One cut for a star

Fold a sheet of paper flat so that a drawing on it lands entirely on one line, cut along that line, and the drawing falls out. The theorem says this is possible for any drawing made of straight segments, which is the most generous statement in the subject: any polygon, any number of them, any arrangement, one cut.

The construction that realises it has two ingredients — the straight skeleton of the drawing, and perpendiculars dropped from the skeleton’s nodes onto the edges that formed them. The first is defined by a process: shrink the shape by moving every edge inward at the same speed, and the traces the corners leave are the skeleton.

That process is implemented here literally, and until now it refused every outline that turned back on itself. A convex shape shrinks by losing edges; a shape with a reflex corner does something else, and the something else had to be written.

What a reflex corner does to a shrink #

Shrinking a convex outline is uneventful. Every corner moves inward along its own bisector, every edge gets shorter, and the only thing that ever happens is that an edge runs out of length and vanishes — an edge event, at which two corners arrive at the same place and one corner leaves.

A corner that turns the wrong way is the exception, and it is the exception in the most literal sense: it is the one place on the outline where the shrinking region gains ground. The two edges meeting at a reflex corner are moving inward, but the corner between them is on the outside of that motion, so it travels outward along its bisector, faster than the edges are travelling in.

Sooner or later it arrives at an edge on the far side of the shape. When it does, the shrinking region touches itself and breaks into two pieces, each of which then goes on shrinking alone. That is a split event, and it is the whole of what the reflex case needs.

Finding the moment, rather than searching for it #

The time a split event happens is solved for, not swept. An edge that has been moving for a time s is the set of points at signed distance s along its inward normal from where it started. The reflex corner at the same moment is at its starting point plus s times its own velocity. Setting the two equal is one linear equation in s, and it has one root.

Three conditions make a root a real event: it must be in the future, the corner must arrive between the two ends of the edge as they are at that moment rather than off the end of it, and the edge must still exist. All three are checked, and the earliest surviving event of either kind is the one that happens.

The events are recomputed from scratch after each one. That is the slower of the two obvious arrangements and it is the one that cannot go stale, because a split changes the structure of the wavefront and every velocity in it.

The order of two events at the same instant #

One detail decided whether an L worked, and it is the kind of detail that is invisible until a shape is degenerate on purpose.

On a rectilinear outline, everything happens at once. An L two units across collapses its notch, its short side and its far corner all at half a unit of shrinking, and the reflex corner reaches the opposite edge at the same moment. With a tie, the order matters: take the edge event first and it removes an edge the split still needed, and the skeleton comes out missing two arcs. Take the split first and the region breaks into two halves, each of which then has its own edge event with nothing in the way.

So splits are preferred at equal times. That is not a tie-break for tidiness — it is the difference between a correct skeleton and one that quietly loses a fifth of itself on the shape everybody draws first.

Two implementations, required to agree #

A new routine that handles more cases can silently change the cases it already handled. The convex shrink is what eight printed patterns were verified against, and the order in which it emits its arcs decides which lettering a search settles on, so it is left exactly as it was and the general routine is checked against it.

On five convex outlines — the triangle, the square, the rectangle, the pentagon and the house — the two produce the same nodes at the same coordinates and the same arcs between the same endpoints, to every digit. They share nothing but the arithmetic of a line: one shrinks a single ring of edges and knows nothing about splits, the other carries a set of loops and computes both kinds of event.

The property the whole construction rests on is checked separately and on every shape: a node of the skeleton is the same distance from every edge that formed it, which is why one fold can carry several edges onto the line at once. It holds on the reflex shapes as it does on the convex ones.

The check that was missing #

Until now nothing here tested the thing the theorem is actually about. The skeleton’s equidistance was asserted, the pattern’s vertices were put through the four conditions, and the sentence the whole drawing lands on one line was carried by the construction’s reputation.

It is now a measurement, which is the collection’s own rule applied to its own machinery: an assertion that has never rejected anything proves nothing. Fold the pattern flat by composing reflections — with one rule that is the whole point, that the outline is drawn on the paper and is not a crease, so crossing it turns nothing over — and carry every sampled point of the outline by the motion of the panel it sits in. The images are then asked whether they are collinear.

