The staircase that is not the diagonal

Take a quarter circle of radius one. Approximate it by a staircase: go across, then up, then across, then up, following the arc. Refine the staircase by doubling the number of steps, again and again. The staircases get closer and closer to the arc — visibly, measurably, and as closely as anyone cares to demand.

Every one of them has length exactly 2. The arc has length π/2, which is 1.5708.

Something has to be wrong, and the tempting conclusion — that π is 4 — is not it.

Why the length never moves #

The staircase is easy to measure and the measurement is exact. Every step consists of a horizontal piece and a vertical piece, all the horizontal pieces go the same way, and all the vertical pieces go the same way. So the total horizontal distance is the width of the quarter circle, which is 1, and the total vertical distance is its height, which is also 1.

The length is 2, at every step count, forever. Doubling the steps halves each piece and doubles their number, and the total is untouched. The figure asserts it to a part in a million million at every panel.

The same thing happens against a straight line.

The diagonal case is the cleaner scandal, because nobody is inclined to believe that √2 is 2. It also shows that the effect has nothing to do with circles or with π; it is about staircases.

Where the argument for π = 4 actually breaks #

It is worth taking the false conclusion seriously enough to say exactly which step fails, because “something must be wrong” is not an answer.

The argument runs: the staircases converge to the circle; the staircases have perimeter 4; therefore the circle has perimeter 4. Two of the three claims are true. The staircases do converge to the circle, in a perfectly respectable sense. Their perimeter is 4, exactly, at every stage.

The step that fails is the therefore, and it is failing on an unstated premise: that a quantity attached to each member of a converging sequence must converge to the same quantity attached to the limit. Written out, that premise is obviously in need of justification, and it is false as stated. It happens to be true for area, which is why the analogous argument about area is sound, and the soundness of the area version is what makes the length version look sound too.

This is the standard anatomy of a paradox on this site. Nothing is wrong with any picture, no arithmetic is mistaken, and one silent assumption is doing all the damage. The rearranged series has the same anatomy, with “the order of the terms does not matter” as the silent assumption.

What does converge #

The area between staircase and curve behaves perfectly. It falls by roughly half each time the steps are halved, and by sixty-four steps it is six thousandths. Any sensible measure of how close the shapes are is going towards nothing.

So the staircases converge to the arc in every reasonable sense of position, and their lengths converge to something that is not the arc’s length. The limit of the lengths is not the length of the limit.

That sentence is the whole content of the essay, and once it is stated the paradox evaporates. Nothing has been shown about π. What has been shown is that length is not a continuous function of a curve, under the notion of closeness these staircases satisfy.

Which notion of closeness #

The staircases are close to the arc in the sense of never being far from it: for any tolerance, all sufficiently fine staircases stay within that tolerance of the curve everywhere. That is uniform convergence, and it is a strong condition — strong enough to preserve continuity, strong enough to allow term-by-term integration.

It is not strong enough to preserve length, and the reason is visible in the picture. Length depends on direction, and the staircases never point in the direction the curve points. Every one of their pieces is horizontal or vertical while the curve runs at forty-five degrees at its midpoint. Positions converge; directions do not converge at all.

The repair is to demand the directions too. A sequence of approximations whose slopes also converge to the curve’s slopes does give the right length, and the standard construction — inscribed polygons, with corners on the curve and straight lines between them — has that property automatically, in the same way a secant becomes a tangent when both of its ends are held on the curve. Inscribed polygons converge to π/2 from below, which is the definition of arc length: the supremum of the lengths of inscribed polygons.

That definition is also the one that makes length behave. An inscribed polygon can only get longer when a corner is added, since a straight line is the shortest route between its endpoints, so the lengths of inscribed polygons form an increasing family and their supremum is a genuine limit rather than a hopeful one. Nothing similar is true of staircases: refining a staircase changes its length by exactly nothing, so the family carries no information about where it is heading. The same monotone-and-bounded argument settles the circle’s circumference and it is the oldest reliable way of pinning a curved length down.

Under that definition the staircase is simply not an inscribed polygon: its corners alternate on and off the curve, and the off-curve corners are what add the extra length. Every one of them is a detour, and there are more of them at every refinement, each smaller — with the total detour exactly conserved.

How much extra the detours account for #

The excess is not mysterious and it can be attributed piece by piece.

Take one step of the staircase, spanning a small horizontal distance dx and a small vertical distance dy. Its length is dx + dy. The piece of curve it replaces has length very nearly √(dx² + dy²). The difference between the two is a fixed proportion of the step’s size, and the proportion depends only on the direction the curve is going.

Refine, and each step is half the size and there are twice as many. The excess per step halves; the number of steps doubles; the total excess is unchanged. That is the arithmetic behind the constant 2, and it makes clear that the excess is not an artefact of coarseness that a finer grid could wear away.

For the diagonal the numbers are exact and worth quoting: every step contributes dx + dy where the curve contributes √2 times as much as either, so the ratio of staircase to curve is 2/√2 = √2, or about 1.414, at every refinement. For the quarter circle it is 2/(π/2) = 4/π, about 1.273. Both are constants, both are greater than one, and neither depends on the number of steps.

The area case, where nothing goes wrong #

The contrast with adding up rectangles is worth drawing out, because the two constructions look almost identical and behave in opposite ways.

