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The same thing twice — page 12

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
Two ways of counting that agree at every number. For n up to 40, the counts of partitions with gaps of at least two against partitions into parts congruent to 1 or 4 mod 5, on a logarithmic scale, equal at every n, with the second identity's counts beside them. Number

Two counts that agree for no visible reason

Write 10 as a sum of whole numbers that differ from each other by at least two, and there are six ways. Write 10 as a sum of numbers that each leave 1 or 4 on division by 5, and there are six ways. The same happens for 20 (thirty-one each), for 40 (three hundred and seventy-four each), for every number anyone has checked and every number there is. The two lists look nothing alike, and no one has found a simple way to turn one into the other.

 How far each estimate can stray, when rare jumps are possible. The error that the sample mean and the median of 12 block means stay within, at confidence levels from 80% to 99.9%, over 60000 repeated samples of 120 draws from normal draws with rare large jumps, with Chebyshev's guarantee for the mean. Probability

The median of many small averages

Knowing only that a quantity has a finite spread, the plain average of n samples can be promised to within σ/√(nδ) with confidence 1 − δ, and no better — Chebyshev's bound is tight, and rare large jumps achieve it. Cut the same samples into a dozen blocks, average each block, and take the median of the averages, and the promise improves to within about σ√(log(1/δ)/n). Nothing about the data has been assumed beyond the spread; only the way of combining it has changed.

 The trefoil and its mirror image. The trefoil beside its reflection, with writhe 3 and −3. The Jones polynomials are t + t³ − t⁴ and −t⁻⁴ + t⁻³ + t⁻¹; the Alexander polynomial of both is t − 1 + t⁻¹. Topology

A polynomial that tells left from right

The trefoil and its mirror image have the same colourings, the same determinant and the same Alexander polynomial, and the first proof that they differ was a hard argument about groups. Smooth every crossing both ways, count the circles in each of the resulting pictures, and add up the counts with the right weights: the total changes when the knot is reflected.

 How much the first player can guarantee, as the coin's bias moves. A plot of the first player's best guaranteed winning chance in Penney's game against the probability of heads, a third on a fair coin and rising past one half only when the coin is heavily biased. Probability

A coin that lets the first player win

On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.

 Removing one end of an ordinary line from a triangle, its midpoints and its centroid. Two panels. Left: 7 points with all 9 connecting lines, one ordinary line solid and one of its ends ringed. Right: the same points with that end removed, 7 connecting lines left. Geometry

At least as many lines as points

Sylvester's theorem says some line through two of the points misses all the rest. Remove one end of that line and the line itself disappears, taking at least one line away with one point. Run that backwards and it proves that n points not all in a line determine at least n lines — and the only sets that manage exactly n are a line of n − 1 points with one point off it.

 9 trees on a cubic, 10 rows of three. The curve y = 1/(1 + x²) with 8 marked points on it and a 9th at infinity. Every line through three of them is drawn: 10 in all. Geometry

Rows of three, planted on a cubic

Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.

 Unifying f(x, g(x)) with f(h(y), g(z)). Three term trees: f(x, g(x)), f(h(y), g(z)), and their common instance f(h(y), g(h(y))) under the most general unifier x ↦ h(y), z ↦ h(y). Logic

Two terms made equal, and no more

Resolution with variables needs two literals to clash, and they clash only after something has been substituted for their variables. There are infinitely many substitutions that would do. One of them is the most general — every other is it followed by something more — and an algorithm of four rewriting rules finds it or proves there is none. That single computation turns the search for instances from guessing into arithmetic.

 From a rational point on x² + y² = 17 to a whole one. The circle of radius √17 on the integer lattice, a rational point on it, and 4 reflections through nearest lattice points, the denominators 3757, 205, 25, 5, 1, ending at (1, 4). Number

A fraction on the circle forces a whole point

If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.

 A ribbon on a lifted figure eight: link 1, twist 0.47, writhe 0.53. A closed ribbon drawn as its core curve and one edge, joined by short ties. The edges have linking number 1; the twist 0.467 and writhe 0.533 add to it. Topology

A whole number split into two that are not

Run a ribbon round a closed loop and its two edges link a whole number of times. That number is shared between two quantities that are nothing like whole numbers: how far the ribbon twists about its core, and how far the core coils about itself. Bend the loop and the twist and the coiling trade continuously, to three decimal places, while their sum stays fixed — the arithmetic behind a coiled telephone cord and a supercoiled loop of DNA.

 Writing 0.3 in base φ. The graph of x ↦ βx mod 1 for the golden ratio φ, the diagonal, and the cobweb staircase of the orbit of 0.3, beside the 8 digits it writes. Dynamics

Counting in a base that is not a whole number

Multiply by β and keep the fractional part, over and over, and the whole parts you throw away are the digits of the starting number in base β — even when β is the golden ratio. Which digit strings can ever appear is decided by one string alone: the way the number 1 is written in that base. In base φ it is .11, so 11 is the only thing forbidden; in base 1.8 it never ends, and no finite list of rules describes what is allowed.

 Where x ↦ βx mod 1 spends its time, the golden ratio φ. A histogram of 2000000 orbit points of the β-map for the golden ratio φ in 50 bins, with Parry's staircase density drawn over it; they agree, and the density steps down at the orbit of 1. Dynamics

Where a base-β orbit spends its time

Follow x ↦ βx mod 1 for a million steps and count how often the orbit visits each part of the interval. For the doubling map the answer is evenly; for the golden ratio it is a staircase with one step, spending 1.17 times the average near 0 and 0.72 times it near 1. The step sits exactly where the orbit of the number 1 lands, and in a base whose 1 never has a finite expansion the staircase has infinitely many steps.

