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The same thing twice — page 11

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
441 factored two ways among the numbers of the form 4k + 1. Two factor trees for 441 among the Hilbert numbers, those one more than a multiple of four: one splits it as 9 times 49, the other as 21 times 21, and every factor is irreducible there. Number

A factorisation that hides its primes

Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.

 Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain. Number

Factoring uniquely with no way to divide

Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

 Permutations that avoid 8 forbidden cells. A 5 by 5 grid with 8 forbidden cells shaded and one permutation that avoids them marked, beside the numbers of ways to place non-attacking rooks on the forbidden cells and the count of avoiding permutations they give. Probability

The cells a permutation must miss

A derangement is a permutation that misses the diagonal of a square grid. Forbid any other set of cells instead and inclusion–exclusion still counts what is left — driven entirely by one list of numbers, the ways to place non-attacking rooks on the forbidden cells. Boards that look nothing alike can share that list, and rooks on a staircase turn out to count the ways to split a set.

 Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row. Algebra

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

 Conway's seven pieces, pulled apart and measured. A triangle with angles 78, 54, 48 degrees divided along its angle trisectors into an equilateral centre, three pieces touching it along its sides and three along the outer sides, spread apart, with each corner's angle labelled as a third of an outer angle plus a multiple of sixty degrees. Geometry

Seven pieces and an equilateral middle

Morley's theorem has two proofs worth knowing, and they run in opposite directions. The trigonometric one starts from the triangle and computes each side of the inner one as 8R sin α sin β sin γ, symmetric in the three angles. Conway's starts from an equilateral triangle, builds six pieces round it from their angles alone, and shows they fit — so the triangle they make is whatever triangle was wanted, and its middle is equilateral because it was built that way.

 Twenty-seven choices of trisector, and the eighteen equilateral triangles. Twenty-seven small panels, one for each choice of trisecting line at each corner of a triangle, each drawing the triangle and the triangle the chosen lines cut out; the eighteen equilateral ones are marked. Geometry

Eighteen equilateral triangles

Every angle of a triangle has three trisectors, not one, once the angle and its outside are both counted. Choosing one at each corner gives twenty-seven ways to cut out a triangle, and eighteen of them give an equilateral one. The nine that fail are exactly the choices whose labels add to 2, 5 or 8 — and all eighteen equilateral triangles have their sides in the same three directions, fixed by a third of the difference between two angles.

 Euclid's game on 34 and 21. A 34 by 21 rectangle tiled by the squares of Euclid's algorithm, each run of equal squares shaded by the player who faces it, with the deciding run outlined. Geometry

The player who meets the first long run

Turn Euclid's algorithm into a game: two players take turns cutting squares off the rectangle, any number from the current run, and whoever cuts the last one wins. The whole game is decided before it starts — by whether the ratio of the sides is more or less than the golden ratio, which is the same thing as how many runs of length one come first.

 Counting the real roots of x⁵ − 5x³ + x² + 3x − 1 with Euclid's algorithm. Above, the graph of a polynomial over an interval with its real roots marked; below, a step function counting sign changes in the polynomial's Sturm chain, which drops by one at each root; beside them the chain of polynomials listed. Geometry

The remainders that count the roots

Run Euclid's algorithm on a polynomial and its derivative, flipping the sign of each remainder, and write down the signs of the whole chain at any point. The number of sign changes drops by exactly one each time the point passes a real root — so the roots in any interval can be counted, exactly, without finding a single one.

 Carroll's babies and crocodiles, on four ellipses. Four overlapping ellipses labelled with the four classes of a sorites, the regions emptied by its premises shaded, and the regions its conclusion requires to be empty outlined. Logic

The conclusion is what survives the erasing

Lewis Carroll's puzzles give three premises about four classes — babies, logical people, the despised, crocodile-managers — and ask what follows. Draw all four, shade what the premises rule out, then erase the classes the conclusion is not about: a region survives as empty only if everything above it was. What is left is the conclusion, and erasing a class turns out to be exactly one step of resolution.

 The three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath. Algebra

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

 Newton's and Halley's basins for z³ − 1. Two squares of the complex plane side by side, each coloured by which root a starting point converges to, the left under Newton's method and the right under Halley's, with non-converging starts marked. Dynamics

A cubic method that is Newton's in disguise

Halley's method, from 1694, uses the second derivative as well as the first and cubes the error at every step where Newton's squares it. It is also, exactly, Newton's method applied to a different function — p divided by the square root of p′ — and that single fact explains why its basins are calmer, why it walks out of the trap that holds Newton for ever, and why its boundaries are still fractal.

 Pollak's circle: one rotation in every n + 1 parks on the line. Several circles of numbered spots, each showing where cars park when every preference in a list is rotated by a fixed amount, with the one rotation that leaves the last spot empty highlighted. Discrete

Cars that park, and trees that grow

Three cars arrive at a one-way street with three spaces; each has a favourite space, drives to it, and takes the first free one from there on. Of the 27 lists of favourites, exactly 16 let every car park — the same 16 as the labelled trees on four points. The reason is a circular street with one extra space, on which every list parks and exactly one rotation of it leaves the extra space empty.

