Matpic
mathematics, in pictures
Concept

Pi — where it appears

The ratio of a circle's circumference to its diameter, and a constant that turns up in a great many places with no circle in them. It is the ratio a circle fixes, and its appearances in counts of lattice points and in Buffon's needle come through that ratio rather than by coincidence.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · Fourier series Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

algebra · Complex numbers A Galton board after 600 balls. 600 balls fall through 12 rows of pegs, each bouncing left or right at random, and pile up in a bell-shaped heap.

A bell curve assembled out of coin flips

Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.

probability · Central limit 120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Getting pi by dropping needles on the floor

Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.

probability · Monte Carlo A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

geometry · Circle area A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Round is not the only way to be the same width

A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

geometry · Constant width The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

number · Sums of two squares Looking for a polynomial with π as a root. A table of the closest an integer polynomial of each degree comes to vanishing at the number, over a bounded search.

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

computation · Constructible numbers The sieve as a product, and the sum over the primes. The whole numbers up to 60, with those built only from 2, 3, 5 marked — the numbers the product of three geometric series multiplies out to. Beside them, the sum of the reciprocals of the primes, which grows without bound.

The sieve written as a product

Multiply out one geometric series for each prime and every whole number appears exactly once, as a single term. That identity turns a statement about factorisation into a statement about convergence, and it is where the analytic study of the primes begins.

number · Prime distribution The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number.

An integral that cannot be a whole number

Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.

number · Irrationality p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

number · Partitions

Named alongside it

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

ConvergenceAreaDerivativeIndependenceIntegral geometryLimitPeriodicityPrimesSineUnique factorisationUnit circleAlgebraic number

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