Matpic
mathematics, in pictures
Concept

Polygon

A closed curve made of finitely many straight segments joined end to end. Almost every statement about curves is easy for polygons and hard in general, because a polygon has finitely many corners, a length, and a triangulation, and a curve need have none of them.

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

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

computation · scissors congruence 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

Named alongside it

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

Closed curveContinuityCounterexampleHomeomorphismJordan curveAreaBarycentric coordinatesCantor setCongruenceDihedral angleDiscDissection

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