 Credit for one prediction, three ways of leaving an input out. Groups of bars, one group for each way of filling in the inputs that are not known, each bar one input's share of the difference between the prediction and the starting value. Applied

What a missing input is worth

A model prices a house at 180 from its size, its garden and its bedrooms, and the question is how much of the price each input is responsible for. Make the inputs the players and the average over orders answers it — once somebody decides what the model says when an input is not known. Three reasonable decisions give bedrooms nothing, nothing, and sixteen, for a model that never reads them.

 Shares on two triangles and a go-between. A network of players with each node labelled by its share of what the whole network earns, and its number of links beneath it. Applied

Cutting a link costs both of its ends the same

Three players, any two of whom can earn 1 together — but only if they are linked. Link all three and each is due a third. Remove one link and the player holding both of the others is due two thirds. Averaging over orders on the game the network allows is the one rule under which breaking any link costs the two players it joined exactly the same, and it pays go-betweens more than their links.

 Loops on a torus that never cross themselves. Squares with opposite edges glued, each carrying one straight loop of a different slope, each labelled with its two crossing counts. Topology

The loops on a torus that never cross themselves

Every loop on a torus is classified by two whole numbers: how often it goes round one way and how often the other. Some classes can be drawn without the loop ever crossing itself and some cannot, and the rule is the oldest in arithmetic — the two numbers must have no common factor. The same two numbers say how often any two loops must meet.

 A twist along the horizontal loop, done once. Squares with opposite edges glued and a shaded horizontal band, showing one loop before and after the torus is twisted along the band. Topology

A twist that carries one loop to another

Cut a torus along a loop, turn one side of the cut once round, and glue it back. Nothing is torn, so every loop that did not cross itself still does not — but a loop of class (0, 1) is now a loop of class (1, 1). Two such twists reach every loop that never crosses itself, by Euclid's algorithm, and the symmetries they generate are exactly the whole-number matrices of determinant one.

 Discs from the rows, and the eigenvalues inside them. The complex plane with one disc per row of a 3×3 matrix — centred on its diagonal entry, with radius the sum of the sizes of the rest of the row — and the 3 eigenvalues marked: 4.14, −5.09, −1.06. Algebra

Discs that fence in the eigenvalues

Draw one disc for each row of a square matrix, centred on the diagonal entry, with a radius equal to the sum of the sizes of everything else in that row. Every eigenvalue lies inside one of the discs, and a group of discs set apart from the rest holds exactly as many eigenvalues as it has discs. Nothing is solved to find them.

 The largest value on each plane, over every direction of space. A longitude–latitude map of directions u in three dimensions, each shaded by the largest value of xᵀAx on the plane perpendicular to u. The smallest such value is the middle eigenvalue 1.829, reached at ±v₁; the largest is 3.622, reached along a great circle. Algebra

The highest point on the sphere is an eigenvalue

For a symmetric matrix, walk a unit arrow over every direction and record the value of xᵀAx. The highest value reached is the largest eigenvalue, the lowest is the smallest, and every eigenvalue in between is a saddle height, a minimum of maxima. From that one description comes a theorem no formula for the roots could give: delete a row and its column, and every eigenvalue of what is left sits between two of the original's.

 A plucked string, released. A string plucked into a tent shape at 0.5, drawn at times 0, 0.1, 0.25, 0.5, 0.75, 1: the corner splits into two corners that run apart, reflect off the ends upside down and meet again. Analysis

A plucked string keeps its corners

Heat smooths a sharp profile at once, because each harmonic decays at a rate set by the square of its frequency. Change one time derivative into two and nothing decays at all: each harmonic swings for ever, the corner of a pluck splits in two and runs along the string, and after one period the shape comes back exactly. The same sines, the same coefficients — and a flow that loses nothing.

 3 orthogonal squares of order 4, read as a code. A table of 16 words of length 5 over 4 symbols, one per cell of 3 orthogonal Latin squares of order 4: row, column and the entry in each square. Any two words agree in at most one position. Computation

Orthogonal squares are a code

Write down each cell of a set of orthogonal Latin squares as a word — its row, its column, and its entry in each square — and no two words agree in more than one place. That is not a pleasant accident of the squares. It is exactly what being Latin and being orthogonal say, it makes the list an error-correcting code as good as any code of its size can be, and the squares a field builds turn out to be a Reed–Solomon code, the one on every compact disc.

 Every residue joined to 2 times itself, on a dial of 199. A circle with 199 equally spaced points and a chord from each point k to the point 2k mod 199, with the 1-cusped curve the chords envelope drawn dashed. Discrete

Multiplying every number on the dial at once

Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.

 Square roots of 2, lifted from one power of 7 to the next. A tree whose rows are the square roots of 2 modulo 7 to the powers 1 to 4, each joined to the root it reduces to; counts 2, 2, 2, 2. Discrete

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

 Every orbit round a square closes. 5 outer-billiard orbits about a square, each drawn as its closed ring of points, with periods 4, 8, 12, 20, 24 growing outwards. Dynamics

The ball that stays outside the table

Turn billiards inside out. A point outside a convex table looks at the corner on its right, jumps straight through it, and lands as far beyond as it started before. Round a square every orbit closes; round a circle every orbit keeps to its own circle; round a regular pentagon the orbits form islands with a fractal between them. Whether some table lets a point wander off to infinity was Moser's question, and the answer — yes, for a kite — took until 2007.

 A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH. Dynamics

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

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