A dissection that never comes apart

The rung below fixed an operation set — cut along a straight line, pick a piece up, put it down somewhere else — and asked what it reaches. The answer in the plane is everything of the right area.

Now narrow the operation set, which is the manoeuvre this field is made of — the same one that turns a compass and a straightedge into a question with an impossibility in it. Forbid picking anything up: the pieces must stay joined at their corners, and the only motion allowed is swinging one about another on a pin. That is a strictly smaller set of moves, so the reach can only shrink, and the question is by how much.

The answer is that it does not shrink at all, and that was open from 1902 until 2007.

The first step was already hinged #

The pleasing part is that nothing had to be redesigned. The construction the rung below used — slice at half the height, cut the small triangle down its own altitude, turn each half a half turn — turns each piece about the end of the slice it stands on, and that end is a point the piece shares with the trapezoid below it.

A point two pieces share and turn about is a hinge. So the standard first step of every plane dissection there is comes hinged, and reading it that way requires no new cuts, no new pieces and no new argument. What it requires is noticing that the rotation centres are shared vertices rather than arbitrary points.

That the two readings agree on the page and differ as claims is worth dwelling on. A cut-and-place dissection asserts that a set of pieces tiles two shapes. A hinged dissection asserts that, and that a continuous motion carries one tiling to the other while the pieces stay connected. The second is strictly stronger and the figures for the two are identical at the ends.

What a hinge is, exactly #

The word wants a definition, because loose readings of it make the theorem either trivial or false.

A hinged dissection of $P$ into $Q$ is a set of pieces, together with a set of pins joining them into a single connected chain or tree, such that some continuous motion of the assembly — each piece rigid, each pin held — takes a configuration tiling $P$ to a configuration tiling $Q$.

Three parts of that are doing work.

Connected. The pieces form one object. A dissection whose pieces fall into two groups is two hinged dissections and not one, and the whole difficulty of the general theorem is keeping everything in one piece.

Continuous. There is a path from one configuration to the other, not merely two configurations. This is the requirement that makes the object physical.

Rigid. Every piece keeps its shape throughout, which the figures check by measuring every edge at every angle drawn rather than by trusting the rotation code.

What the definition deliberately does not require is that the motion avoid self-intersection. A hinged dissection in which two pieces pass through each other on the way is still a hinged dissection by the standard definition, and whether every one can be made to avoid overlap is a separate and harder question that is open in general.

What it costs #

A hinge is a constraint, and the natural expectation is that satisfying it costs pieces. Sometimes it does and the amount is not well understood.

The triangle-to-rectangle step costs nothing — three pieces either way. The two-piece staircase does not hinge at all in its natural form, because the motion carrying one piece to the other is a translation, and a translation is not a rotation about any point.

That is the general obstruction and it is worth stating plainly: a hinged assembly moves by rotations, so a dissection whose pieces are related by translations has to be rebuilt. A translation can be written as two half turns about two different points, which suggests a repair, and the repair costs extra pieces to carry the intermediate positions.

So the price of hinging is real and is not a fixed overhead. For the pairs of shapes anybody has looked at closely, hinged dissections use the same number of pieces as the best known unhinged ones or one or two more, and there is no theorem saying they must be close.

The repair is worth working through once, because it is the standard move and it shows exactly where the extra pieces come from. A translation by a vector $v$ is the composition of a half turn about any point $p$ with a half turn about $p + v/2$: turning twice about two different centres returns every piece to its own orientation and shifts it by twice the distance between the centres. So a piece that needs to be slid can instead be swung twice — but the two swings need two pins, the second pin sits at a point that must belong to some piece, and if no piece reaches that point one has to be introduced.

That is the whole cost, and it is why the staircase is awkward rather than impossible. Its two pieces meet along a long jagged boundary and the translation is by a single step; the intermediate configuration of a double swing has the sliding piece out at an angle, and the point it needs to pivot about the second time is off the assembly entirely. Adding a small carrier piece there fixes it and takes the count from two to three.

