The obstruction that was the only one

Two polygons of the same area can always be cut into each other. The same sentence for solids is false: a cube and a regular tetrahedron of the same volume cannot be cut into each other with finitely many flat cuts, however cleverly the pieces are chosen and moved. Max Dehn proved it in 1901, within a year of Hilbert posing the question, by finding a second number that every dissection must preserve and showing that it differs for the two solids.

That settles one direction and opens another. Dehn’s number is an obstruction: when it disagrees, no dissection exists. It says nothing about the case when it agrees. Two solids with the same volume and the same Dehn number might still be impossible to cut into each other, blocked by some third quantity nobody had found.

Jean-Pierre Sydler proved in 1965 that there is no third quantity. Two solids can be cut into each other exactly when their volumes and their Dehn invariants agree. The obstruction Dehn found was the only one.

A number built from edges and angles #

A dissection changes a solid completely except for two kinds of thing: what is inside, which is the volume, and what happens along edges, which is what Dehn’s number records.

For each edge of a solid, take its length $\ell$ and its dihedral angle $\alpha$ — the angle between the two faces that meet there. The Dehn invariant is the sum over all edges of $\ell \otimes \alpha$, where the symbol $\otimes$ means that lengths and angles are combined without being multiplied, and one rule is imposed: any angle that is a rational multiple of $\pi$ counts as nought.

That rule is the whole content. When a solid is cut, the cuts create new edges and split old ones, and the bookkeeping at each has a fixed shape. A new edge running across the interior of a face has two pieces meeting along it at angles summing to $\pi$. A new edge inside the solid has pieces around it whose angles sum to $2\pi$. An old edge split by a cut is shared between pieces whose angles sum to the old angle. Every contribution the cutting makes is a multiple of $\pi$ — or cancels — so a quantity that ignores multiples of $\pi$ cannot change, and the invariant of a solid is the sum of the invariants of its pieces, whatever the cuts were.

The cube’s dihedral angles are all right angles, which are half of $\pi$, so its invariant is nought. The regular tetrahedron’s are all $\theta = \arccos(1/3)$, about $70.53°$, and its invariant is $6\ell \otimes \theta$ — six edges at that angle. So the cube and the tetrahedron differ in their invariants unless $\theta$ happens to be a rational multiple of $\pi$, and it is not.

Measured, then recognised #

The table at the top does not take any solid’s angles from a reference. It builds each solid from coordinates, computes each face’s outward direction, and measures every dihedral angle as the angle between neighbouring faces — and then does the step that turns a measurement into an exact statement.

Each measured angle is recognised: written as a rational multiple of $\pi$, with denominator at most twelve, plus a whole multiple of $\theta$, and the match is required to be exact to twelve places and unique. The cube’s twelve edges all come out as $\pi/2$. The prism’s nine come out as seven at $\pi/2$ and two at $\pi/4$, because its triangular ends have angles $\pi/2$, $\pi/4$ and $\pi/4$. The sixth of a cube has two edges at $\pi/4$, three at $\pi/2$ and one at $\pi/3$. The tetrahedron’s six are all $\theta$; the octahedron’s twelve are all $\pi - \theta$.

With every angle recognised, the invariant is exact: the multiples of $\pi$ drop out, and what remains is the sum of each edge’s length times its multiple of $\theta$. A measurement becomes a theorem at the moment the recognition is unique, because from there on nothing is approximate — the invariant of the octahedron is exactly $-12\sqrt2 \otimes \theta$, and the minus sign comes from the $-\theta$ inside $\pi - \theta$.

That also explains a feature of the table that looks like an accident: the octahedron’s invariant is negative. An angle of $\pi - \theta$ contributes the same as $-\theta$, because the $\pi$ counts as nought. Negative invariants are perfectly meaningful — they are what cancels positive ones in a sum — and the octahedron is about to be used for exactly that.

What vanishes, and what can be cut #

The first three rows have invariant nought, and for all three a dissection into a box is not a theorem to be believed but a construction that can be carried out.

A prism is a polygon extruded straight up. Its end faces meet its sides at right angles, and its vertical edges carry the polygon’s own angles, which add to a multiple of $\pi$. So every prism has invariant nought, and the dissection is the plane theorem, extruded.

Cut the prism’s triangular end into a rectangle and carry the cuts through its whole height, and the prism becomes a box. The plane theorem then turns that box’s end into a square of the same area, again extruded, and a box on a square end can be turned into a cube by the same trick applied from the side. Every prism can be cut into a cube, and the argument never needed to know that its invariant was nought — the invariant is nought because this argument works, which is the direction Sydler’s theorem runs in general.

Six tetrahedra that make a cube #

The sixth of a cube is a tetrahedron, and a tetrahedron with invariant nought is the case that matters, because a tetrahedron is not a prism and no extrusion argument reaches it directly.

