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The same thing twice — page 8

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
The map moved to the other side of the product. Two panels over the unit curve of the product u₁v₁ + u₂v₂. The left applies A to u and measures it against v, giving 2.328; the right applies the adjoint to v and measures it against u, giving the same number. Algebra

Moving a map across a product

The transpose looks like a fact about a matrix: reflect its entries in the diagonal. It is a fact about the inner product. Measure lengths and angles differently and the map that slides to the other side of the product is a different matrix, and a matrix that was symmetric stops being so.

 ln(1 + x) past its radius, with a denominator allowed. ln(1 + x), its Taylor sum of degree 8, and its Padé approximants of order 2 and 4. At x = 3 the Taylor sum is out by 5.95e+2 and the highest-order approximant by 2.97e-4. Analysis

A denominator that reaches past the radius

The Taylor series of ln(1 + x) is useless beyond x = 1 however many terms it is given. The same coefficients spent on a numerator and a denominator converge at x = 3, and at x = 100, because a polynomial cannot imitate a singularity and a quotient of two polynomials can.

 Perpendicular to both, and as long as their parallelogram. The vectors (2, 0.4, 0.2) and (0.6, 1.8, 0.3), the parallelogram they span, and their cross product (−0.24, −0.48, 3.36), drawn perpendicular to both with a length of 3.403, which is the parallelogram's area. Algebra

A plane disguised as an arrow

The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.

 Announcing a mixture is worth 5/3 more than any equilibrium. The leader's payoff against the probability it announces for its first action, for a leader with a dominant action that is better off not being seen to play it, with the follower's reply switching where the follower is indifferent. The best announcement is worth 11/3; the best equilibrium of the simultaneous game is worth 2. Applied

Worth more for being seen first

Moving first sounds like a disadvantage, since the other side gets to see the move and answer it. When the move is a mixture that is announced and believed, it is never a disadvantage, it is worth exactly nothing in a game of pure conflict, and in other games it is worth more than any equilibrium — sometimes by announcing an action that would never be played in secret.

 A spherical triangle becomes a quadrilateral with two right angles. A triangle on a sphere with the arc through the midpoints of two of its sides, the perpendiculars dropped from its three corners, and the quadrilateral with right angles at its base that the same area makes when the two corner pieces are moved. Computation

Equal area on a sphere, without a rectangle

On a sphere, two polygons of the same area can still be cut into each other, exactly as in the plane. Almost nothing in the plane proof survives the move: a sphere has no rectangles, no parallel strips and no similar triangles of different sizes. What carries the theorem instead is a quadrilateral with two right angles, built from a triangle's midline.

 Pascal's triangle modulo 4, where one digit at a time is not enough. 32 rows of Pascal's triangle coloured by remainder modulo 4 — hue for the last base-2 digit, depth for the second. The digit-by-digit product that gives every remainder modulo 2 gets the remainder modulo 4 wrong at 100 of the 243 entries 2 does not divide. Discrete

A remainder read two digits at a time

Lucas' theorem reads a binomial coefficient's remainder on division by a prime off its digits one at a time. On division by the prime's square the same reading is wrong at four odd entries in ten. What replaces it still reads digits — in overlapping pairs, with the prime taken out first and a sign that the carries decide.

 A polygon of 4 points averaged down to its curve at t = 0.40. A control polygon of 4 points, the 3 rounds of weighted averaging at t = 0.40 drawn as nested polylines, the single point they end at, and the whole curve those points trace. The weights on the control points are 0.216, 0.432, 0.288, 0.064. Discrete

Averaging down the triangle

Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.

 A mass split 10 times, 0.3 to the left and 0.7 to the right. A self-similar measure on the unit interval: the mass is split 10 times, 0.3 of each piece's share going left and 0.7 right, and each of the 1024 pieces is drawn as a bar as tall as its mass. The heaviest carries 0.0282 and the lightest 5.9 × 10⁻⁶. Dynamics

A dimension for every rate of crowding

Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.

 A graph of dimension 1.585, built by raising midpoints. The Takagi–Landsberg graph with w = 0.75 on the unit interval, drawn with 16 columns shaded by how far the graph rises and falls across each. Counting boxes at seven widths gives a slope of 1.582. Dynamics

The room a jagged graph takes up

The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.

 Six points on an ellipse, and the line their opposite sides meet on. A hexagon with its six corners on an ellipse. Its three pairs of opposite sides are extended until they meet, and the three meeting points lie on one straight line. Moving one corner off the ellipse breaks the alignment. Geometry

Six points on a conic, and the line they share

Put six points on an ellipse, join them into a hexagon, and extend each pair of opposite sides until they meet. The three meeting points always lie on one straight line. The statement uses no length, no angle and no focus — which is why it holds for every conic at once, and why a straightedge alone can draw the curve through any five points.

 Two confocal ellipses, two confocal hyperbolas, and equal diagonals of 1.399. Two ellipses and two hyperbolas with the same foci cut out a four-sided region with curved sides. Its two diagonals, drawn as straight segments, both measure 1.3987. Geometry

Equal diagonals in a curved quadrilateral

Two ellipses and two hyperbolas sharing the same foci cut out a four-sided region with curved sides and no symmetry to speak of. Its two diagonals are nevertheless exactly equal. The reason is a stretch that carries one ellipse onto the other and moves every pair of points so that crossed distances match — a property only confocal curves have.

