Finitely many, and nobody says how many

The plane theorem says finitely many pieces and says nothing else about the number. That is an honest statement of what is proved, and it leaves open the question anybody actually asks, which is how many.

The proof is a chain of constructions, so the chain can simply be run.

Thirty-nine, for a hexagon into a square. The best dissection anybody has found for that pair uses five, and nobody has proved that five is optimal or that four is impossible.

Where the thirty-nine comes from #

The chain has four stages and each contributes.

Fan the polygon into triangles. An $n$-gon falls into $n-2$ of them, and the count is not in doubt — it is the same count Euler’s formula produces for a planar subdivision, and it does not depend on which diagonals are chosen. A hexagon gives four.

Each triangle into a rectangle. Three pieces apiece, by the slice-and-turn construction of the rung below. Four triangles give twelve.

Each rectangle into a square. Three more apiece in the good case, and more when the rectangle is elongated enough to need halving first. Twelve again.

Combine the squares in pairs. Two squares become one by the Pythagorean dissection, which takes five pieces, and combining $k$ squares takes $k-1$ such steps. Four squares give fifteen.

Total: $4 + 12 + 12 + 15 = 43$ by that reckoning, and the figure reports thirty-nine because the first stage’s four triangles are not pieces of the final answer — they are consumed by the second stage. The bookkeeping is where a piece count goes wrong, and it goes wrong in the same direction every time: stages are counted that produce intermediate objects rather than final pieces.

Linear growth is the right description and it is generous. The count is bounded by a constant times $n$, which is a real guarantee and is what makes the theorem constructive; it is also thirty-nine on a shape a schoolchild would dissect in five.

What ingenuity finds #

The gap is not a factor of two. For the pairs anybody has studied, the best known dissections are small numbers and the chain’s counts are dozens.

An equilateral triangle into a square takes four pieces, by Dudeney’s dissection of 1902. A regular hexagon into a square takes five. A regular pentagon into a square takes six. A Greek cross into a square takes five. The chain would take dozens for every one of them.

Those numbers were found by hand, by a small number of people over a century, mostly using the strip technique the previous rung described. There is no algorithm that produces them, and there is no algorithm that would recognise one as optimal if it were handed over.

That last figure is the honest summary of the situation. A construction that ignores its input gives a bound that ignores its input, and a bound of that kind can be arbitrarily far from the truth on any particular instance.

The lower bound that is missing #

Every other impossibility on this ladder has an invariant behind it. A cube does not become a tetrahedron because Dehn’s quantity differs. A triangle does not slide into a rectangle because the direction quantity differs. A cube is not doubled by straightedge and compass because a degree does not divide.

For piece counts there is nothing of the kind, and it is worth being precise about why.

An invariant works by being preserved by the moves. Every dissection of $P$ into $Q$ preserves area, and area is therefore available to say that no dissection exists when the areas differ. But a piece count is not a property of $P$ and $Q$; it is a property of a particular dissection, and there are infinitely many dissections. To prove that five pieces are needed, one must rule out every four-piece dissection there is, and the set of four-piece dissections is not finite: the cuts may be anywhere, at any angles, and the pieces may be placed anywhere.

So a lower-bound proof would have to be an argument about a continuum of possibilities, and the only general tool for that is an invariant, which is exactly what is missing. There is no quantity attached to a pair of shapes whose value bounds the number of pieces below.

What exists instead is a handful of ad-hoc arguments for very small counts. Two pieces are easy to rule out: a two-piece dissection means one cut, so the two shapes share a boundary structure that can be enumerated. Three is harder and has been done for particular pairs by case analysis. Four has been done essentially never, which is why Dudeney’s dissection has stood unproved-optimal since 1902.

Two lower bounds that do exist #

The situation is not completely empty, and the two results that exist say something about the shape of the difficulty.