The six are the regions $0 \le x_a \le x_b \le x_c \le 1$, one for each of the six orders of the three coordinates. Every point of the cube, away from the dividing planes, satisfies exactly one such chain of inequalities, which is why the six fill the cube without overlapping — and the figure checks that claim by testing points rather than by trusting it.

Their angles are the evidence that the invariant is doing something real. Two edges at $\pi/4$, three at $\pi/2$ and one at $\pi/3$, all rational parts of $\pi$: the invariant of each piece is nought, as it must be, since six copies make a cube whose invariant is nought and a regular tetrahedron’s invariant would not have allowed it. A solid with an irrational angle cannot tile a cube, and these six can because every angle they have is rational.

That kind of tetrahedron has a long history. Hill described in 1895 a family of tetrahedra that can be cut into cubes — before Hilbert asked whether every tetrahedron could — and this is the simplest member of it.

A ledger that must balance #

The additivity of the invariant is the property every argument above rests on, and it is worth watching it hold on a case where it could visibly fail.

Four small tetrahedra alone would carry twice the invariant of the large one — each has six edges of length 1 at angle $\theta$, so four of them carry twenty-four, where the large solid’s six edges of length 2 carry twelve. The octahedron makes up the difference, and it can only do so because its angle is $\pi - \theta$, which counts as $-\theta$. The octahedron’s obtuse angles are exactly what the corner tetrahedra’s acute ones need to cancel.

The volumes add the same way and are easier to believe: the large tetrahedron has eight times the volume of a small one, the four corners take four of those eight, and the octahedron has the other four.

The cancellation can be watched happening edge by edge, and doing so is the clearest way to see why the rule about multiples of $\pi$ is the right rule rather than a convenient one.

Along the original edges. Each edge of the large tetrahedron is split at its midpoint into two halves of length 1, and each half belongs to exactly one corner tetrahedron, at the same angle $\theta$ the whole edge had. So the six original edges, each now two halves, contribute $12 \otimes \theta$ to the pieces’ ledger — exactly the large solid’s invariant, and nothing yet has been created or lost.

Along the cuts inside each face. Each face of the large tetrahedron is divided into four small triangles, and the three segments joining its midpoints are new edges. Along each of them a corner tetrahedron meets the octahedron, and the two dihedral angles there are $\theta$ and $\pi - \theta$, which together make a flat $\pi$ — the face was flat before the cut and is still flat after it. The pair contributes $1 \otimes \theta + 1 \otimes (\pi - \theta)$, which is $1 \otimes \pi$, which counts as nought.

And that is every edge. The corner tetrahedra’s other edges are the face cuts again, and the octahedron has no edges but those twelve. So the ledger’s $24 - 12$ is not an arithmetic coincidence: twelve of the corners’ twenty-four come from the old edges, the other twelve are cancelled one for one by the octahedron at the twelve new ones, and each cancellation is a flat face refusing to bend. A cut inside a flat face always contributes a multiple of $\pi$, a cut through the interior of the solid always contributes $2\pi$, and an invariant that ignores multiples of $\pi$ is the largest thing a cut cannot touch.

And the ledger has a consequence the table records in its last row. Two regular tetrahedra and one regular octahedron of the same edge have invariants $6\sqrt2$, $6\sqrt2$ and $-12\sqrt2$ times $\theta$, which sum to nought. By Sydler’s theorem, that collection can be cut into a box of volume $2$ — although neither the tetrahedron nor the octahedron can be cut into a box on its own. Two solids that cannot be tamed separately can be tamed together, because the obstruction is a number and numbers can cancel.

What Sydler proved, and why it took sixty-four years #

Dehn’s direction is a bookkeeping argument: follow the invariant through an arbitrary dissection and show it cannot change. Sydler’s direction is the opposite kind of statement. It says that for every pair of solids with equal volumes and equal invariants, a dissection exists — so it must produce one, or at least prove one exists, without knowing anything about the solids beyond two numbers.

The shape of the proof is a reduction. Every polyhedron can be cut into tetrahedra; every tetrahedron can be cut into a small number of special tetrahedra of a standard shape, determined by three angles; so the question becomes whether a combination of those special tetrahedra with total invariant nought can always be cut into a prism. Prisms, as above, are the solved case.

That last step is where the sixty-four years went, and it is not a question about cutting at all. The special tetrahedra satisfy relations — one can be cut into others — and the question is whether those relations are enough to cancel every combination whose invariant vanishes, which is a question about which combinations of angles can add to a rational multiple of $\pi$ and in how many ways. The hard part of a theorem about dissections is an algebraic statement about angles, and Sydler’s contribution was a functional equation that captured exactly the relations the cuts provide.

