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Concept

Winding number

How many times a closed curve goes round a chosen point, counted with a sign for direction.

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

zeroa tangent field98 arrows, every onechecked perpendicularto the radius2 zerosand the sum of theirindices is forced to be 2

Nothing on a sphere can be combed flat

Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.

topology · fixed points radius 0.40 turns0 roots insideradius 10 turns0 roots insideradius 1.42 turns2 roots insideradius 23 turns3 roots insidep(z) = z³ − 2z + 2 — as the circle grows, the number of turns steps up from 0 to 3the count can only change by the image crossing the origin, and a crossing is a root

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 insidethe ray crosses the curve 9 times, so the point is insidefired in any of 360 directions from the same point, the count changes and its parity does not

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

Named alongside it

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

ContinuityClosed curveNonconstructiveTopological invariantBoundaryBrouwerComplex numbersConnectednessCounterexampleDegreeEuler characteristicExistence proof

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