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 20 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 Partial sums of x − x²/2 + x³/3 − … on [0, 1]. Partial sums of the power series x − x²/2 + x³/3 − … drawn on the interval from 0 to 1 with the function the series sums to inside the interval, and the values at x = 1 marked.

A sum read from inside

The series 1 − 1/2 + 1/3 − 1/4 + … adds to log 2, and the reason is not in the series. Its power series equals log(1 + x) inside the interval, and the value at the edge is read off by continuity. Abel's theorem says when that reading is honest — and the series 1 − 1 + 1 − …, which has no sum, is read the same way as a half.

analysis · Uniform convergence A circle trapped between two 12-sided polygons. A circle with a regular polygon of 12 sides inscribed in it and another circumscribed about it, beside a table of the bounds on pi obtained by doubling the side count.

Pinned between two sequences

The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.

geometry · Circle area 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 Area out to infinity, for three powers. Left: the curves 1/√x, 1/x, 1/x² from x = 0 to 10, with the region beyond x = 1 shaded under the lowest. Right: the area from 1 to T for each, on logarithmic scales, for T up to 10^6. 1/√x keeps growing, 1/x keeps growing, 1/x² levels off at 1.

An endless region with a finite area

A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.

analysis · The integral Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

A rectangle cut by a curve

Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

analysis · The integral The tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles.

The fraction Lambert built for the tangent

The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

number · Irrationality Measuring a closed curve with inlets by throwing lines at it. A curve inside a disc crossed by a sample of random lines with each crossing marked, beside the mean number of crossings and the length it implies against the true length.

A length counted by the lines that cross it

Throw straight lines at random across a curve and count how often they cross it. The average count, times π times the radius of the target, is the curve's length — for a wiggly closed curve, a spiral or a snowflake alike, with no following of the curve and no derivative anywhere. It is Crofton's formula of 1868, and it measures length the way a map-reader's ruled transparency does.

analysis · Arc length Random partitions of 1,000, scaled, against their limit shape. The outlines of 4 uniformly random partitions of 1000, scaled by the square root of 1000, lying close to the curve e^(−cx) + e^(−cy) = 1 with c = π/√6.

The shape a random partition takes

There are about twenty-four thousand billion billion billion ways to write 1,000 as a sum of whole numbers. Pick one at random, draw its Ferrers diagram, shrink it by the square root of a thousand, and it is almost exactly the curve e^(−cx) + e^(−cy) = 1 with c = π/√6. So is the next one, and the next. A random partition of a large number has a shape, and the shape is known exactly.

number · Partitions

Named alongside it

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

ConvergenceLimitAreaIntegralIntegral geometryLogarithmApproximationArchimedesAsymptoticsCircle areaContinuityConvergence rate

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