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Series — page 6

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.
The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

Permutation parity

  1. 1 The crossings that will not come out even
  2. 2 The only bit that survives
  3. 3 The puzzle that is exactly half solvable
  4. 4 When the label may be a matrix
  5. 5 Thirty-one moves from solved
  6. +1 more
6 essays · algebra Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

Prime distribution

  1. 1 The primes on a spiral, and a pattern nobody ordered
  2. 2 The primes are what is left over
  3. 3 Counting what has no formula
  4. 4 Always one before the double
  5. 5 The sieve written as a product
  6. +1 more
6 essays · discrete j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Quantifiers

  1. 1 Every row, or one column
  2. 2 Six sentences from two quantifiers
  3. 3 A quantifier is a shadow
  4. 4 Arithmetic with addition alone
  5. 5 A machine that carries one bit
  6. +1 more
6 essays · logic The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

Quaternions

  1. 1 A multiplication that remembers the order
  2. 2 A rotation of four-space takes two of them
  3. 3 The identity that multiplies sums of squares
  4. 4 The integers among the quaternions
  5. 5 What is lost at eight
  6. +1 more
6 essays · algebra The chance of being connected, against the chance of an edge. Curves of the exact probability that a random graph on three to six labelled points is connected, plotted against the probability of each individual edge.

Random graphs

  1. 1 The moment everything joins up
  2. 2 The moment a giant appears
  3. 3 Two thresholds, not one
  4. 4 Finding a threshold with two moments
  5. 5 The window where the giant is born
  6. +1 more
6 essays · probability The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

Roots of unity

  1. 1 The polygon an equation forces
  2. 2 Every third coefficient
  3. 3 On the circle and never home
  4. 4 The sums of roots of unity that add to nothing
  5. 5 The sums that obey a smaller equation
  6. +1 more
6 essays · algebra The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order.

Symbolic dynamics

  1. 1 The orbit written as a word
  2. 2 A matrix that counts the returns
  3. 3 The folds that measure chaos
  4. 4 Counting in a base that is not a whole number
  5. 5 Where a base-β orbit spends its time
  6. +1 more
6 essays · dynamics Every way of choosing one thing from each of 4 pairs. A table with one row per choice function on a small family of pairs, each row giving what it takes from each pair, with the row a stated rule names picked out.

Axiom of choice

  1. 1 The choice nobody can write down
  2. 2 A function that adds and is nowhere a line
  3. 3 Infinitely many guessers, finitely many wrong
  4. 4 The choice inside a countable union
  5. 5 A spanning tree for every graph
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 Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square

  1. 1 Completing the square, by completing a square
  2. 2 Where two roots run into each other
  3. 3 One number under every bell
  4. 4 The signs no completion can change
  5. 5 Regression is a square completed halfway
5 essays · algebra The most triangle-free edges on 6 points. A graph on 6 points carrying 9 edges and no triangle, found by examining every graph on those points, with the two sides its edges cross between drawn apart.

Extremal graphs

  1. 1 The edge that forces a triangle
  2. 2 Moves that only ever add edges
  3. 3 The densest graph without a square
  4. 4 The fewest triangles an edge density allows
  5. 5 As many points as two steps allow
5 essays · discrete The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

Field extensions

  1. 1 A tower whose degrees multiply
  2. 2 Seven powers in a space of six
  3. 3 The integers a field contains
  4. 4 How a polynomial breaks modulo the primes
  5. 5 Units that form a lattice
5 essays · algebra Odd numbers as square shells. Nested L-shaped shells of 1, 3, 5 … 11 cells stack into a 6 by 6 square.

Figurate numbers

  1. 1 Every square is a stack of odd numbers
  2. 2 Sums of powers, read off a staircase
  3. 3 Three triangular numbers, and no fewer
  4. 4 Six numbers that need five pentagons
  5. 5 Fifteen numbers decide every number
5 essays · geometry The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

Galois correspondence

  1. 1 The lattice that runs the other way
  2. 2 The group that will not come apart
  3. 3 Three ways to pair four roots
  4. 4 The quintics that have a formula
  5. 5 Three real roots and no real radicals
5 essays · algebra A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

Geometric series

  1. 1 The sum that fits in one square
  2. 2 The series everything else is measured against
  3. 3 The repair at the boundary
  4. 4 A geometric series whose ratio is a matrix
  5. 5 A series that converges to minus one
5 essays · analysis 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 A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Jordan curve

  1. 1 Which side of the line is inside
  2. 2 A curve that has area
  3. 3 Every loop is a circle in disguise
  4. 4 Two pieces, in every dimension
  5. 5 A ball whose outside is not one
5 essays · topology A largest flow of 5 through two wide ends and a narrow middle. A network with a capacity on every road, the amount a largest flow sends along each, and the cut whose capacity equals that flow's value drawn as a line separating the places.

Network flow

  1. 1 The bottleneck is the whole story
  2. 2 When several pairs share the roads
  3. 3 One tree for every cut
  4. 4 Three places cut apart
  5. 5 The cheapest way to send
5 essays · discrete The Goodstein sequence from 4, with the ordinal beside each term. A table of the Goodstein sequence with each term's hereditary representation and the ordinal obtained by replacing the base with omega.

Ordinals

  1. 1 A sequence that explodes and still stops
  2. 2 One step in front of infinitely many
  3. 3 Every ordinal in base omega
  4. 4 Reached from below, or not at all
  5. 5 An ordinal as a growth rate
5 essays · logic The nine-point grid, and the lines they force. 9 points with all 20 of their connecting lines drawn. The 12 carrying exactly two points are drawn solid and the rest faintly; the count is computed from the coordinates rather than read off the drawing.

Ordinary lines

  1. 1 The line with only two points on it
  2. 2 Three ordinary lines from a count
  3. 3 The fewest ordinary lines a polygon allows
  4. 4 At least as many lines as points
  5. 5 Rows of three, planted on a cubic
5 essays · geometry

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