A two-piece dissection is therefore the hardest kind to hinge, which is a pleasing inversion: the more pieces there already are, the more places a pin can be put, and the more likely it is that the motions needed are rotations about points that already exist.

Two pins, and how many a dissection needs #

The assembly on this page has three pieces and two pins, and the arithmetic behind that is worth stating because it is the whole of what “connected” costs.

$n$ pieces joined into a chain need exactly $n-1$ pins: each new piece is attached to the last one and to nothing else. Joined into a tree they still need $n-1$, since a tree on $n$ nodes has $n-1$ edges — the same count Euler’s formula produces for a graph with no cycles. So the number of pins is decided by the number of pieces and not by any choice, and the only choice is the shape of the joining.

The shape matters. A chain folds like a strip and a tree does not, and Dudeney’s model is a chain because a chain is what somebody can pick up and fold. Whether every hinged dissection can be arranged as a chain rather than a general tree is not settled, and the 2007 construction produces trees.

Adding a pin beyond $n-1$ creates a cycle in the joining, and a cycle is a linkage rather than a folding: the assembly then has fewer degrees of freedom than pieces, and may have none at all. An extra pin can freeze the whole thing, which is why hinged dissections are always trees and never anything else.

There is one more count worth having. A chain of $n$ pieces has $n-1$ angles that can be set independently, so its configuration space is an $(n-1)$-dimensional torus with the overlapping configurations removed — and the question of whether two given foldings are connected in that space is the question of whether the removed set separates it. For three pieces that is a statement about a square with some holes in it. For four hundred it is not a question anybody has answered.

The strip, and why it folds twice #

Dudeney’s four-piece model is a strip, and the strip is a method rather than a piece of luck.

The idea is to take the two shapes and tile the plane with each of them in a particular way: cut the first shape into pieces that tile an infinite strip of a chosen width, and do the same for the second at the same width. Superimpose the two strip tilings, offset so that their boundaries cross well, and the overlaps are pieces that assemble into either shape — because each strip’s tiling was made from one of them.

That is the strip technique, and it explains why so many classical dissections come in fours and fives rather than in dozens: the piece count is decided by how many times the two tilings’ boundaries cross in one period of the strip, and for two well-chosen strips that is a small number.

The technique also explains the hinges, which is the part that matters here. A strip tiling repeats, so the pieces produced by superimposing two of them come in a cyclic order along the strip, and consecutive pieces share a boundary point. Joining them at those points gives a chain, and folding the chain one way recovers one strip’s tiling and the other way recovers the other. The hinges are not added to the dissection; they are what the periodicity of the strip already provides.

So Dudeney’s model is not a lucky arrangement of four pieces. It is what the strip technique returns on the triangle and the square, and the same method returns hinged chains on a great many other pairs. What it does not do is work on every pair, which is why the general theorem needed something else and took a hundred years.

The theorem, and how long it took #

Dudeney published his four-piece dissection of an equilateral triangle into a square in 1902, and presented it as a physical model with the pieces hinged in a chain — a strip of four pieces that folds one way into a triangle and the other way into a square. That single example is where the whole subject starts, and it is a demonstration rather than a theory.

For a century the question of whether every pair of equal-area polygons admits a hinged dissection was open. Individual pairs were found by ingenuity, the constructions were beautiful, and no general method existed.

Abbott, Abel, Charlton, Demaine, Demaine and Kominers settled it in 2007: any finite set of polygons of equal area has a common hinged dissection. Every pair of equal-area polygons can be cut into pieces joined in a chain that folds from one into the other.

The proof is not a refinement of the classical chain. It works by taking an ordinary dissection and repairing it — replacing each piece by a “hinged bubble” that can be manoeuvred into the needed position — and the piece counts it produces are astronomical. The theorem establishes existence and gives up entirely on economy, which is exactly the relationship the rung below described between the plane theorem and Dudeney’s four pieces, arriving one level up.