 A derivation of ((p → q) ∧ (q → r)) → (p → r), and the term it is. A natural-deduction derivation of ((p → q) ∧ (q → r)) → (p → r) with a lambda term written under each formula. The whole derivation is the term λh₁. λh₂. (π₂ h₁) ((π₁ h₁) h₂), and at every node the type of the term is the formula the derivation proves there. Logic

Every derivation is a term

Write a variable for each assumption, an abstraction where one is discharged, and an application where an implication is used, and a natural-deduction derivation becomes a term. The formula it proves is the term's type, and checking the one is checking the other. A detour in the proof — a lemma introduced and at once used — is a term that simplifies, and simplifying it is removing the detour.

 The Farey tessellation, and a line down to √2 − 1. Semicircles over the unit interval joining every pair of Farey neighbours with denominators up to 13, and a vertical line at √2 − 1. The 6 arcs it crosses are the intervals of the Stern–Brocot descent to √2 − 1, and their turns spell LRRLL. Number

The arcs a line crosses on its way to a number

Draw a semicircle over every pair of neighbouring fractions and the half-plane above the number line is cut into curved triangles that never overlap. A straight line dropped towards any number crosses those arcs one after another, and the arcs it crosses, and the side it leaves each triangle by, are exactly the steps of the Stern–Brocot descent towards that number.

 Minkowski's question-mark function. The graph of Minkowski's function ?(x) on the unit interval: continuous and increasing, sending each Stern–Brocot fraction to the binary fraction in the same position. It sends √2 − 1 to 2/5 and φ − 1 to 2/3. Number

The function that sends fractions to binary

The Stern–Brocot tree and the tree of binary fractions have exactly the same shape, so there is a function that sends each fraction to the binary fraction in the same position. It is continuous and increasing, it turns every quadratic irrational into an ordinary fraction, and it does all of its rising on a set of numbers so thin that at almost every point its slope is nought.

 3 sheets over a surface of genus 2: a surface of genus 4. A 3-sheeted covering of the closed surface of genus 2, drawn as 3 copies of its 8-sided face with each side coloured by its generator and numbered with the sheet it glues to. The Euler characteristic −6 is 3 times −2, and the cover has genus 4. Topology

Covering a surface multiplies its count

A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.

 The coefficients of (1 + x)¹² sorted by remainder mod 3. The binomial coefficients of the 12th power coloured by the remainder of their index on division by 3, beside the 3 points one plus a root of unity, whose powers averaged pick out each colour's total. Algebra

Every third coefficient

Add every third number in the twelfth row of Pascal's triangle and the answer is 1366 — a third of 4096, rounded up. Which way the rounding goes is decided by two arrows of length one in the complex plane, and the same average over the roots of unity counts dice totals, subsets and necklaces.

 Fences of 300 against a straight wall. Fences made of two, three and four straight pieces with both ends on a wall, beside a half-circle with the same length of fence, each labelled with the area it holds. Geometry

Half a circle against a wall

Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.

 5 two-literal clauses drawn as 10 arrows. The literals of four variables arranged in a circle with an arrow for each implication a two-literal clause contains, the arrows forming a path from one literal to its negation and back highlighted, beside the list of clauses and their arrows. Logic

Two literals make an arrow

A clause of two literals, p ∨ q, says that if p is false then q is true, and if q is false then p is true — two arrows. A set of such clauses is a directed graph on the literals, and it is unsatisfiable exactly when some variable and its negation reach each other. Each of those two paths is a chain of resolution steps, and finding them takes time proportional to the size of the input.

 Bhāskara's cyclic method on x² − 61y² = 1, step by step. A table of the cyclic method's rows: the helper m chosen at each step and the near miss a² − Db² = k it produces, ending at k = 1 with the fundamental solution. Number

A method that is allowed to miss

Bhāskara's cyclic method solves x² − Dy² = 1 by aiming at the wrong target. It keeps a pair a, b with a² − Db² = k for some small k, combines it with a helper chosen so that k can be divided out, and repeats until k is 1. For D = 61 it reaches the ten-digit fundamental solution in 13 steps, where walking the convergents of √61 takes 22 — and for every D up to 100 it is faster.

 The 6 ways to deal 4 labels between pieces of size 2 and 2. Every way of splitting 4 labels between a piece of size 2 and a piece of size 2, listed as two rows of boxes each. The count is the binomial coefficient that distinguishes a labelled product from an ordinary one. Discrete

The product that deals the labels

Multiplying two counting series pairs one choice with another. When the things being counted carry labels, the labels have to be dealt out as well, and the only series that survive the extra bookkeeping are the ones divided by n factorial.

 Permutations of n things, by how many pairs they put in the wrong order. A table whose row n and column k hold the number of permutations of size n whose inversions is k, with each row's total beside it — the plain count the one-variable series gives. Discrete

The coefficient that is a polynomial

Add a second variable to track a statistic and each coefficient stops being a number. Set the new variable to one and the old count comes back untouched; leave it in and the mean of the statistic is a derivative rather than an average.

 The splits of any two of three decide that nobody can out-argue. The triangle of all splits of a joint gain, with the splits marked at which every player's loudest complaint against every other is matched by an equally loud complaint back. Applied

An objection one player makes to another

The core lets a coalition object to everybody at once. Narrow it to one player objecting to one other, require every such objection to be met by an equally loud one coming back, and exactly one split survives — with no dictionary order anywhere in the argument.

 One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths. Analysis

The length belongs to the journey

Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.

 The coordinate change that makes an unlocked map a rotation. The conjugating map built from one orbit, plotted as a staircase from the circle to itself, with the rigid rotation it turns the circle map into. Dynamics

A rotation in different coordinates

At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.

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