A curve that has area

A closed curve with no self-crossings divides the plane into an inside and an outside. The statement is short, it is believed instantly, and the crossing rule settles it on any polygon in six lines of arithmetic. What that essay could not show, because no drawing can, is the reason the general theorem took forty years to prove and is still argued about: the curves the theorem is stated for are not polygons, and almost nothing a reader assumes about a curve is part of its hypotheses.

The hypotheses are three words long. A curve is continuous, it is closed, and it never meets itself. That is the whole list. Anything else — that it has a length, that it has a direction at each point, that it is thin, that its inside contains a disc — is an extra assumption, and each of them is false of some curve satisfying all three. This essay is those counterexamples, and it exists because they are what makes the theorem hard.

What a curve is, and what it is not #

A curve is a continuous map from the circle into the plane. That is all. The curve is the map, or if one prefers, the image of the map — and the two readings differ, which is the first sign that the subject has more room in it than the word suggests. The same distinction decides what a loop on a surface can be pulled tight to, where the map matters and the image does not.

Nothing in that definition mentions a formula, a length, a smooth join or a finite number of pieces. The map has to be continuous: nearby angles go to nearby points. It has to be injective: distinct angles go to distinct points. There is no third condition.

A polygon meets all of this and a great deal more. It has finitely many straight edges, so it has a length; it has a tangent everywhere except at its corners, and only finitely many of those; a ray meets it in finitely many points, which is what makes the crossing count possible at all; and its inside contains discs of a definite size. Every one of those extras is used by the polygonal argument, and none of them is available in general.

The theorem is stated where the argument is not. That gap is the subject of this essay, and every figure in it is a stage of something whose limit no figure can draw.

Removing crosses, and the area that stays #

Take the unit square. Remove a cross through its middle whose arms have width $c_1$, leaving four squares in the corners. Remove a thinner cross from each of those, leaving sixteen. Keep going, thinning the cross at every stage.

After $n$ stages, $4^n$ squares are left, and their total area is a product: each stage keeps a fraction $(1 - c_k)^2$ of what it was given, so the area is $\prod_{k \le n} (1 - c_k)^2$. With $c_k = 2^{-(k+2)}$ that product does not go to zero. It converges, and the figure above computes it: 0.5931, checked against the arithmetic of the stages rather than reported from a drawing.

That the area survives is the easy half. The hard half is that the surviving set is a curve, and the visiting order in the figure is where that comes from. The squares are visited in Hilbert’s order, in which consecutive squares always share a side of their parent — a fact the generator checks for all $4^n$ of them rather than assuming. Fatten the joins between consecutive squares into corridors, and the region at each stage is connected; each stage sits inside the one before; the diameters halve. A nested sequence of connected compact sets with diameters going to zero, indexed by a parameter that runs from 0 to 1, is exactly the data that defines a continuous map from an interval — and with the ends identified, from the circle.

The construction is Osgood’s, from 1903, and its content is the sentence that follows from the two halves together: there is a simple closed curve whose points have positive area. Not the region it encloses — the curve itself. It is a set of measure at least 0.59 in a square of measure 1, and it is the image of an injective continuous map of a circle. Injectivity is what separates it from the older curiosity in the same family: a map of the interval onto the whole square exists and is continuous, and it necessarily visits some points more than once.

The measure-theoretic half of that has a relative elsewhere in this collection. A set can be nowhere dense and still take up most of the interval — the fat Cantor set is built by removing middles that shrink fast enough, and the arithmetic above is the same arithmetic in two dimensions. What Osgood adds is that the leftovers can be strung onto a curve.

A tangent at no point of it #

The second failure is a different one, and it is easier to draw because it does not need a limit to become visible.

The recipe is Weierstrass’s, wrapped round a circle. A function whose graph has a corner at every point is built by adding cosines whose frequencies grow faster than their amplitudes shrink; the sum is continuous because the amplitudes are summable, and it has no derivative anywhere because the difference quotients at scale $h$ grow like $h^{\alpha-1}$.

The graph of such a function is not a closed curve, and turning it into one is the step this rung needs. Use the function as a radius: let $r(\theta)$ be one plus a small multiple of Weierstrass’s sum. Provided the multiple is small enough that $r$ stays positive, the map $\theta \mapsto (r\cos\theta, r\sin\theta)$ is injective — two different angles lie on two different rays out of the origin and cannot land on the same point. It is continuous because $r$ is. So it is a simple closed curve, and it has a tangent at no point of it.

This is worth pausing on, because the injectivity argument is the whole reason the construction works and it costs one line. A curve that wanders can meet itself, and checking that it does not is in general the difficult part; a curve given in polar form with a positive radius cannot, whatever the radius does.

There is an honest caveat under that figure and it belongs in the text as well. What is drawn is a truncation — five terms, or three — and a truncation is smooth. The roughness in a picture is always the roughness of a finite sum. The measurement reported is taken of the limit, at scales finer than the drawing resolves, which is the only way a picture of this object can be evidence for anything about it.

Room, and the lack of it #

The third failure needs no limit at all, and it is the one most likely to be missed, because it is about the inside rather than about the curve.

