Which curves have a length at all

The staircase that is not the diagonal ends with a repair and does not carry it out. The staircases fail to compute a length because their corners are off the curve; the construction that works puts every corner on it. That essay uses the value 1.5708 for a quarter circle and takes it as known.

Here is what it means for the number to exist.

Why inscribing is the repair #

A polygon inscribed in a curve has all its corners on the curve and straight segments between them. Two properties follow immediately and between them they are the whole definition.

It is shorter than the curve. A straight segment is the shortest route between its endpoints, and the curve between two consecutive corners is some other route, so each segment is at most as long as the arc it replaces. Adding up, the polygon is at most as long as the curve — whatever “as long as the curve” turns out to mean.

Adding a corner makes it longer. Replace one segment by two, through a new point of the curve, and the triangle inequality says the two together are at least as long as the one. So refining a partition never shortens the polygon.

Those two facts make the family of inscribed lengths increasing and bounded above, which is the one shape of family whose limit is guaranteed to exist. So define the length of the curve to be the supremum of the lengths of its inscribed polygons, and the definition is not a hopeful one: the supremum exists because a bounded increasing family has one.

The staircase has neither property. Its corners are off the curve, so it is longer than the curve rather than shorter; and refining it changes its length by exactly nothing, so the family carries no information about where it is heading. The difference between a definition and a paradox is which of the two families is used, and it is one word.

Where the supremum is infinite #

Nothing in the argument above says the supremum is finite. Bounded above was assumed, and for some curves it is false: the polygons get longer without limit, there is nothing to converge to, and the curve has no length.

Such a curve is called non-rectifiable, and the examples are not exotic. A curve with a corner at every point has infinite length between any two of its points — not merely a large length, and not merely at fine scales. A curve that has area is worse and is still a continuous closed curve with no self-crossings.

The question is what separates the two cases, and for a graph the answer is a single computation.

The sum that decides it #

Take a function on an interval and a partition of that interval. Add up the absolute changes: $\sum |f(t_k) - f(t_{k-1})|$. Refining the partition can only increase that sum, for the same triangle-inequality reason as before — splitting one change into two, $|a + b| \le |a| + |b|$. So the sums form an increasing family, and their supremum is the function’s total variation.

The theorem is that the graph of $f$ has finite length exactly when $f$ has finite total variation, and the inequality relating the two is immediate: the length of the graph’s inscribed polygon at a partition is $\sum \sqrt{(\Delta t)^2 + (\Delta f)^2}$, which lies between $\max(\text{width}, V)$ and $\text{width} + V$. Finite variation and finite length are therefore the same condition.

The pair of functions in the figure is the standard separator and the separation is a single exponent. Both $t\sin(1/t)$ and $t^2\sin(1/t)$ are continuous on $[0,1]$ when given the value nought at the origin, and both oscillate infinitely often as the origin is approached. The $n$-th hump of the first has height about $1/n$ and there are humps at every $n$, so the total rise and fall is about $\sum 1/n$, which diverges. The second’s humps have height about $1/n^2$, and $\sum 1/n^2$ converges.

So the boundary between having a length and not is the boundary between the harmonic series and a convergent one, which is the same boundary the harmonic series is famous for sitting on. That is not a coincidence of construction; it is the reason those two functions are the examples.

What the figure had to get right to be evidence #

The variation is computed over a partition, and which partition matters more here than anywhere else in this collection.

A uniform partition of $[0,1]$ into a million pieces samples the interval near the origin exactly once, in its first cell — and all of the trouble is in that cell. So a uniform refinement reports a finite variation for $t\sin(1/t)$, however fine, and the number it reports grows so slowly that a reader would conclude the sum settles. The figure partitions geometrically instead, so that the refinement actually reaches the oscillation, and it says so.

That is a general hazard rather than a detail of this figure. A quantity defined as a supremum over partitions is only as informative as the partitions tried, and a sampling scheme that misses where the action is produces a confidently wrong answer with no symptom. The same trap catches any numerical attempt at a supremum: the answer is a lower bound, always, and how good a lower bound depends on the choices.

The figure’s claim is written against the increments rather than the values, for the same reason. The variation of $t\sin(1/t)$ grows like the logarithm of the refinement — slowly, and without ever settling — so comparing two values at different refinements does not separate it from a convergent case. Comparing the increase does: a finite variation has increases falling to nothing, an infinite one does not.

