Every loop is a circle in disguise

Two things are believed about a closed curve in the plane, and they are usually believed as one thing. The first is that it has an inside and an outside. The second is that the inside is a disc — that it could be pushed and stretched into a round one without tearing anything, taking the curve to a circle as it goes. Only the first of those is the Jordan curve theorem. The second is stronger, it is due to Schoenflies, and this is where it is earned.

The distinction matters because in three dimensions the two statements come apart. A closed surface in space separates it, exactly as a curve separates the plane; the inside of that surface need not be a ball. So separation and the inside is what it looks like are two theorems, and the second one is the one that fails when there is room for it to fail. In the plane it does not, and the reason is that a polygon can always be cut into ears.

The two statements, and which one is being proved #

Write them out, because the difference is easy to lose.

Separation. The complement of a simple closed curve has exactly two connected components, the curve is the boundary of each, and one of them is bounded.

Schoenflies. There is a homeomorphism of the whole plane onto itself carrying the curve to the unit circle.

The second implies the first at once: a homeomorphism of the plane preserves how many pieces anything’s complement falls into, and the complement of a circle falls into two. The interest is entirely in the other direction, and there is none — separation does not imply straightening, which is exactly what the horned sphere shows one dimension up.

The second statement is not about the curve. It is about the whole plane, and it says the curve was never in a complicated position to begin with.

That last reading is the useful one. A curve wound into a tight spiral is not intrinsically knotted or difficult; it is a circle that has been distorted, and the distortion can be undone. Whether the same is true of a sphere in three-space is a question with a different answer, and it is the reason this rung and the two above it exist.

Every polygon has an ear #

The whole of the polygonal case rests on one small fact.

Call three consecutive corners of a simple polygon an ear when the corner turns the right way and the triangle it cuts off contains no other corner of the polygon. Cutting an ear off leaves a simple polygon with one fewer corner. If every polygon of four or more corners has an ear, the process terminates and produces a triangulation.

It does, and the proof is two sentences. Take the lowest-leftmost corner of the polygon, which is necessarily convex. If the triangle it makes with its two neighbours is empty, that is an ear. If not, some corner of the polygon lies inside it; take the one furthest from the line joining the two neighbours, and the triangle it makes with its neighbours is empty. Meisters proved the sharper statement in 1975: every simple polygon of four or more corners has at least two non-overlapping ears, which is why a polygon can be triangulated in more than one way and why the ordering the figure draws is a choice rather than a fact.

The two-ears statement is sharper than the argument above needs and it is the one that makes ear clipping a method rather than an existence proof: at every step there is a choice, so a search that keeps the fattest available ear produces a better-conditioned triangulation than one that takes the first it finds. That is a computational remark rather than a mathematical one, and it is why ear clipping survives in graphics libraries beside asymptotically faster algorithms — the triangulation of a region defined by nearest neighbours is wanted quickly and repeatedly, and a quadratic method with a small constant wins at the sizes that occur.

The number of triangles never depends on that choice. Each clip removes one corner and produces one triangle, and the last three corners produce one more, so the count is $n - 2$ whatever route was taken. That is the same accounting that makes the angles of any polygon add to a number depending only on its corners, and it is the first hint that a triangulation is carrying information rather than being a convenience.

The map, one triangle at a time #

Now the construction, and it is short.

A triangulation of an $n$-gon is a set of diagonals, and each diagonal is a pair of corner indices. Whether two diagonals cross is decided by those indices alone — the pairs $\{a,b\}$ and $\{c,d\}$ cross exactly when one of $c,d$ lies strictly between $a$ and $b$ in cyclic order and the other does not. Nothing geometric enters.

So take the same set of index pairs and draw them on a regular $n$-gon. They are non-crossing there for the same reason, so they cut it into the same $n-2$ triangles with the same corner labels.

Now match the two triangulations triangle by triangle. On each triangle, send a point with barycentric coordinates $(\lambda_1, \lambda_2, \lambda_3)$ to the point with those barycentric coordinates in the matching triangle. That is an affine map, it is a bijection of triangle to triangle, and there are $n - 2$ of them.

The one thing to check is that the pieces fit together. Two triangles meeting along a diagonal must agree on the diagonal, or the pieces are $n-2$ maps rather than one. They do, and the reason is that an affine map is determined on a segment by its values at the two ends: both triangles send the diagonal’s ends to the same two image corners, so both send the point a fraction $s$ of the way along to the point a fraction $s$ of the way along the image. The figure checks this rather than asserting it, at four interior points of each of the seventeen shared diagonals.

The result is a continuous bijection from the closed polygon onto the closed regular $n$-gon, affine on each piece, carrying corner $k$ to corner $k$. A continuous bijection between compact sets has a continuous inverse, so it is a homeomorphism — and composing it with the obvious radial map from the regular $n$-gon to the disc finishes the job.

There is one more thing the count settles. A triangulation of the inside gives it a combinatorial description — a list of triangles and which edges they share — and that description contains everything: the inside is a disc precisely because its triangulation is a disc’s triangulation, glued in a fan. The same move classifies surfaces one dimension up, where a polygon with its edges identified in pairs is the whole datum and two numbers read off the gluing decide which surface it is.

Getting the rest of the plane #

That map takes the inside to a disc. Schoenflies’s statement is about the whole plane, and the extension is worth a paragraph because it is where the “in disguise” of the title is earned.

