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mathematics, in pictures
Concept

Geometric series

A sum whose terms shrink by a fixed ratio each step, finite in total exactly when that ratio is under one. Its total is the first term divided by one minus the ratio, and the picture of that is a square dissected into ever smaller pieces.

Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.

The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

The shape of a number's divisors

Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

number · Unique factorisation A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

The sum that fits in one square

Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

analysis · Geometric series Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

analysis · Measure 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 tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

number · Irrationality The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

number · Partitions A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison.

No interval in it, and length to spare

The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.

analysis · Measure The rationals covered by intervals of total length 0.1800. Intervals of rapidly shrinking length placed around the rationals of the unit interval in the order they are listed, with the union of them drawn as a single band beneath.

Covering a set from outside

To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.

analysis · Measure One walk on the whole numbers, three chances, three different fates. The relative weight of each state for three step-up chances, drawn as bars, with the running total of those weights and what each case means beneath.

Where the shares have nowhere to go

On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.

probability · Markov chains 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 n / 2ⁿ held under a geometric series. A bar for each term of the series n / 2ⁿ with a decaying geometric curve above them, the curve lying above every bar from term 2 onwards.

The series everything else is measured against

A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.

analysis · Geometric series Runs of doubling length in 1/n^2, and the geometric series that bounds them. A bar for the total of each run of terms of 1/n to the 2, with an outlined bar above it for the bound obtained by replacing every term in the run with its largest, the bounds forming a geometric series.

The repair at the boundary

Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.

analysis · Geometric series xⁿ is uniform off a strip, with N = 14, 29, 59 for 3 strips. The functions x to the n on the unit interval with a band of half-width 0.05 around zero. For each of 3 strips next to 1, the member at which every later one stays in the band away from the strip.

Uniform, except on a small set

The functions xⁿ settle on their limit at every point and never uniformly — the trouble is all in a strip next to 1. Throw the strip away and the convergence is uniform on what is left, however thin the strip. Egorov proved that this always happens on an interval, Lusin proved the matching fact about a single function, and a bump sliding off along the whole line shows why both need a set of finite length to start from.

analysis · Uniform convergence Where a coin-signed geometric series lands, for λ = 1/3, 1/2, 1/√2, 1/φ. Histograms of the exact distribution of the sum of plus or minus λ to the k, one panel per value of λ: a dust of separated pieces below one half, a flat block at one half, and smooth-looking overlapping shapes above.

A coin in front of every power

Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.

analysis · Harmonic series Powers of a matrix that shrink in the end. Three curves of the norm of the n-th power of a two-by-two matrix against n on a logarithmic axis: one decays steadily, two rise to peaks of about 18 and 7 before decaying.

A geometric series whose ratio is a matrix

1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.

analysis · Geometric series 1 + 2 + 4 + … in two notions of size. Two sets of points against the number of terms up to 20 on an axis whose gridlines are powers of 2: the partial sums' ordinary size rising, their 2-adic distance from −1 falling.

A series that converges to minus one

1 + 2 + 4 + 8 + … runs off to infinity, and yet the formula for a geometric series says it should equal 1/(1 − 2) = −1. Measure size by how many factors of 2 a number has, instead of how large it is, and powers of 2 become small: the series converges, and to exactly −1. The same change of ruler explains why every repeating decimal is a fraction, and why repeating binary digits running off to the left are fractions too.

analysis · Geometric series Trisecting 120° by bisection, step after step. Partial sums θ(1/4 + 1/16 + …) for θ = 120°: 30.000, 37.500, 39.375, 39.844, 39.961 degrees, approaching 40.000.

A third reached only in the limit

No compass-and-straightedge construction trisects every angle. But a quarter, plus a quarter of a quarter, plus a quarter of that, and so on, adds up to a third — and each of those pieces is two bisections away. So an angle can be trisected by bisecting forever, every stage exact and the shortfall shrinking to a quarter each time. The construction never ends, and that is precisely what Wantzel's proof forbids: a construction is a finite thing, and a third of a general angle is only reached in the limit.

computation · Neusis

Named alongside it

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

ConvergenceLimitConvergence rateHarmonic seriesMeasureCantor setMeasure zeroPower seriesSelf-similarityBijectionBinary expansionContinuity

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