Børge Jessen recast the argument in the language of algebra a few years later, and in 1972 extended it to four dimensions, where volume and a small family of Dehn-type invariants are again enough. In five dimensions and above the question is open.

Only translations, and other restrictions #

The theorem allows the pieces to be moved by any rigid motion, including turning them over. Restricting the motions changes the answer, and in the plane that essay showed how: forbid turning and a new invariant appears, attached to each direction of edge.

The same happens in space. For dissections using translations alone, further invariants attached to directions of edges and faces appear alongside volume and Dehn’s number, and Jessen and Thorup showed in 1978 that those together are again a complete list — the translational version of Sydler’s theorem. The pattern is the one a dissection that never comes apart and its sequels kept finding: narrowing the operation set adds obstructions, and the work is in proving that the added ones are all there is.

A complete list of invariants is the strongest kind of classification theorem there is — it says a question about existence, which in principle ranges over infinitely many constructions, is decided by computing finitely many numbers. That is what the table at the top is: the whole of the three-dimensional problem, in two columns.

Where the account needs care #

The $\otimes$ is not multiplication. The invariant lives in a space where $\ell \otimes \alpha$ can be added to $\ell' \otimes \alpha'$ but not simplified into one number in general, and $\ell \otimes \alpha$ is nought only when $\ell$ is nought or $\alpha$ is a rational multiple of $\pi$. The table can write each invariant as a single real number times $\theta$ because every irrational angle it meets is a multiple of the same $\theta$. Solids with two unrelated irrational angles need two such numbers, and nothing about them reduces to a single column.

Recognition is a check, not a proof. Matching a measured angle to $p\pi/q + b\theta$ to twelve places is extremely strong evidence and is not a proof that the angle is exactly that. For the solids drawn the exact angles are known in closed form and the recognition agrees with them; for an arbitrary solid the recognition could, in principle, be fooled by a coincidence, which is why it is required to be unique among small denominators.

The theorem says a dissection exists and says nothing about its size. Sydler’s proof, like the plane theorem’s, is not a practical construction, and the number of pieces it would produce for two particular solids is not something anybody computes.

And the solids must be polyhedra. A sphere has no edges and no Dehn invariant, and it cannot be cut into a cube with finitely many flat cuts for a much simpler reason: flat cuts never make a curved face.

Hilbert’s third problem #

Hilbert’s list of twenty-three problems, presented in Paris in 1900, included as its third a question about exactly this: whether two tetrahedra of equal base and height can always be cut into each other. The motivation was Euclid’s treatment of the volume of a pyramid, which needs a limiting argument — a staircase of infinitely many slices — where the area of a triangle needs only cutting and rearranging. Hilbert suspected, correctly, that the limit could not be avoided.

Dehn answered within a year, making it the first of the twenty-three problems to be solved. His invariant was stated in the language of the time and later polished into the tensor form above.

Sydler’s converse took until 1965 and appeared in a Swiss journal. The easier half of the problem was solved at once and the harder half took two generations, which is the usual order for a pair of statements where one says something is impossible and the other says everything else is possible.

What a table of numbers cannot show #

The table reports that the cube and the tetrahedron-plus-octahedron collection can each be cut into a box, and no figure here shows either dissection being performed. For the prism and the six tetrahedra the cutting is drawn; for the collection it exists only because Sydler’s theorem says so, and that is the honest limit of the page.

The icons in the table are drawn at one size, so a sixth of a cube looks as large as a cube. Volume is in its own column because the pictures are not to scale, and a reader comparing the drawings rather than the numbers would compare the wrong thing.

And the invariant itself has no picture. It is a sum over edges of something that is not a number, recorded here by its coefficient on $\theta$; the figures show edges and angles and the ledger shows arithmetic, and the object in between is algebra.

Still open: other geometries, and higher dimensions #

In five or more dimensions nobody knows whether volume and the Dehn-type invariants are enough. In curved spaces — the three-dimensional sphere and hyperbolic space — the corresponding question is also open, and it is tied to deep problems about the homology of groups of isometries. In two dimensions, on the sphere, the plane theorem’s conclusion holds even though almost every step of its proof fails, and equal area on a sphere is the account of what replaces the rectangles a sphere does not have.

An obstruction that turns out to be complete #

The habit is about the relationship between a reason something is impossible and a guarantee that everything else is possible.

An obstruction is usually found first, because it only has to be checked against the cases it rules out. Its completeness — that nothing else can go wrong — is a claim about every case it does not rule out, and it is almost always harder. The degree argument for constructions and its converse, the essay on what two points can build, have exactly this shape, and so does every classification by invariants.

When an obstruction is found, ask whether it is complete, and expect the answer to be a separate and harder theorem. The table at the top of this page is what the answer looks like when it is yes: two columns, and nothing hidden in a third.