A Riemann sum approximates a region by rectangles and its area converges to the region’s area. A staircase approximates a curve by segments and its length does not converge to the curve’s length. Both approximations get uniformly close to the thing approximated; one quantity survives and the other does not.

The difference is what each quantity depends on. Area depends on position: two shapes that are everywhere close have nearly the same area, and the error is bounded by the closeness times the size. Length depends on the first derivative, and two curves that are everywhere close may have wildly different slopes at every point.

A quantity is preserved by a limit when it depends on no more than what the limit controls. That is the general rule, and this essay is one instance of it.

The rule also predicts which other quantities survive. The area enclosed by the staircase converges to the area enclosed by the arc, for the same reason as the Riemann sum: it depends on position. So does the position of the curve’s centroid, and so does the maximum distance from the origin. Anything that can be computed from where the curve is, without asking which way it is heading, comes through the limit intact.

Anything that consults the direction does not. Length is the first such quantity anyone meets; curvature is worse, since it depends on the second derivative, and the staircases have no curvature at all — they are flat everywhere and turn through a right angle at each corner. A sequence converging uniformly to a smooth curve while having no curvature anywhere is a stronger version of the same scandal, and it is the same scandal.

Where the limit does commute #

The derivative is a limit of slopes, and it is defined that way precisely because the slopes do converge. The secant construction differs from the staircase in exactly the way that matters: each secant is a chord of the curve, with both endpoints on it, so it is an inscribed segment rather than a detour.

That is not a coincidence of exposition. The definition of the derivative was chosen, historically, from among several candidates, and the reason chords won is that the resulting limit exists and behaves. A definition built out of staircases would have produced a quantity that was always 2 and told nobody anything.

Another limit that refuses #

The same failure, in a different costume, appears in building a square wave out of round ones. Partial sums of the Fourier series converge to the square wave at every point, and near the jump each partial sum overshoots by about nine per cent. The overshoot narrows and never shrinks.

So the maximum of the limit is not the limit of the maxima, exactly as the length of the limit was not the limit of the lengths. In both cases a quantity that seems like it ought to pass through the limit does not, and in both cases the reason is that the convergence, while genuine, is not of the kind that controls the quantity in question.

Three failures of this shape appear on this site and they are worth collecting: length under uniform convergence, maximum under pointwise convergence, and the sum of a rearranged conditionally convergent series under reordering. A fourth is nearby: the function whose graph no set of rectangles settles on, which appears beside the rectangles themselves and fails one step earlier, at whether the quantity exists rather than at whether it survives. Each is a case where an operation that commutes with finite manipulation stops commuting with an infinite one.

What the picture cannot show #

The panels stop at sixteen steps. A staircase with a thousand steps would be indistinguishable from the arc at any printable resolution, and would still be 2 long. That is the whole point and it is the thing the drawing is least able to convey, since the drawing’s persuasiveness increases with exactly the refinement that changes nothing.

The convergence is measured as a maximum distance and reported as a number. The figure computes, for each staircase, the largest distance from any point of the curve to the staircase, and asserts that it falls. That number going to nothing is what “converges” means here, and it is a computation rather than a picture — the panels at four and sixteen steps look similar and the numbers differ by four.

And nothing here defines arc length. The essay uses the value 1.5708 for the quarter circle and √2 for the diagonal, and takes both as known. Defining arc length properly requires the supremum over inscribed polygons, showing it exists, and showing it agrees with the integral of the speed — none of which is drawn, and all of which is what makes the staircase’s 2 a wrong answer rather than a second opinion.

The rule this leaves behind #

The practical lesson is small and it is worth stating plainly, because it survives long after the paradox stops being surprising.

An approximation is an approximation to something, and it has to be checked against the quantity actually wanted. A staircase is an excellent approximation to a curve if the question is where the curve goes, a useless one if the question is how long it is, and a perfectly good one again if the question is what area it encloses.

Anything computing a length from a sampled curve is in exactly this trap. Sample a curve at points, join them with straight lines, and the length comes out short of the truth; sample it onto a grid and trace the grid, and the length comes out long by the factor above. Neither error shrinks with resolution in the way an area error does.

The same caution applies to any quantity built from derivatives — curvature, slope, direction, speed. Position converging says nothing about them, and this essay’s figures are what that sentence looks like when it is drawn rather than asserted.

Where the ladder goes next #

The rung above asks which curves have a length at all. A curve that wiggles infinitely much can have infinite length while fitting in a small box — the boundary of a snowflake curve is the standard example, and it has infinite length and finite area. Such curves are called non-rectifiable, and for them the supremum over inscribed polygons is unbounded and there is nothing to converge to.

There is also a question about what a curve is. Everything here treats a curve as a set of points in the plane, and under that reading the staircase and the arc are close because their point sets are close. Treat a curve instead as a map from an interval — a parametrisation, with a speed at every moment — and the staircases stop converging to the arc in any useful sense, because their speeds are wrong everywhere. Length is a property of the parametrised object rather than of the point set, which is the technically correct diagnosis of everything above and which no picture on this page can show, since a picture only ever draws the point set.

The other direction is towards what does pass through a limit and what does not, systematically. Uniform convergence preserves continuity and integrals; it does not preserve derivatives, lengths or maxima. Stronger notions preserve more and are harder to satisfy. The whole of that bookkeeping is what turned nineteenth-century analysis from a subject where results were discovered into one where they were proved, and the staircase is the cheapest possible demonstration of why the bookkeeping was needed.