On the triangle, fifteen sampled points, none further than 2 × 10⁻¹⁶ of a sheet width from the line through the first two. On the house, twenty-five points at 4 × 10⁻¹⁵. On the L, thirty points at 2 × 10⁻¹⁶. On the star, fifty points at 3 × 10⁻¹⁶. That is a claim about the construction rather than about the theorem, and it is the claim a reader with scissors is going to test.

The star, which needs no perpendiculars at all #

The five-pointed star is the surprise of the three new shapes. Its skeleton is a single node — the centre — with ten arcs running out to the ten corners, and not one perpendicular survives: every foot dropped from the centre onto an edge lands off the end of that edge, and is dropped.

The pattern is therefore ten creases through one point and nothing else. It folds: four hundred and twenty of its letterings satisfy every vertex condition, an ordering of its panels exists, and the outline lands on one line to a part in 10¹⁶.

The reason is symmetry rather than luck. All ten edges of a regular star are the same distance from its centre — which is why the shrink collapses to a point in one event — and the ten sectors between consecutive creases are congruent. Folding the sectors onto one another therefore carries all ten edges onto the same segment. It is the same structure the regular pentagon has, with twice as many sectors and half of them reflex.

Four hundred and twenty is one binomial #

The star’s lettering count is not a number a search happened to arrive at. It is Maekawa’s theorem evaluated once, and writing it out is worth doing because it explains why the count is so large.

Ten congruent sectors means every sector equals every other, so no sector is strictly smaller than both its neighbours and the smallest-sector lemma has nothing to act on. Kawasaki’s alternating sums are five sectors each and equal by symmetry. Maekawa is the only condition doing anything, and it admits the letterings with six of one letter or four:

$$\binom{10}{6} + \binom{10}{4} = 210 + 210 = 420$$

Exactly the reported number, out of $2^{10} = 1024$ — so forty-one per cent of all letterings of the star pass every condition, which is an enormous share by this collection’s standards and is entirely because the pattern is one vertex with equal sectors.

A dropped foot is a parity, not a defect #

The dart’s refusal deserves the same treatment, because the essay’s account makes a lost perpendicular sound uniformly bad and it is not.

A node’s degree is the arcs meeting there plus the perpendiculars that survive. Maekawa needs the mountains and valleys to differ by two, which requires an even degree — so what a lost foot does is flip one node’s parity.

An even number of losses at a node changes nothing. Two feet falling off the same node leaves its degree even and its foldability intact; one foot falling off makes it odd and unfoldable whatever letters are chosen.

The dart loses four feet across three nodes and ends with two nodes odd, so the losses were distributed one, one and two rather than evenly. Had the same four losses fallen two-and-two on two nodes, every node would still be even and the pattern would have been drawn.

That is a better diagnostic than a count. The question to ask of a refused outline is not how many perpendiculars were lost but whether any node lost an odd number of them, and that is one pass over the nodes rather than a search over sixty-five thousand assignments.

It also says where the missing machinery bites. The full construction reflects an escaping perpendicular into the neighbouring region, which adds a crease back at the node it left — restoring the parity as well as the crease. So the reflection step is not an optional refinement; it is exactly what keeps every node even, and an implementation without it fails on precisely the outlines where the losses are odd.

The L, which needs eight of them #

The L is the ordinary case and is worth reading beside the star for contrast. Three skeleton nodes, eight arcs, eight of the ten perpendiculars its nodes want, sixteen creases, three interior vertices of degree six.

Its lettering was found by search rather than enumerated: sixteen creases is sixty-five thousand assignments, past the point where every one is checked, so the first that satisfies the conditions is taken. Whether its panels can be ordered is not decided — twenty-one panels is past what an exhaustive ordering search will finish — so the honest verdict on it is that it satisfies every condition this collection can check and that no folded state has been exhibited.

The dart, which is refused #

The third new outline does not work, and the reason is precise enough to be worth more than the two that do.