An inside can have a great deal of area and no room. The width of a shape and the area it holds are independent quantities in a stronger sense than that essay needed, and the corridor is the demonstration. Narrow the corridor and lengthen it; the area stays where it is and the largest inscribed disc shrinks with the pitch. Nothing stops that process, and in the limit there is a simple closed curve whose inside has positive area and contains no disc of any radius whatever.

That matters because several plausible-looking proofs of the Jordan curve theorem quietly need a disc. Any argument that says take a small ball around an inside point and note that it is entirely inside is using a fact about polygons. Any argument that shrinks the curve toward a point, or approximates it by a polygon and transfers the conclusion, needs to know that the approximation does not change which points are inside — and the inradius is exactly the quantity that would control that, going to zero.

Why the crossing rule cannot be the proof #

Put the three failures beside the rule from the rung below, and the rule stops working three times over.

A ray need not cross the curve finitely often. On the Osgood curve a ray can meet the curve in a set of positive measure. There is no count to take the parity of, and no way to define one by a limit that does not beg the question.

A ray need not cross transversally. With no tangent anywhere, the notion of the ray crossing rather than touching has nothing to be defined in terms of. On a polygon a crossing is decided by which side of the edge the ray enters and leaves; here there is no edge and no side.

Approximating by polygons does not transfer the answer. The natural repair is to replace the curve by a nearby polygon, apply the rule there, and pass to the limit. That works if the classification of a fixed point is stable under small changes of the curve — and the inradius result above is exactly the statement that it need not be. A point can be inside a curve and outside every polygon within $\varepsilon$ of it, for every $\varepsilon$.

So the rule is an algorithm for polygons, and it was never a proof for curves. What a real proof needs is a way of assigning an integer to a point that does not count anything: the winding number, defined by lifting the curve to the line rather than by counting crossings, and a compactness argument to get from local statements to global ones. That is the machinery, and it is why the theorem sits where it does in a course rather than in the first week.

What the theorem still says #

Everything above is a list of things a simple closed curve can do, and none of it damages the conclusion. That is the striking part.

The curve can have positive area — and it still separates the plane into exactly two pieces, and it is still the boundary of each. It can have no tangent anywhere; two pieces. Its inside can contain no disc; two pieces, and the inside is still connected and still open. Camille Jordan’s statement survives every counterexample that has ever been aimed at it, and the counterexamples are not exotic constructions dreamed up to be difficult. They are what a curve is like when the extra conditions are dropped.

This is the difference between a theorem that is obvious and a theorem that is easy. The conclusion is obvious for the curves anyone pictures; the theorem is about a class of objects for which nothing is obvious, and the sentence in the middle — it holds for those too — is doing all the work. Four colours suffice is the same shape and the same trap: a statement so easy to check on examples that it took a century and a computer to establish in general.

There is a converse to draw from that, and it is the reason to have read this far. When a theorem seems too obvious to need proving, the useful question is not how would one prove it but what are its hypotheses, and what satisfies them that nobody pictured? The three curves here are the answer to that question for this theorem, and each of them was found by someone asking it.

What the pictures cannot show #

Not one figure here is the object it describes. The first draws stage two and stage three of a construction whose curve is the limit of every stage; the second draws a sum of five cosines where the curve is the sum of infinitely many; the third draws two corridors where the claim is about a sequence of them. A limit is not drawable, and the honest response is to draw a stage and report the number the stages converge to, which is what each of these does.

The area is a lower bound, not a measurement. The product 0.5931 is the area of the squares that survive every stage. The corridors joining them are drawn and deliberately not counted, because they only add. What the figure asserts is that the limit has at least that much area, which is what the claim needs; the exact measure of Osgood’s curve depends on details of the joins that the picture does not fix.

Nowhere is not a property of a picture. The steepness measured under the second figure is measured at five particular angles and over nine scales. That the curve has no tangent at every one of its uncountably many points is Weierstrass’s theorem, and the figure is an illustration of it rather than evidence for it.

And the third figure shows two corridors, not a limit. The claim that some curve’s inside contains no disc at all needs the sequence continued forever, and each of its terms is a perfectly ordinary polygon whose inside contains a perfectly ordinary disc.

Where the ladder goes next #

Everything here has been about what can go wrong. The rung above is about how much nevertheless goes right, and it is more surprising than any of the counterexamples: in the plane, a simple closed curve is not merely a separator but a round circle in disguise. There is a homeomorphism of the whole plane carrying any simple closed curve onto a circle, so the inside is a disc, the outside is the outside of a disc, and the picture in the reader’s head is correct after all — for the plane.

That is Schoenflies’s theorem, and its polygonal case is a construction rather than an argument. The rung after it asks what happens one dimension up, where the separation statement survives and the straightening one does not, and the reason it does not is a surface with horns.

Sideways, the failure of the crossing rule connects to the surfaces on which the theorem is simply false, where a closed curve can fail to separate anything at all, and to the count of what a closed surface’s shape costs, which is the invariant that decides which curves on a surface separate it.

What is worth carrying away #

A hypothesis list is a definition of the objects a theorem is about, and reading it as a description of the objects one has in mind is the most common way to misunderstand what a theorem says.

Three words — continuous, closed, injective — admit curves with area, curves with no tangent, and insides with no room. Every one of those was found by someone who took the list literally and asked what else satisfies it. The habit generalises past this theorem and past this subject: the interesting objects in any definition are the ones nobody drew when writing it down.