A function with a length is two increasing functions #

There is a structural description of the functions with finite variation, and it is much more useful than the definition.

Accumulate the rises and the falls separately: let $P(t)$ be the total of everything the function has gained up to $t$ and $N(t)$ the total of everything it has lost. Both are increasing by construction. Their difference is $f(t) - f(0)$, because every change is counted in exactly one of them. And their sum is the total variation up to $t$.

So a function of bounded variation is a difference of two increasing functions — Jordan’s decomposition, from 1881 — and conversely a difference of two increasing functions has variation at most the sum of their total rises, so it is of bounded variation. The class is exactly the differences of monotone functions, which is a description in terms of a familiar object rather than in terms of a supremum.

That description is what makes the class useful. An increasing function has a derivative almost everywhere, has at most countably many discontinuities, and can be integrated against — and a difference of two inherits all three. So every function of bounded variation is differentiable almost everywhere, jumps at most countably often, and defines an integral. None of that is visible in the definition and all of it follows from the decomposition in a line.

A third function, to show what settling looks like #

The cleanest case is an increasing function, where the variation is the total rise and the coarsest partition already computes it: every intermediate point adds nothing, because a sum of rises telescopes. So the supremum is attained at the two-point partition, and refining is wasted effort.

That is worth having as a baseline because it says exactly where the difficulty in the general case comes from. The variation of a monotone function is $f(b) - f(a)$ and needs no partition at all. The variation of anything else is larger than that difference by however much the function doubles back, and the whole content of the supremum is measuring the doubling back. A partition is a device for catching reversals, and the finer it is the more reversals it catches — which is why a function reversing infinitely often is the hard case and a monotone one is free.

The middle case, $t^2\sin(1/t)$, does the doubling back infinitely often and the reversals shrink fast enough to add up. So it needs a partition, and it needs a fine one, and the answer it converges to is genuinely larger than the difference between its endpoints. Three behaviours, one apparatus, and the apparatus reports which is which.

What rectifiability is not #

Three things it is not, and each is a mistake somebody makes.

It is not smoothness. A rectifiable curve need not have a tangent anywhere in particular: a polygon has corners, and a curve can have a dense set of corners and still have finite length — any monotone staircase does. Conversely a curve can be differentiable at every point and still have infinite length if the derivative is unbounded in the right way.

It is not boundedness. $t\sin(1/t)$ lives inside a box of side one and has no length. Nothing about fitting in a small region limits the total distance travelled inside it, which is the observation the snowflake’s boundary is built on.

And it is not a property of the picture. Two curves whose images are indistinguishable at any printable resolution can differ in whether they have a length, because the difference is in the oscillation below the resolution. Every figure on this page reports a computation rather than showing a shape, and that is not a limitation of these particular drawings.

What a finite length then buys #

Once a curve has a length, several things become available at once, and they are the reason the condition is worth isolating.

The curve can be parametrised by its own length: assign to each point the length of the arc up to it, and the assignment is increasing, so it can be inverted. That gives a parametrisation travelling at unit speed, which is the canonical one, and it exists exactly for rectifiable curves.

Integration along the curve becomes defined, which is what a line integral is, and the definition needs the arc-length parametrisation the previous paragraph supplies.

And the length becomes additive: the length from $a$ to $c$ is the length from $a$ to $b$ plus the length from $b$ to $c$, for every $b$ in between. That is nearly obvious and it is not free — it needs the supremum over partitions of the whole to equal the sum of the suprema over the parts, which is true because any partition of the whole refines to one containing $b$, and refining never shortens.

The condition on a curve, rather than on a graph #

Everything so far has been about a graph, where one coordinate increases and the variation of the other decides. A curve in the plane is a pair of functions and the condition is about both.

A parametrised curve $t \mapsto (x(t), y(t))$ is rectifiable exactly when both coordinate functions have bounded variation, and the reason is the same inequality as before applied twice. Each inscribed polygon’s length is $\sum \sqrt{\Delta x^2 + \Delta y^2}$, which lies between $\max(\sum|\Delta x|, \sum|\Delta y|)$ and their sum — so the polygon lengths are bounded exactly when both variations are.

