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mathematics, in pictures
Concept

Centroid

The average position of the points of a region, weighted by area. It is the point about which the mean squared distance to the region is smallest, and the parallel-axis identity is what makes that true.

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

A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

analysis · Convexity A scatter walking towards its own centres. 4 panels of the same 24 sites: the initial clumpy scatter and the Voronoi diagram after 1, 3, 12 rounds of Lloyd's iteration, with the cost falling to 52% of the scatter's as the cells even out.

Every site in the middle of its own cell

Move each point to the centre of mass of its own Voronoi cell, then redraw the diagram, then do it again. The rule is two lines long, it never mentions hexagons, and what it settles into is a honeycomb.

geometry · Voronoi The roots of the derivative inside the hull of the roots. Three polynomials of degree five, each drawn as its roots with their convex hull shaded, and the four roots of its derivative marked. Every root of the derivative lies inside the hull.

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

algebra · Polynomial roots

Named alongside it

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

ConvexityWeighted averageArithmetic meanChordComplex numbersConvex hullDerivativeEllipseFixed pointFocusGeometric meanHexagonal packing

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