A five-cornered dart — a quadrilateral with a spike pushed into one side — has three skeleton nodes wanting nine perpendiculars between them, and four of the nine fall off the ends of the edges they were dropped onto. A perpendicular whose foot is not on its edge is not a crease of the pattern, so it is not drawn, and the node it came from is left with five creases instead of six.

An odd vertex cannot fold. The mountains and valleys at a flat-foldable vertex differ by two, and an odd number of creases cannot produce an even difference — so two of the dart’s three nodes are unfoldable whatever letters are chosen, and the search comes back empty. The construction is refused rather than drawn.

That refusal is where the general theorem’s remaining machinery lives. The full construction reflects a perpendicular that leaves its own region of the skeleton back into the next one, adding creases that this implementation does not have; the missing feet are exactly the places that would happen. So the honest position is that the skeleton is now general and the pattern is not, and the boundary between them is a measurable count rather than a caveat.

The star was cut long before it was proved #

The five-pointed star is the shape this theorem is always advertised with, and the reason is historical rather than mathematical. A one-cut star appears in an American conjuring book of 1721 and in Japanese collections long before that, folded by a sequence somebody worked out at a table, two and a half centuries before anybody proved that every straight-line drawing could be done.

That gap is the ordinary shape of this subject and it is worth noticing which half is which here. The traditional star fold is a sequence — bring this corner to that crease, fold in half, cut — and it produces one shape. What the construction gives is a pattern, derived from the outline by a rule, which produces whatever outline it was given. The traditional fold is not this pattern: it comes from halving the square and bisecting angles, and it lands the star’s edges on a line by a different route.

So the two agree on the object and on nothing else, which is exactly what a rediscovery looks like from the inside: a construction reaching a result that a craft tradition had already reached, with the value of the construction being everything else it also reaches.

What the pattern costs to fold #

Three numbers say what an evening with each of these is like, and they separate the three new shapes more sharply than their pictures do.

The star is 2.43 sheet widths of crease across ten creases, all of them straight lines through one point — the cheapest of the three to fold and the easiest to fold accurately, because every crease can be made by bringing one edge of the fan onto the previous one. The L is 3.07 across sixteen creases at three separate vertices, which is a harder fold: the three vertices have to collapse together and nothing aligns them for the folder. The house, the most complicated convex outline here, is 4.13 across eighteen.

Crease length rather than crease count is what a hand feels, and the ordering by the two measures is not the same: the star has ten creases and the L sixteen, a ratio of 1.6, while their lengths are 2.43 and 3.07, a ratio of 1.3. The difference is that the star’s creases all run from the centre to a corner, which on a square sheet is a short distance, while the L’s perpendiculars run from three interior nodes to edges scattered round the outline.

None of that is visible in the crease pattern, and all of it is visible in the ten minutes after a reader prints one.

What a reader can do with it #

The star is the one to fold. It is ten creases through the centre of a square, its letters are four of one and six of the other, it is printed here at true scale, and one straight cut along the folded edge releases a five-pointed star whose points are as sharp as the paper allows.

It is also the one that best shows what the theorem is claiming, because nothing about it looks like a star while it is being folded. The pattern is a fan; the fold is an accordion around a point; and the shape appears only when the cut is made. That gap between the pattern and the object is the whole reason the crease pattern rather than the sequence is the artefact worth publishing.

What is not claimed #

Not that the construction is now general. It handles reflex corners in the skeleton and refuses outlines whose perpendiculars leave their edges. Three of the eight outlines here are non-convex, two of them produce patterns and one does not.

Not that the L folds. Local conditions at every vertex do not decide a sheet, and it satisfies every condition at every vertex and its ordering is undecided, which is the same verdict half the printed shelf carries and is a statement about the size of the search rather than about the paper.

And not that the star’s pattern is the theorem’s pattern. The general construction would give the star perpendiculars; this one gives it none, because the feet miss. The pattern that results is simpler than the theorem’s and is verified on its own terms — the outline lands on one line, which is the property being claimed, checked rather than inherited.