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Series — page 3

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 triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Scissors congruence

  1. 1 Equal area is enough, and equal volume is not
  2. 2 A dissection that never comes apart
  3. 3 Slid, but never turned
  4. 4 Finitely many, and nobody says how many
  5. 5 The obstruction that was the only one
  6. +1 more
6 essays · computation The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants.

Stern brocot

  1. 1 Every fraction, exactly once
  2. 2 Two matrices that generate the tree
  3. 3 Every rational in one sequence
  4. 4 The fractions that beat every smaller one
  5. 5 The arcs a line crosses on its way to a number
  6. +1 more
6 essays · number Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

Taylor series

  1. 1 One point's worth of information
  2. 2 An error with an unknown in it
  3. 3 The centre is a choice
  4. 4 The points that ruin the fit
  5. 5 A denominator that reaches past the radius
  6. +1 more
6 essays · analysis A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

Voting rules

  1. 1 The majority that goes in a circle
  2. 2 Five rules and five winners
  3. 3 Four conditions, and no rule that has all of them
  4. 4 A lie that pays
  5. 5 How often the majority goes in a circle
  6. +1 more
6 essays · applied Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

Arc length

  1. 1 The staircase that is not the diagonal
  2. 2 Which curves have a length at all
  3. 3 The length belongs to the journey
  4. 4 The length the derivative never sees
  5. 5 A length counted by the lines that cross it
5 essays · analysis 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
5 essays · applied 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
5 essays · probability A closed interval and an open one, matched point for point. Two number lines, one closed and one open, with arrows showing the countable sequence of points that has to move.

Cardinality

  1. 1 Two injections make a bijection
  2. 2 The arithmetic that loses subtraction
  3. 3 A line with as many points as a square
  4. 4 Countable, and everywhere
  5. 5 The size that cannot be pinned down
5 essays · logic Every triangulation of a 6-gon. All 14 ways of cutting a convex 6-gon into triangles with non-crossing diagonals — the 4th Catalan number, counted by drawing them.

Catalan numbers

  1. 1 One sequence, counting everything
  2. 2 Counting the paths that go wrong
  3. 3 One word, and four objects
  4. 4 The equation a sequence satisfies
  5. 5 The solid whose corners are triangulations
5 essays · discrete The dihedral group of a 4-sided shape, drawn as a map. A Cayley graph: one dot per motion of the shape, with one arrow per generator, so that multiplying by a generator is following an arrow of that colour.

Cayley graph

  1. 1 The group drawn as a map
  2. 2 How fast the ball fills
  3. 3 The edge that is as big as the ball
  4. 4 What is left when the middle is taken out
  5. 5 The polygon a lattice becomes from far away
5 essays · algebra The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

Collatz

  1. 1 The question nobody can answer
  2. 2 Almost every number comes down
  3. 3 The heuristic that cannot be a proof
  4. 4 How short a cycle could be
  5. 5 Every pattern happens exactly once
5 essays · dynamics five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

Concentration

  1. 1 How far from the average a thing can be
  2. 2 The bound is the answer to a search
  3. 3 No single input can move it far
  4. 4 The median of many small averages
  5. 5 A sphere that is nearly all equator
5 essays · probability The de Bruijn graph on 2 letters and words of 3, and the cycle through it. A graph whose vertices are short words and whose arrows are words one letter longer, with a closed walk using every arrow exactly once marked, and the cyclic sequence it spells.

De bruijn

  1. 1 Every word once, around a cycle
  2. 2 Every necklace, in order
  3. 3 A memory of four bits
  4. 4 A page that knows where it is
  5. 5 A cycle for every pair
5 essays · computation The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

Determinant

  1. 1 The number that says how much room is left
  2. 2 The only function that behaves like a volume
  3. 3 One point in every big enough shape
  4. 4 A determinant that counts trees
  5. 5 The same sum without its minus signs
5 essays · algebra 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
5 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
5 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
5 essays · number All 24 arrangements of 4 objects, and the 9 that move every one. Every permutation of 4 objects drawn as a grid of cells, with the diagonal — where an object stays where it began — shaded, and the arrangements that avoid it entirely marked.

Inclusion exclusion

  1. 1 Nobody gets their own hat
  2. 2 A sum stopped early still says something
  3. 3 How many get their own hat
  4. 4 The cells a permutation must miss
  5. 5 A round table with no couple together
5 essays · probability Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

Infinitude of primes

  1. 1 There is no last prime
  2. 2 Infinitely many of one kind
  3. 3 Which infinitudes are proved
  4. 4 Every class, and in equal shares
  5. 5 The sieve that cannot finish
5 essays · number Inversion in a circle of radius 1. Three points and their images under inversion in a circle: each image lies on the same ray from the centre, at the distance whose product with the original is the squared radius. Beside it, the tangent construction that finds the image with compass and straightedge.

Inversion

  1. 1 The map that trades circles for lines
  2. 2 Eight circles touching three
  3. 3 Curvatures that stay whole
  4. 4 The number four points agree on
  5. 5 Every flat graph is a pile of circles
5 essays · geometry

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