Where the extra strength shows #

It is worth asking what the stronger statement buys, since a mathematical claim about a continuous motion sounds like a claim about a toy.

It rules out a class of cheats. A cut-and-place dissection may reflect a piece — turn it over — and some famous dissections do. A hinged one cannot: the motion is continuous and orientation is preserved throughout, so a hinged dissection is automatically one that works with pieces that have a top and a bottom. The rung below noticed that the classical chain happens not to use reflections; hinging makes that a consequence rather than an observation.

It makes the pieces an object. A cut-and-place dissection is a statement about two tilings by the same shapes. A hinged one is a single physical thing with a configuration space, and asking about that space — is it connected, how many folded states does it have — is a question the unhinged version cannot pose.

And it is the version a physical model demonstrates. A hinged strip is a proof somebody can hold, which is not a mathematical virtue and is a large part of why the subject exists at all. Dudeney’s model, and the hundreds since, are the reason anybody looked.

Where it stops #

In three dimensions the question is not merely open but obstructed. The rung below’s Dehn invariant already forbids most pairs of solids from being dissected at all — it is the reason a cube will not become a regular tetrahedron and the same shape of obstruction as the degree count that says a cube will not be doubled — and hinging is a further constraint on top of that. Where a solid dissection exists, whether it can be hinged is unknown in general.

On the sphere and in the hyperbolic plane the classical theorem holds and the hinged version is open. Equal-area polygons on a sphere are scissors-congruent — the Bolyai–Gerwien argument survives the change of geometry with care — and nothing is known about hinging them.

The 2007 theorem says nothing about overlap. Its motions are allowed to pass pieces through each other, and whether every hinged dissection can be realised by a motion that keeps the pieces disjoint is open. For the small examples on this page the motion is visibly clean; for a construction with millions of pieces nobody has checked.

And the piece counts are unusable. The general construction is a proof of existence with no useful bound attached, in the way the general chain is. Every hinged dissection anybody would draw was found by hand.

What the pictures cannot show #

The figures draw four or five moments of a swing and the swing is continuous. Nothing on the page establishes that the motion between two drawn frames is clean, that the pieces do not overlap on the way, or that a piece does not leave the table — all of which are checked by the assertions and none of which is visible.

More sharply, no still picture can distinguish a hinged dissection from an unhinged one. The two differ in whether the pieces are joined, and a drawing of pieces in contact looks the same either way; the pins are drawn as dots because there is no other way to say where they are. A reader shown only the two end states has been shown a cut-and-place dissection, and the swing is the claim.

And the figures show one dissection. That every pair of equal-area polygons has a hinged dissection is a theorem about all of them, proved by a construction nobody has drawn, and the four frames here are one instance of the easiest case.

The ladder from here #

Rungs above: what survives when the pieces may only be slid, where narrowing the operation set further does cost something and the cost is a new invariant. The piece count, where the chain’s guarantee is measured against what ingenuity finds. Twisted and piano hinges, which allow a piece to be flipped over about an edge rather than turned about a point and change what is reachable again. Hinged dissections of three-dimensional solids, where the Dehn invariant bites first. And the configuration space of a hinged assembly, which is a question about linkages rather than about dissections.

Narrowing the operation set #

The habit is the field’s own and this rung is the clean case of it: take a construction that works, remove one of the moves, and ask what is lost.

Removing “pick the piece up” from a plane dissection loses nothing, and it took a century to know that. Removing “turn the piece” loses something, and the next rung measures what. Removing “the pieces must be measurable” gains everything and destroys the subject, which is where the rung below ended. Adding a move rather than removing one has the same character: a marked ruler buys cube roots that a compass cannot reach, and the buying is not proportional to how large the addition looks either.

The three answers are different in kind and that is the point of asking. A narrowing can be free, can cost a measurable amount, or can be catastrophic, and nothing about how large the restriction looks predicts which. Forbidding a lift sounds severe and is free; forbidding a turn sounds mild and is not.