Surround the polygon by a large circle, and consider the region between the two — an annulus with a polygonal inner boundary and a round outer one. Triangulate it, which the same ear-based machinery does after cutting it open along a segment from the polygon to the circle. Do the same to the region between the unit circle and a large circle. Match those two triangulations as before, agreeing with the first map on the polygon and with the identity on the far circle. Outside the far circle, do nothing.

Piecing the three together gives a homeomorphism of the plane to itself, the identity far away, that carries the polygon to the unit circle. Everything a reader believes about the picture follows: the inside is a disc, the outside is the outside of a disc, and any two simple polygons are carried to each other by a motion of the plane.

The identity far away is worth noticing. It says the straightening is local in the sense that matters: nothing at a distance had to move, so the theorem is about the curve and its immediate surroundings rather than about the plane’s behaviour at infinity — which is the property that fails for a curve wandering out to infinity, and the reason the theorem is stated for closed curves.

Where “polygon” was used, and what the general case costs #

Every step above used finiteness. Corners were tested one at a time, an ear was found by search, the triangles were counted. A curve with no corners has no ears to clip, and the argument does not merely become harder — it has nothing to grip.

The general statement is nevertheless true, and Schoenflies proved it in 1906. The modern route goes through analysis rather than combinatorics. By the Riemann mapping theorem there is a conformal bijection from the open unit disc onto the inside of the curve; Carathéodory’s theorem says that when the boundary is a simple closed curve, that map extends continuously to the closed disc and is a homeomorphism there. The extension to the outside is then a second application of the same pair of results.

That is a heavy machine for a statement about a loop of string, and the weight is not accidental. The curves of the previous rung — the one with positive area, the one with no tangent anywhere — are all covered by it, and any proof has to cover them. The polygonal argument is the case where the picture in the reader’s head is literally the proof; everything else is the price of the hypotheses being as weak as they are.

There is a third route, worth naming because it is the one that generalises: approximate the curve by polygons, straighten each, and show the straightening maps converge. It works in the plane. It is exactly what fails in three dimensions, where the approximating polyhedra can be straightened and the limit of the straightenings is not a homeomorphism.

What a straightening is good for #

The theorem’s practical content is permission. Once a curve is known to be a circle in disguise, every statement about circles transfers.

The inside is simply connected, so any loop drawn inside it can be pulled tight without leaving it — a fact used constantly and almost never justified. A continuous map from the inside to itself has a fixed point, because the disc has that property and homeomorphic spaces share it. Any two points inside can be joined by a path inside, and any point of the curve can be reached from inside along a path, which is the statement that the curve is accessible and which is false for the horned sphere’s outside.

It also settles a question the rung below raised and could not answer. The crossing rule classifies points by parity; the winding rule classifies them by an integer. On a simple closed curve the two agree, and one clean way to see it is that both are unchanged by a homeomorphism of the plane and both are trivially computed on a circle. That is the winding number doing what invariants are for: it is defined without reference to any particular position of the curve, so a theorem that moves the curve to a convenient position cannot change it.

The permission runs the other way too, and this is where the theorem stops being a formality. A curve in space is not a circle in disguise — three moves are enough to change a knot’s diagram and not enough to untie it — so nothing in this section transfers to a loop that has been allowed a third dimension to wander in. The plane’s flatness is doing the work, and the horned sphere is what happens when it is taken away.

What the pictures cannot show #

Every polygon here is a polygon. The construction shown is the whole proof only for curves with finitely many straight edges. For every other curve the theorem is true and the picture is an analogy.

The map is drawn at four points. A homeomorphism is uncountably many correspondences and a figure can mark a handful. What the figure can and does establish is the property those four points illustrate — agreement across shared edges — because that is checkable at a finite sample of a finite set of diagonals.

The extension to the whole plane is described and not drawn. The annulus triangulation is a second application of the same machinery and it would double the figure without adding an idea.

And the tinting is an order, not a canonical one. Meisters’s theorem says at least two ears are available at every step; the shading shows the route this particular search took, and a different search gives a different picture of the same theorem.

Where the ladder goes next #

The rung above takes the separation statement — the weak half — into every dimension, where it remains true: a topological sphere in $n$-space has a complement with exactly two pieces, and a corollary of that is the fact that makes dimension meaningful at all.

The rung after it takes the strong half into three dimensions, where it is false. Alexander’s sphere separates space into two pieces, its inside is an ordinary ball, and its outside is not — there is a loop out there that cannot be pulled tight, and no homeomorphism of space can carry the surface to a round sphere. The construction is a tree of clasped horns, and it exists precisely because the polygon-approximation route above has no third dimension to run in.

Sideways: surfaces are classified by two numbers, and the classification is proved by cutting them into triangles and rearranging — the same tool as here, doing more work.

What is worth carrying away #

A theorem that says the picture is right is worth more than a theorem that says the picture is not wrong, and the two are easy to confuse.

Separation says the curve does not fail to have an inside. Schoenflies says the inside is the thing anyone would have drawn — and the difference between the two is a whole dimension’s worth of counterexample. The polygonal case makes the reason visible: the straightening exists because the inside can be cut into finitely many pieces, each simple enough to be carried by an affine map, and every difficulty in the general theory is about what to do when finitely many pieces are not available.