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Counting the same thing twice — page 9

One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.
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.

 The core of a market with two houses and two buyers. The core of an assignment game drawn in the plane of buyer A's payoff against buyer B's, a polygon with vertices (0, 1), (0, 0), (1, 0), (4, 3), (2, 3); the buyers' best corner is (4, 3). Applied

The prices nobody can break away from

When houses are sold to buyers who value them differently, there is a whole range of prices at which nobody wants to walk away, and it has a remarkable shape — one corner best for every buyer at once, one best for every seller at once, and the buyers' corner pays each buyer exactly what the market would lose without them.

 Selberg's parity example: the same sieve data, the primes on one side and nothing on the other. Horizontal bars for the numbers up to 100000 with an odd and an even count of prime factors, before sifting (50144 and 49855) and after sifting by the primes below √100000 (9527 and 0). Number

Two halves a sieve cannot tell apart

Split the whole numbers into those with an even number of prime factors and those with an odd number. To a sieve — which sees only how many numbers each small divisor divides — the two halves look almost exactly alike. Yet sifting out the small primes leaves every prime in one half and nothing at all in the other. That is Atle Selberg's parity example, and it is the reason no sieve can prove there are infinitely many twin primes.

 Prime pairs a fixed distance apart, counted, against Hardy and Littlewood's prediction. Bars for the number of prime pairs (p, p + d) up to 2000000 for even d from 2 to 60, jagged with peaks at multiples of 6 and 30, each with the Hardy–Littlewood prediction marked. Number

The patterns primes are allowed to make

Primes two apart are twin primes; primes six apart are about twice as common; three primes two apart happen exactly once, at 3, 5, 7. Which patterns of primes can recur for ever is decided by remainders alone — a pattern must leave some remainder free modulo every prime — and how often each one recurs is predicted, to within a few per cent, by a constant computed from nothing else.

 Integration error against the number of points: lattice, Halton and random, on a smooth periodic function. Log-log plot of integration error against N for random points (slope -0.47), Halton points (-0.96) and Fibonacci lattices (-1.83). Probability

Points on a lattice that see almost nothing

Average a function over the points of a carefully tilted lattice and the error can fall like one over the square of the number of points — far faster than random sampling, and faster than the most evenly spread sequences. The reason is that a lattice rule is blind to only a thin set of frequencies, its dual lattice, and a smooth periodic function has almost nothing there.

 Every count of corners and faces a convex solid can have. A grid of pairs (V, F) from 4 to 16, with the 85 realisable pairs filled between the lines F = 2V − 4 and V = 2F − 4, and the regular solids labelled. Topology

Every count a solid can have

Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.

 Counting overlapping targets from sensor counts alone, by integrating against χ. A 36 by 32 grid of sensor counts from 7 overlapping discs; the sum of the Euler characteristics of the level sets is 7. Topology

Counting targets by their holes

The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.

 A map that folds the plane over itself, with every point still counted once. The map (u, v³ + uv): its domain shaded by the sign of the Jacobian determinant and its image with the grid carried across. At 5 marked target points the preimages number 3, 3, 1, 1, 1 and their signed counts are all 1; the determinant integrates to 4.447, equal to the integral of the signed count. Algebra

The count a fold cannot change

A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.

 12 points, a cup of 5 and a cap of 5. 12 points in general position with the longest convex-upward chain (5 points) and the longest convex-downward chain (5 points) marked. Discrete

Every crowd holds a bowl or a dome

Among enough points in the plane, some k of them always bend upward like a bowl or some l bend downward like a dome. The number that forces it is a binomial coefficient, it is exactly right, and it proves that every large enough crowd contains a convex polygon — with a bound nobody could lower to the true answer for eighty years.

 The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region. Number

Counting the classes that break factorisation

The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

 How far a deck of 52 is from random after each riffle shuffle. A bar chart over one to 12 riffle shuffles of the exact total variation distance from a uniformly random deck. The bars stay near one for the first few shuffles and drop sharply around 8. Probability

The forgetting that happens all at once

A single small chain forgets its start gradually, a little more with every step. A family of large ones can do something different — stay almost perfectly informed about where it began, and then lose all of it inside a window far shorter than the wait. That cliff is the cutoff phenomenon, and it is why "seven shuffles" is an answer rather than a convention.

 Every way to cut a genus-2 surface into pairs of pants. 2 thickened graphs, one for each type of pants decomposition of the closed surface of genus 2, each with 2 coloured junctions and 3 cutting circles. Topology

Every surface is sewn from pants

A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.

 The seven-vertex torus, unrolled onto a lattice. A triangular lattice with every point labelled a + 3b mod 7 and fourteen triangles shaded as one copy of the torus. Topology

The fewest corners a surface needs

Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.

 A loop that is a hole, and the same loop filled in. a hollow triangle: 3 points, 3 edges, 0 triangles; holes in each dimension 1, 1, 0; a filled triangle: 3 points, 3 edges, 1 triangles; holes in each dimension 1, 0, 0. Algebra

A hole is a cycle that bounds nothing

A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.

