Matpic
mathematics, in pictures
Concept

Closed curve

A curve that returns to where it began, so that it has no loose ends anywhere along it.

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

ABCDABABCBCDADCDABCD4 circles cut the plane into 14 pieces — Euler's count is 1414 of the 16 patterns appear; missing: A¬BC¬D, ¬AB¬CD

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 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.

ContinuityRegion countWinding numberArrangementBoundaryComplex numbersConnectednessConvexityCounterexampleDegreeEuler formulaExistence proof

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