 Evidence as steps on a scale of decibans. A horizontal decibel scale of odds with a starting dot at the prior and one arrow per piece of evidence, ending at a posterior probability of 44.2%. Probability

Evidence measured in decibans

Write a probability as odds and take the logarithm, and every piece of evidence becomes a length. A positive result on a good test is thirteen decibans; a negative one is minus twenty. Lay the lengths end to end from the prior and the posterior is where they stop, in any order. The rule fails in exactly one way — when two pieces of evidence share a cause — and Turing built a code-breaking method on the arithmetic.

 Switching envelopes when the largest amount is 64. A bar chart of the probability-weighted gain from switching at each amount that might be seen, 1 to 64: small positive bars and one large negative bar, adding to zero. Probability

The envelope that always looks better

Two envelopes, one holding twice as much as the other. Open one, see an amount, and reason that the other holds double or half with equal chance — so switching gains a quarter on average. By symmetry the same argument says switch back. The step that fails is not the arithmetic; it is the claim that double and half are equally likely whatever amount is seen, which no honest prior allows — and there is one prior under which the other envelope really does look better at every amount.

 A random ranking on the triangular graph, 8 a side. A bipartite graph in which arrival t is joined to places t through 8. One random ranking of the places is used to match greedily; 5 of 8 arrivals are matched. Discrete

Matching as they arrive

Applicants arrive one at a time, and each must be given a post or turned away on the spot. Any sensible rule matches at least half as many as the best assignment chosen with hindsight, and an adversary arranging the arrivals can hold every fixed rule to exactly half. Shuffle the posts once, at random, and give each arrival its best-ranked free post: the guarantee rises to 1 − 1/e, about 63%, and no rule of any kind can do better.

 Dividing a rent of 90 three ways, on a 9-step grid. A triangle of possible rent splits, triangulated into 81 small triangles, with each grid point coloured by the room its housemate would pick; 3 small triangles have all three rooms. Applied

A rent nobody envies

Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.

 How often each factor pattern occurs, against the Galois group. Bars for four polynomials — x³ − 3x + 1, x³ − 2, x⁴ − 10x² + 1, x⁴ − 2 — giving the share of primes up to 20000 with each factorisation pattern, beside the predicted share from each Galois group. Algebra

How a polynomial breaks modulo the primes

Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.

 The units of a cubic field, as a lattice of logarithms. Points for 66 units of the field of a root of x³ − 3x + 1, plotted by the logarithms of two of their conjugates, lying on the lattice spanned by the logarithms of θ and θ − 1. Algebra

Units that form a lattice

In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.

 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. Analysis

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.

 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. Analysis

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.

 Hippias's quadratrix, traced by two uniform motions. A unit square with a quarter circle, several positions of a turning radius and a falling horizontal line, their crossings, and the curve through them ending on the base at 2/π. Computation

A curve that divides any angle

Let a radius turn at a steady rate while a horizontal line falls at a steady rate, both finishing together, and mark where they cross. The curve they trace turns heights into angles, so dividing a height — which a ruler and compass can always do — divides the angle in the same ratio. The same curve meets its base at 2/π of the side, a length from which a square with the area of a circle follows. It reaches what no marked ruler or conic can, and the ancient objection to it is exact.

 A rectangle of the parity table, nearly balanced. A 32 by 32 grid of +1 and −1 entries — the parity of the inner product of row and column — with a 15 by 14 rectangle of chosen rows and columns highlighted; its entries sum to 8. Computation

Two weak sources make one fair bit

No fixed rule can turn every weakly random source into fair bits: for any rule, some source with almost full unpredictability makes it constant. Two independent sources are different. Multiply their bits in pairs, add, and keep the parity — and if the two together carry more unpredictability than the length of one, the result is nearly fair, whatever else the sources do. The reason is that the table of those parities is balanced on every large rectangle.

 The trace–determinant plane and the flows it sorts. The plane of trace against determinant, divided by the horizontal axis and the parabola tr² = 4 det into saddle, node, spiral and centre regions, with 6 matrices marked and their phase portraits drawn alongside. Analysis

Two numbers decide the flow

A linear system in the plane has four coefficients, and what its solutions do forever afterwards — spiral in, race out, swing round, or split along two lines — is decided by two of the numbers made from them. The plane of trace against determinant is a complete map of the possibilities, and the only places it cannot decide are the lines where it changes its mind.

 Which real 2 × 2 matrices have a real logarithm, read off their eigenvalues. Eigenvalues of 7 matrices plotted in the complex plane with the negative real axis emphasised, beside a table saying for each whether a real logarithm exists and why: A yes, B yes, C yes, D no, E no, F yes, G no. Analysis

The matrix that has no logarithm

Every square matrix has an exponential, and a matrix exponential is always invertible. The converse fails, and it fails in a way that can be read straight off the eigenvalues — a real matrix with eigenvalues −1 and −2 is no exponential at all, while minus the identity is the exponential of a whole continuum of matrices that do not even commute with each other.

 An auction for 4 objects, bid by bid, with increment 1/5. A 4 by 4 table of values beside the log of 12 bids of an auction with increment 1/5: bidder, object, the raise, and the prices after each bid; it ends with total value 26. Applied

Prices the bidders raise

The cheapest assignment is certified by a price on every task, and those prices can be found without anyone in charge. Let each unassigned person bid for the task that suits them best at current prices, raise its price by a little more than it is worth to them over the next best, and wait. The bidding ends, and when the increment is small enough the prices it ends at are a proof of optimality.

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