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mathematics, in pictures
Concept

Region count

How many pieces a drawing cuts its surface into, counted including the unbounded one. It enters Euler's relation as the face count, which is why the unbounded region has to be included.

Named by 2 essays across 2 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 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

Named alongside it

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

Closed curveArrangementBoundaryConnectednessContinuityConvexityCounterexampleEuler formulaMembership patternNon-orientableParitySet operations

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