A dissection needs at least as many pieces as the target has corners that the source cannot supply. If $Q$ has a corner of angle $\theta$ and $P$ has no corner of that angle, some piece must contribute it — either a piece with that angle, made by cutting, or two pieces meeting there. Counting the corners that must be manufactured gives a bound, and it is usually one or two, which is far below the truth.

A dissection of a convex shape into a convex shape has a bound from the boundary. Every straight portion of the target’s boundary is covered by edges of pieces, and every such edge was either part of the source’s boundary or made by a cut, and a cut contributes to two pieces. That gives a linear relation between edges and pieces, and for two convex polygons it gives a bound proportional to the number of sides — again far below what the chain produces and often below the best known dissection.

Neither is close, and both share a feature worth naming: they bound the pieces from below by counting something on the boundary. The reason they are weak is that a dissection’s difficulty is not concentrated on the boundary; most pieces exist to move area around inside.

The one count that is decidable #

Two pieces is the case where the question can be settled, and seeing how says exactly why three cannot.

A two-piece dissection of $P$ into $Q$ is one cut. The cut is a path from one boundary point of $P$ to another, the two pieces are what it separates, and reassembling them means applying a rigid motion to one of them so that the pair tiles $Q$. Both pieces keep every one of their edges, so the multiset of edge lengths and angles of $P$ plus the cut has to match that of $Q$ plus the same cut — and that is a finite condition on a finite amount of data.

So the question “do these two polygons have a two-piece dissection” is answered by an enumeration: try every way of matching a chain of $P$’s boundary against a chain of $Q$’s, and check whether the leftover is a single cut consistent with both. Finitely many matchings, each checked in finite time.

Three pieces is already outside that. Two cuts can meet each other in the interior, so the pieces need not each carry a whole chain of the original boundary, and the cuts’ meeting point is a free parameter ranging over a region rather than a finite set. The enumeration becomes a search over a continuum, and the finite argument stops.

That is the honest reason the subject stalls at four rather than a claim that nobody has tried. A decision procedure for $k$ pieces would need to handle $k-1$ cuts with $k-2$ free interior parameters, and no way of reducing that to a finite check is known for any $k$ above two. Whether the general question is even decidable is open.

What the theorem does guarantee, and it is not nothing #

The chain’s count is embarrassing beside a good dissection and it is not worthless, and the difference is worth stating because dismissing a constructive bound is as common a mistake as over-reading it.

It is uniform. The chain works on any pair of equal-area polygons, including pairs nobody has looked at and pairs with a thousand sides, and it produces a dissection every time without any search. The five-piece hexagon dissection works on one pair.

It is computable. Every stage is a construction with a formula, so a program can be handed two polygons and will return an explicit dissection with explicit pieces and explicit motions. Nothing comparable exists for the small counts.

And it certifies the theorem. The theorem’s content is that a dissection always exists, and the chain is why anybody believes it. A proof that produced no construction would establish the same statement and would leave the reader with nothing to check, which is the situation for the space case above dimension four.

So the right reading is the one this collection keeps arriving at: an existence proof that is also an algorithm gives an upper bound on the cost and says nothing about the true cost, and the two statements are both true and about different things. What is missing is not a better construction but a different kind of theorem, and the kind that is missing is the kind every other impossibility on this ladder has.

Why the chain is so wasteful #

It is worth asking where the chain’s thirty-nine actually goes, because the answer says what a good dissection is doing differently.

The chain works by normalising: every triangle is brought to a rectangle, every rectangle to a square, every pair of squares to one square. Each normalisation is a construction that works for every input of its kind, and the price of that generality is that it uses no information about the particular shapes.

A good dissection does the opposite. Dudeney’s four pieces exist because a triangle and a square of equal area happen to admit two strip tilings whose boundaries cross four times, and that is a fact about those two shapes and no others. The chain is an algorithm and a good dissection is a coincidence, and there is no reason for an algorithm to find a coincidence.

The same relationship appears wherever a constructive proof is available. The chain of Euclidean constructions produces a compass-and-straightedge procedure for anything reachable, and the procedures it produces are long; the elegant constructions for particular figures were found by looking at those figures.

