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Series — page 2

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
A table of shares written as a lottery over 3 whole assignments. A doubly stochastic table of shares, and beneath it the permutation matrices and weights that add up to it exactly, each drawn as a grid with one marked cell per row.

Assignment

  1. 1 A lottery over whole assignments
  2. 2 The corners are whole assignments
  3. 3 A price for every person and task
  4. 4 One table, two lotteries
  5. 5 Where the corners stop being whole
  6. +3 more
8 essays · applied A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

Billiards

  1. 1 A bounce is a fold of the table
  2. 2 A table folded into a surface
  3. 3 A room that cannot be lit
  4. 4 The obstacle that makes a table chaotic
  5. 5 The triangle nobody can settle
  6. +3 more
8 essays · dynamics When a shared birthday becomes likely. The chance that some pair in a group shares a birthday, against group size. It passes a half at 23 people, where the probability is 50.7%.

Birthday problem

  1. 1 Twenty-three people
  2. 2 Any unevenness brings the match sooner
  3. 3 A collision that finds a factor
  4. 4 Fourteen people within a day
  5. 5 A room where nobody is alone
  6. +3 more
8 essays · probability One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

Borsuk ulam

  1. 1 One line that halves them both
  2. 2 Two opposite points that agree twice
  3. 3 As many cuts as colours
  4. 4 Opposite labels that have to meet
  5. 5 The colours a circle forces
  6. +3 more
8 essays · topology A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

Circular functions

  1. 1 A sine wave is a circle seen from the side
  2. 2 The angle that is really an area
  3. 3 When two circular motions come home
  4. 4 Three sines tie a knot
  5. 5 The sine of a figure eight
  6. +3 more
8 essays · analysis Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

Conic sections

  1. 1 One cone, four curves
  2. 2 Every ray comes back to the other focus
  3. 3 Aimed at one focus, turned towards the other
  4. 4 Two families that cross at right angles
  5. 5 One sign decides which curve
  6. +3 more
8 essays · geometry The line spiralling over the circle. A circle with a helix drawn above it: the helix is the real line, and the map that sends each of its points straight down onto the circle covers the circle once per turn. Above one marked point sits a column of points, one per turn.

Covering spaces

  1. 1 The same loop, unrolled
  2. 2 The subgroup that is freer than the group
  3. 3 The symmetries a cover has of its own
  4. 4 A covering is a permutation
  5. 5 Folding a graph until it decides
  6. +3 more
8 essays · topology The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.

Diagonalisation

  1. 1 The row that is not on the list
  2. 2 A list that cannot contain itself
  3. 3 The sentence that says it has no proof
  4. 4 The word that cannot describe itself
  5. 5 The program that prints itself
  6. +3 more
8 essays · logic Two polytopes, two optima, one number. The feasible regions of a linear program and of its dual, side by side, each with its optimal vertex, and a number line on which the gap between the two optima closes to nothing.

Duality

  1. 1 Two numbers that have to meet
  2. 2 What a constraint is worth
  3. 3 When one of the two numbers is missing
  4. 4 The lines the optimum lies under
  5. 5 Prices at every corner
  6. +3 more
8 essays · applied Waiting for all 6 kinds. One bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.

Expectation

  1. 1 How long until every one turns up
  2. 2 The one that hardly ever comes up
  3. 3 Two patterns, one chance, different waits
  4. 4 A coin that lets the first player win
  5. 5 Three patterns in a circle
  6. +3 more
8 essays · probability One cake, one halving cut at 4/9, and two measures of it. A cake as a bar with two step valuations above and below it, the cutter's halving cut marked, and a table of both people's exact value of each piece.

Fair division

  1. 1 One cuts and the other chooses
  2. 2 Three people and a trimmed piece
  3. 3 Envy-free, up to one item
  4. 4 How many cuts a fair share costs
  5. 5 The product that makes a division fair
  6. +3 more
8 essays · applied Necklaces of 5 beads in 2 colours. Every string of beads, grouped by the rotations that carry one onto another.

Fermats little theorem

  1. 1 Necklaces that prove a theorem
  2. 2 One residue whose powers are all of them
  3. 3 The exponent that is smaller than Euler's
  4. 4 An order that proves a prime
  5. 5 Two primes where Fermat holds twice
  6. +3 more
8 essays · number The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Finite geometry

  1. 1 Seven points, seven lines
  2. 2 A schedule where every pair meets once
  3. 3 More blocks than points
  4. 4 A plane in a list of numbers
  5. 5 A plane no field built
  6. +3 more
8 essays · computation A matching that covers all 5 of one side. A bipartite graph with every possible pairing drawn thin and one complete matching drawn thick, so that each vertex on the left is joined to a distinct vertex on the right.

Halls theorem

  1. 1 One bottleneck and nothing else
  2. 2 What the search has when it fails
  3. 3 The piece that cannot pair off
  4. 4 Enough partners in every finite group
  5. 5 Matching as they arrive
  6. +3 more
8 essays · discrete 3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

Homotopy

  1. 1 A loop that cannot be pulled tight
  2. 2 The group a space has at a point
  3. 3 Every cover is a subgroup
  4. 4 Cutting a space to find its group
  5. 5 Why the second group commutes
  6. +3 more
8 essays · topology Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

Irrationality

  1. 1 The square that cannot shrink
  2. 2 Which roots refuse to be fractions
  3. 3 A tail too small to be a whole number
  4. 4 An integral that cannot be a whole number
  5. 5 Approached too fast to be algebraic
  6. +3 more
8 essays · number the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

Iteration

  1. 1 The staircase that shows the whole orbit
  2. 2 How fast the staircase arrives
  3. 3 The same map in different coordinates
  4. 4 The histogram an orbit leaves
  5. 5 The orbit a computer draws
  6. +3 more
8 essays · dynamics A rule for moving between 3 states. 3 states drawn as circles with an arrow for every move the rule allows, labelled with its chance; a dashed loop is the chance of staying put.

Markov chains

  1. 1 The rule that forgets where it came from
  2. 2 The chain that stops
  3. 3 The chain that runs the same backwards
  4. 4 The time spent and the share held
  5. 5 Where the shares have nowhere to go
  6. +3 more
8 essays · probability A line, a point, and many parallels. A disc whose lines are arcs meeting the boundary at right angles, showing several lines through one point that never meet a given line.

Models

  1. 1 Two worlds that both obey the rules
  2. 2 An infinite tree has an infinite path
  3. 3 A number larger than every number
  4. 4 A countable field that passes for the line
  5. 5 Two lists that are one order
  6. +3 more
8 essays · logic 120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Monte Carlo

  1. 1 Getting pi by dropping needles on the floor
  2. 2 The error that does not care how many dimensions
  3. 3 Sampling where the answer lives
  4. 4 A walk that samples a distribution
  5. 5 Points too even to be random
  6. +3 more
8 essays · probability

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