That makes the condition checkable one coordinate at a time, and it makes one thing clear that the graph case hides: a curve can be non-rectifiable because of either coordinate alone. A curve whose $x$ marches steadily and whose $y$ is $t\sin(1/t)$ has infinite length for the reason above; so does one where the roles are swapped, and so does one where each coordinate is separately fine and they conspire — except that they cannot, since the bound is the sum of the two variations and a sum of two finite numbers is finite.

So non-rectifiability is always attributable to a coordinate, which is a more useful statement than it looks. It means the property is not a genuinely two-dimensional phenomenon: the pathology lives in one real function at a time, and the curve inherits it. The curve that has area is the extreme case, and both of its coordinates have infinite variation.

What the pictures cannot show #

The variations are computed at five partitions and the claim is about a supremum over all of them. Five points of an increasing family cannot prove it unbounded; what they can do is show that the increases are not shrinking, which is the shape of evidence available and which the figures check rather than assume.

The graphs are drawn on the part of the interval where the two functions differ visibly, and the interesting behaviour is at the origin where neither can be drawn. At any resolution the two look alike near nought; the difference is the rate at which the humps shrink, and a rate is not a picture.

And nothing here establishes the theorem relating variation to length. The figures compute variations and the essay computes an inequality between the variation and the polygon length; the equivalence in general is two lines of algebra above and is not in any drawing.

Still open: what a length measures when there is none #

A non-rectifiable curve has no length and it is not featureless. The Hausdorff dimension measures how it fills space, and a dimension that is not a whole number is what that quantity gives for curves whose length is infinite — the snowflake at about 1.26, a curve that fills a square at 2.

What the dimension does not do is refine the class of rectifiable curves. Every curve with finite length has dimension exactly one, and dimension one does not imply finite length, so the two measurements agree on one side and not the other. Whether there is a natural quantity interpolating between them — finer than dimension and defined where length is not — is a live subject, and the candidates behave badly in ways the plain length does not.

The question with the clearest statement is about which sets are rectifiable in the sense of being covered, up to a negligible remainder, by countably many curves of finite length. That is a subject in its own right, the characterisations are recent, and the condition turns out to be about how the set looks at small scales in a way the one-dimensional case here does not hint at.

Who arrived at it, and in what order #

The definition is Archimedes’ in substance and nobody’s in form for two thousand years. Measuring a circle by inscribed polygons is the method of exhaustion, it is how $\pi$ was first bounded, and it is exactly the construction above — the increasing family, the bound from outside, and the number trapped between.

What was missing was the question of whether the supremum exists for an arbitrary curve, and that question could not be asked until there were arbitrary curves. Through the eighteenth century a curve was something given by a formula and its length was an integral; the integral was evaluated or it was not, and no case arose in which there was nothing to evaluate.

The cases arrived with the functions. Weierstrass exhibited a continuous function with no derivative anywhere in 1872, Jordan defined bounded variation in 1881, and the second was a response to the first: something had to separate the curves a length could be defined for from the ones it could not, and Jordan’s condition is that separator. The definition is ancient and the question it answers is modern, which is the usual order in analysis — the constructions come first and the hypotheses that make them legitimate come when somebody finds a case where they fail.

Peano’s space-filling curve in 1890 settled how bad it could get. A continuous map from an interval onto a square has both coordinates of infinite variation, has infinite length, and shows that continuity alone constrains a curve far less than anybody had assumed. After that the apparatus was no longer a technicality; it was the thing keeping the subject’s statements true.

What the monotone family was for #

The whole of this essay is one observation about which limits are safe. A quantity defined as a limit of approximations is at the mercy of whether the approximations converge; a quantity defined as a supremum of an increasing family is not, because an increasing family bounded above always has one.

So the work in a definition is often in arranging the monotonicity, and the inscribed polygon is what that arrangement looks like here. It is why the length exists when it exists, why it is additive, why the whole thing does not depend on a choice of partition, and why the staircase — which has no monotonicity in it at all — produced a number that never moved and meant nothing.

The same arrangement is what makes the integral work in the essay next door: the lower sums increase, the upper sums decrease, and a function is integrable exactly when the two meet. Two subjects, one device, and in both cases the device is what turns a hopeful limit into an existence theorem.