What a piece count is for #

The question deserves an answer, because “how many pieces” is not obviously a mathematical question at all.

It is a question about the operation set’s cost, and the field this essay sits in is the field of operation sets. What two points can build asks what is reachable; the natural second question is how many moves it takes, and for straightedge-and-compass constructions that question has been studied and has partial answers. For dissections it is the same question and it has almost none.

It is also the question a physical model asks. A dissection with four pieces can be made out of wood and handed to somebody; one with thirty-nine cannot, and the difference is not one of degree. The subject’s whole popular life — Dudeney’s puzzles, the models in museums — is about small piece counts, and the mathematics that would justify any of those counts as optimal does not exist.

And it is a case where the existence proof and the interesting question have come apart completely. The theorem is settled, has been since 1833, and is proved by a construction; the question everybody asks about it is open in every instance.

Where the account needs care #

“Best known” is a moving target and a weak claim. The five-piece hexagon dissection is the best anybody has published, which is a statement about a small literature rather than about the mathematics. Several long-standing records have been broken by amateurs in the last thirty years.

Piece counts depend on what a piece may be. The counts quoted assume pieces are polygons and may be turned over. Forbidding reflection raises some counts; allowing curved cuts lowers none, since a curved dissection can always be refined to a straight one; allowing pieces to be disconnected would make the question meaningless.

The chain’s count depends on choices inside it. Fanning from a different corner gives the same number of triangles but different ones, and a rectangle whose sides are in an awkward ratio needs a halving stage that the count above did not include. The figures report what the drawn chain produces on the drawn polygon and no more.

And the linear growth is a bound rather than a rate. Nothing says a good dissection of an $n$-gon into a square needs to grow with $n$ at all, and for regular polygons the best known counts grow much more slowly than the chain’s.

What the pictures cannot show #

The figures draw the first two stages of the chain — the fan and the three cuts per triangle — and count all four. The squaring stage and the combining stage are arithmetic in the caption rather than drawings, because drawing twelve rectangles becoming twelve squares and then combining them would be a figure of thirty-nine small polygons in which nothing is legible.

More importantly, no figure here shows a good dissection. The five-piece hexagon-to-square dissection exists and is not drawn, because reproducing it means reproducing somebody’s construction rather than running one, and this collection draws what its generators compute.

And nothing here shows the absence the essay is about. A missing lower bound is a missing theorem, and the picture of a theorem nobody has is the same picture as the picture of a theorem that is false — which is the standing difficulty with every open question this collection touches.

The ladder from here #

Rungs above: the strip technique in full, which is the method behind most known dissections and is nearly an algorithm. Sydler’s theorem, which completes the classification in space and says that volume and Dehn’s quantity together are sufficient. Dissections into pieces of a restricted kind — all triangles, all convex — where the counts are larger and the lower bounds are slightly better. The complexity of deciding whether a $k$-piece dissection exists, which is not known to be decidable at all. And dissections in spherical and hyperbolic geometry, where the area of a triangle is determined by its angles and the whole question changes shape.

A guarantee is not a measurement #

The habit is worth carrying because the confusion it prevents is constant.

A constructive existence proof produces an object and therefore a bound on that object’s size. It is easy to read the bound as a measurement of how hard the problem is, and it is nothing of the kind — it is a measurement of how hard the construction is, and the construction was designed to be general rather than to be small.

The gap between the two is the whole of the practical subject and none of the theoretical one. Every essay on this ladder has a version of it: the chain produces thirty-nine and ingenuity produces five; the hinged general theorem produces astronomical counts and Dudeney produced four; the classical chain would take twenty pieces where Dudeney takes four.

The corollary is that a bound from a construction should be read as an upper bound and never as an estimate, and that the absence of a matching lower bound should be stated rather than glossed. On this ladder every upper bound is available and no lower bound is, and that asymmetry is a fact about the tools rather than about the shapes.