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Jordan curve — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

    Which side of the line is inside

    A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

    part 1 · topology
  2. Stage 3 of a curve that has area. A square split into 64 smaller squares by removing crosses of decreasing width, the squares joined in Hilbert order; the kept area is 63.2 per cent and its limit is 0.5931.

    A curve that has area

    The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.

    part 2 · topology
  3. Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge.

    Every loop is a circle in disguise

    Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

    part 3 · topology
  4. A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on.

    Two pieces, in every dimension

    A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

    part 4 · topology
  5. Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge.

    A ball whose outside is not one

    Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.

    part 5 · topology

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