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Concept

Sphere

The set of points at a fixed distance from a centre, in any number of dimensions. Its defining property in topology is that it separates: a sphere in n-dimensional space cuts the complement into exactly two pieces, and that is true in every dimension whether or not the sphere sits tamely.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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.

topology · jordan curve Two inversions, and the number four points agree on. Four points, their images after one inversion and after a second in a different circle, with the cross-ratio computed at each stage; it is conjugated once and restored twice.

The number four points agree on

One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.

geometry · inversion

Named alongside it

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

DimensionHomeomorphismJordan curveKnotCantor setClosed curveComplex numbersConformal mapConnectednessCounterexampleCross ratioDegree

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