases meet at the incircle’s centre and there is nothing between two branch points to be too short. From inside the span there are three ways out: forward until the new crease is long enough, back across the event and on until the dying crease is, or back onto the event and stop. The third is always shorter than the second, and it is shorter than the first for every design in the half of the after-span nearest the event. With a left side 0.85 high, a design a millimetre past the event carries a crease of 0.29 mm; the event is a millimetre back, the far edge of the span 0.7 mm on, and the near edge 1.36 mm back. For half of every span the nearest molecule with no unfoldable crease is the degenerate one, and the degenerate molecule is the simplest the polygon can have.

In tree-method terms the degenerate case is a polygon whose flaps’ circles are arranged so that one circle touches every side of the polygon they leave. Each flap already costs a circle on the sheet, and the polygon between the circles is whatever the packing leaves; packing is the hard part because those leftovers are not chosen directly, which is also why a designer finds a molecule near an event by accident rather than on purpose. That is a condition a designer can impose by choosing flap lengths, and the last free parameter is the observation that once the packing is fixed there is exactly one number left in a molecule. Near an event, spending that freedom on reaching the event rather than avoiding it is the move that removes the short crease.

What this picture cannot show #

Whether the degenerate molecule folds. All five creases of the pentagon’s molecule meet at one point at the event, with the hinge creases from the tangent points running in to meet them. The standing account is that a molecule always folds; this essay does not check a flat lettering of the single-point molecule, and a vertex where many creases meet is where a lettering is most constrained.

Whether the molecule is the skeleton. The molecule drawn here 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. When one does, the universal molecule splits the polygon along that path, and the creases near the split are different from the skeleton’s. None of the polygons here was checked against a tree, since none of them came from one.

What a real sheet does with a short crease. Half a millimetre is taken from the alignment error of hand folding; a crease slightly shorter than that is not impossible, only undetermined in direction. The spans are the shapes where a crease falls under a stated scale, not the shapes where the model fails to fold.

And a random polygon is not a design. The census draws polygons as hulls of random points, which favours some shapes over others and has no reason to resemble the polygons a tree method produces. It says near-events are common among polygons in general; how common they are among real molecules is a question about real designs.

The idealisations underneath #

The paper is of zero thickness and the creases are lines. The polygon is a hundred millimetres across for every quoted span, and each span scales in proportion. The slopes are measured a five-hundredth of a side either side of the event, where the shortest crease is still straight in the corner’s position to many figures, and the spans assume that straightness; the shortest crease of 0.5 mm is well inside it. The molecule is Lang’s, from the tree method set out fully in Origami Design Secrets; the incircle reading of its events is the straight skeleton’s geometry and is not a claim about the tree method’s own history.

How the claims were checked #

The event is found twice. Halving on the molecule’s crease count finds where it jumps, and the answer is required to agree with $1 - 1/(2h)$ to a millionth of a side — so the incircle is a checked identity for every left-side height on the dial, not a formula fitted to one.

At the event the molecule is required to have a single branch point, and it is required to sit at the circle’s centre, $(\tfrac12, \tfrac12)$, to nine figures.

The born crease is required to be slower than the dying one at every left-side height from 0.53 to 0.975, and the ratio is required to exceed two.

The quadrilateral bound is required on every quadrilateral drawn, crease at least half the difference of the opposite sums, so a single quadrilateral below the line would stop the figure; and the census is required to rise with the corners at its smallest limit.

Still open: whether the degenerate molecule is the right one #

Two measurements follow directly. The first is the lettering of the single-point molecule: a pentagon with an incircle has one vertex where five skeleton creases and five hinge creases meet, and a tree cannot argue found molecule letterings free of contradictions on seven outlines. Whether the degenerate one keeps that property, and how many letterings it has against its neighbours either side, would say whether moving onto the event costs anything the crease count does not show.

The second is a pentagon test on the sides. Pitot’s condition makes the quadrilateral’s near-events readable from four lengths. A pentagon with an incircle satisfies a condition on its sides and angles together, and a single number measuring how far a pentagon is from having one — the analogue of the Pitot difference — would give a designer the same ruler check for five-sided molecules, with a bound to put on it.

Sideways from here, the corner that splits the shrink is about the other kind of event, the split a reflex corner causes, and its molecule does not exist rather than being short. A polygon near a split is a polygon nearly reflex; the census here counts only convex shapes, and whether near-splits are as common in real layouts as near-incircles are in random ones is the next thing a population of tree-method polygons would show. The molecule that does not exist and from packing to crease pattern are where such a population would come from.

The habit worth carrying is about degeneracies. When a construction changes at an isolated shape, find what is special about that shape before measuring around it. A jump sampled at 0.2872 looked like a feature of the pentagon; recognised as a circle touching five sides, it became a condition with a formula, a classical theorem for four sides, and a place to go rather than a place to avoid.