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Concept

Closed curve

A curve that returns to where it began, so that it has no loose ends anywhere along it. It divides the plane into an inside and an outside, which is obvious to look at and requires a real theorem to establish.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

4 circles, and the 14 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

logic · Class diagrams The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

algebra · Polynomial roots 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.

topology · Jordan curve 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.

topology · Jordan curve 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.

topology · Jordan curve 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.

topology · Jordan curve

Named alongside it

The objects these essays reach for when they reach for this one.

ContinuityCounterexampleHomeomorphismJordan curveConnectednessDegreeParityPolygonRegion countWinding numberArrangementBarycentric coordinates

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