Matpic
mathematics, in pictures
Series

Arc length — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

    The staircase that is not the diagonal

    A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

    part 1 · analysis
  2. Inscribed polygons in a quarter circle, and the length they climb towards. Four polygons inscribed in a quarter circle with increasing numbers of corners, each drawn over the curve, with its length beneath it — the lengths increase towards the curve's own.

    Which curves have a length at all

    A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.

    part 2 · analysis
  3. One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths.

    The length belongs to the journey

    Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.

    part 3 · analysis
  4. The Cantor function has length 2. The graph of a singular or partly singular increasing function on the unit interval with an inscribed polygon, beside a table of inscribed lengths by stage and the value the derivative formula gives.

    The length the derivative never sees

    The Cantor function climbs from 0 to 1 with a slope of zero almost everywhere, so the formula ∫√(1 + f′²) dx says its graph has length 1 — the length of a flat line. The inscribed polygons say 2. Mix it half and half with the diagonal and the length becomes exactly the golden ratio, while the formula still reports only the part the slope can see.

    part 4 · analysis
  5. Measuring a closed curve with inlets by throwing lines at it. A curve inside a disc crossed by a sample of random lines with each crossing marked, beside the mean number of crossings and the length it implies against the true length.

    A length counted by the lines that cross it

    Throw straight lines at random across a curve and count how often they cross it. The average count, times π times the radius of the target, is the curve's length — for a wiggly closed curve, a spiral or a snowflake alike, with no following of the curve and no derivative anywhere. It is Crofton's formula of 1868, and it measures length the way a map-reader's ruled transparency does.

    part 5 · analysis
  6. Four stages of the four-corner Cantor dust. Four panels showing stages 1 to 4 of the four-corner Cantor set: 4, 16, 64 and 256 squares kept at the corners.

    A dust that almost every line misses

    Keep the four corner squares of a square, then the four corners of each of those, and so on. What is left has a length, in the sense that measures length, and yet almost every straight line misses it entirely. Stage by stage the chance that a random line hits the dust falls — to 48% by the eighth stage — and how fast it falls to nothing is one of the few questions about a simple picture that is still open.

    part 6 · analysis

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