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The same thing twice — page 5

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way. Number

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.

 the right triangle at an eighth of a turn: 16 directions, and a surface of genus 2. A polygonal billiard table with a long trajectory drawn on it, the finite set of directions that trajectory takes, and the arithmetic of the surface it unfolds into. Dynamics

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

 A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem. Dynamics

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

 Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge. Topology

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

 A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on. Topology

Two pieces, in every dimension

A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

 6 necklaces, concatenated into a de Bruijn sequence. The Lyndon words of length dividing 4 over 2 letters, listed in lexicographic order and written end to end; the result is the lexicographically least de Bruijn sequence of order 4. Computation

Every necklace, in order

The graph construction needs the whole graph in memory and finds one sequence among hundreds of millions. Listing the necklaces in alphabetical order and writing them end to end needs no graph at all, and produces the smallest of them.

 A 4-bit register that visits all 15 nonzero states. The first 15 states of a 4-bit linear feedback shift register with taps at 4 and 1, with the bit that leaves the register at each step; the output shows every nonzero window of 4 bits exactly once. Computation

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

 Five rules, one dial. Seats for each of 5 regions at 21 settings of the rounding threshold, with the three settings that are the named methods marked; the largest region gains and the smallest loses as the threshold rises. Applied

Five rules and one dial

Adams, Webster and Jefferson are usually taught as three rules for rounding a share. They are one rule with a number in it, and turning that number from nought to one moves seats from the smallest region to the largest, one at a time.

 What each rule is answering. A table of five apportionments against three measures of inequality between two regions, with a tick where no transfer of a seat reduces the measure; each measure certifies exactly one of the five. Applied

Choosing what unfair means

Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.

 An Apollonian gasket, 125 circles in. The Apollonian gasket generated from four mutually tangent circles of curvature −1, 2, 2 and 3, drawn to 4 generations; every curvature in it is a whole number. Geometry

Curvatures that stay whole

Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.

 Two inversions, and the number four points agree on. Four points, their images after one inversion and after a second in a different circle, with the cross-ratio computed at each stage; it is conjugated once and restored twice. Geometry

The number four points agree on

One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.

 Nineteen circles, one per vertex. A circle packing of a triangulation with seven interior vertices and twelve on the boundary: two circles touch exactly when their vertices are joined, and the radii were solved for rather than chosen. Geometry

Every flat graph is a pile of circles

A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.

 The cube cut into 6 symmetric chains. The subsets of a set of 4 partitioned into 6 chains by the bracket rule, each chain running from size k to size 4 − k and passing once through the middle layer. Discrete

The cube cut into chains

Write a subset as a string of brackets, match them the ordinary way, and the unmatched ones say which chain it is on. Six chains cover all sixteen subsets of a four-element set, and the bound and the example arrive together.

 The two supplements, and the residue classes that decide them. A table of odd primes with the Legendre symbols of minus one and two beside the residue of p modulo four and modulo eight. Number

The two supplements, and where the eight comes from

The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.

 Multiplication by 3 modulo 11, and the sign of the shuffle. Residues in two rows joined by strings showing where multiplication sends each one, with a strip beneath comparing the sign of the shuffle to the Legendre symbol for every multiplier. Number

The symbol is the sign of a shuffle

Multiplying every residue modulo p by a fixed number rearranges them. That rearrangement is a permutation, permutations have a sign, and the sign is exactly the Legendre symbol — so a question about squares becomes a question about crossings.

 The Gauss sum for 13, added one root at a time. Partial sums of the p-th roots of unity signed by the Legendre symbol, drawn as a walk closing on a point at distance root p from the origin. Number

One sum, squared two ways

Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.

 A rotation of four-space, and the two angles it turns through. The four by four matrix of the map sending x to p x conjugate q, beside two dials showing the angle it turns through in each of its two invariant planes. Algebra

A rotation of four-space takes two of them

One unit quaternion, conjugating, turns three-space about an axis. Two of them, multiplying from the left and the right, turn four-space — and a rotation of four-space has no axis at all, but two independent angles and two planes it spins in.

 The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it. Algebra

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

 The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one. Algebra

The integers among the quaternions

The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.

 The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair. Algebra

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

 A game that stops, over totals 0 to 5. States in a row with arrows up and down between them and the two ends absorbing, above a table of the expected number of steps and the chance of ending at the top from each start. Probability

The chain that stops

Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.

 A walk on a weighted graph, and a cycle whose traffic goes one way. A weighted graph with the long-run share of each state read off its total weight, beside a three-state cycle whose shares are equal and whose traffic circulates. Probability

The chain that runs the same backwards

Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.

 The time a single walk spends in each state, against the share it should hold. Paired bars for each state, one the fraction of a long run's time spent there and one the computed stationary share, above a table of expected return times. Probability

The time spent and the share held

Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.

 A model whose worlds are sets of sentences. Worlds labelled by which of a fixed finite set of formulas they accept, with an arrow wherever every boxed formula accepted by one has its inside accepted by the other. Logic

Worlds built out of sentences

A Kripke model needs worlds, and nothing so far has said where worlds come from. They can be made of the syntax: a world is a set of formulas it commits to, one world sees another when the boxed commitments line up, and in the model that results every formula is true exactly where it was assumed.

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