 The core of a group worth the square of its size: the outline of its six arrival orders. Splits of 9 among three players; core corners (1, 5, 3), (5, 1, 3), (1, 3, 5), (5, 3, 1), (3, 1, 5), (3, 5, 1); arrival-order splits (1, 3, 5), (1, 5, 3), (3, 1, 5), (5, 1, 3), (3, 5, 1), (5, 3, 1); average 3, 3, 3. Applied

The corners are the orders of arrival

Line the players up, let each join in turn, and pay each what it adds on arrival: every order gives a split. When a newcomer always adds at least as much to a bigger group, those splits are exactly the corners of the core — so the core is never empty, it is the outline of the orders, and the average over all of them lies inside it. For a group worth the square of its size the outline is a hexagon whose corners are the six orderings of 1, 3 and 5.

 The complete graph on 7 points, drawn straight with 9 crossings. A straight-line drawing of the complete graph on 7 vertices with 9 crossings, the minimum for straight drawings; each crossing comes from four vertices in convex position. Discrete

The crossings a graph cannot avoid

Five points all joined need one crossing, six need three, seven need nine. Euler's formula gives a lower bound that grows like the square of the number of points and is badly wrong; a sampling trick turns the same formula into a bound that grows like the fourth power and is right to within a constant. And for drawings with straight edges the count turns out to be something else entirely: the number of quadrilaterals the points make.

 How many windows of each period, on two maps and as a count of necklaces. period 1: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 2: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 3: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 4: 2 (logistic), 2 (sine), 2 (necklaces), 2 (polynomials); period 5: 3 (logistic), 3 (sine), 3 (necklaces), 3 (polynomials); period 6: 5 (logistic), 5 (sine), 5 (necklaces), 5 (polynomials); period 7: 9 (logistic), 9 (sine), 9 (necklaces), 9 (polynomials); period 8: 16 (logistic), 16 (sine), 16 (necklaces), 16 (polynomials); period 9: 28 (logistic), 28 (sine), 28 (necklaces), 28 (polynomials); period 10: — (logistic), — (sine), 51 (necklaces), 51 (polynomials); period 11: — (logistic), — (sine), 93 (necklaces), 93 (polynomials); period 12: — (logistic), — (sine), 170 (necklaces), 170 (polynomials); period 13: — (logistic), — (sine), 315 (necklaces), — (polynomials); period 14: — (logistic), — (sine), 585 (necklaces), — (polynomials). Dynamics

Windows counted like necklaces

The logistic map has one window of period three, two of period four, three of period five, five of period six, nine of period seven — and the sine map, which shares no algebra with it, has exactly the same numbers, in exactly the same order along the parameter. The counts are 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, and they are the number of ways to thread a necklace of beads in two colours.

 The volume of the unit ball and the area of its sphere, dimension by dimension. Unit ball volumes for dimensions 0 to 20, largest at 5 (5.2638); sphere areas largest at 7 (33.0734). Geometry

The ball that is largest in five dimensions

A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.

 Three sizes of infinity, and the collections that have each. ℵ₀: the fractions, the algebraic numbers, every finite text, every definition of a number; 2^ℵ₀: the real numbers, the continuous functions, the open sets, the closed sets, the Borel sets; 2^(2^ℵ₀): all functions, all sets of reals, the non-Borel sets, the measurable sets. Logic

What counting can prove exists

There are only as many continuous functions as points of a line, only as many Borel sets, only countably many definitions — and more sets of reals than any of those. So counting proves that discontinuous functions, non-Borel sets and undefinable numbers exist without exhibiting one. It cannot prove that a set with no length exists, because the sets that have a length are exactly as numerous as all sets, and that difference turns out to be the difference between what the axioms force and what they merely allow.

 The necklaces of 7 beads with 3 black. C(7, 3) = 35 arrangements in 5 rotation classes of sizes 7, 7, 7, 7, 7. Number

The row that proves a prime

Fermat's little theorem can be fooled: 561 passes it for every base and is not prime. Thread necklaces with a fixed number of black beads instead of any colours at all, and the count becomes a statement about a whole row of Pascal's triangle — every middle entry of row n is a multiple of n exactly when n is prime — which no composite can fake. Written as polynomials it is (x + a)ⁿ = xⁿ + a, and cut down to size it is the first proof that primes can be recognised in polynomial time.

 What a rule collects when every value comes from the same distribution. uniform on [0, 1]: best rule 0.995, one threshold 0.635 of E[max] at n = 200; exponential: best rule 0.905, one threshold 0.678 of E[max] at n = 200; Pareto, tail exponent 2: best rule 0.802, one threshold 0.714 of E[max] at n = 200; Pareto, tail exponent 1.2: best rule 0.802, one threshold 0.682 of E[max] at n = 200. Probability

When every value comes from the same hat

A rule that sees values one at a time and must keep or discard each on the spot can guarantee half of what a prophet collects, and no more, when the values come from different distributions. When they all come from the same one, the guarantee rises to 0.745 — and a single fixed threshold, set so that each value crosses it with chance 1/n, already secures 1 − 1/e. For bounded values the best rule collects nearly everything; only a heavy tail, where one enormous value carries the prize, keeps the gap open.

 The two-sided cover of a sphere with 3 cross-caps, built from aabbcc. Two copies of the polygon aabbcc with opposite senses of turning, glued within or across copies; 2 vertices, 6 edges, 2 faces, characteristic −2: a surface with 2 handles. Topology

Two copies of the polygon, cross-matched

The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.

 The 3 periods of the 13th roots of unity, and the equation they solve. Roots of unity modulo 13 grouped by the cosets of the index-3 subgroup; the periods 0.274, 1.377, −2.651 are the roots of x³ + x² − 4x + 1. Algebra

The sums that obey a smaller equation

The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.

 The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8. Algebra

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

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