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mathematics, in pictures
Concept

Quadratic polynomials

Expressions whose highest power is a square, and the curves those expressions describe. Completing the square solves them all, and the discriminant decides how many real roots there are before any solving is done.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

discrete · Prime distribution Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · Completing the square Eight circles touching three. Three given circles and the eight circles tangent to all of them, each labelled by which of the three it contains and which it lies outside.

Eight circles touching three

Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.

geometry · Inversion An Apollonian gasket, 125 circles in. The Apollonian gasket generated from four mutually tangent circles of curvature −1, 2, 2 and 3, drawn to 4 generations; every curvature in it is a whole number.

Curvatures that stay whole

Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.

geometry · Inversion Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

algebra · Completing the square A quadratic in the exponent, completed. Two panels sharing an x-axis. Above, the parabola −x² + 2x with its top at x = 1 marked. Below, e raised to that parabola: a bell centred at the same x = 1, with peak height e^1, beside the faint unmoved bell e^(−x²).

One number under every bell

The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.

algebra · Completing the square

Named alongside it

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

Completing the squareTangencyCircleInversionRootsAlgebra tilesApollonian gasketApollonius problemAreaComplex numbersConformal mapConjecture

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