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Series — page 5

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 rotation number against the parameter at K = 1. The measured rotation number of a circle map plotted against its parameter, forming a staircase that is flat over an interval at each simple rational.

Mode locking

  1. 1 The staircase that is flat almost everywhere
  2. 2 How a lock comes apart
  3. 3 A rotation in different coordinates
  4. 4 A whole interval of speeds
  5. 5 The last circle to break
  6. +1 more
6 essays · dynamics An angle of 60° cut in three with one mark. A circle with a marked point on it, a straightedge laid through that point so the segment between the extended diameter and the circle equals the radius, and the third-angle it makes.

Neusis

  1. 1 The mark that changes what is reachable
  2. 2 Two instruments with one reach
  3. 3 A quintic a sliding mark reaches
  4. 4 A curve that divides any angle
  5. 5 The spiral that measures its own circle
  6. +1 more
6 essays · computation Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle.

Pascals triangle

  1. 1 Pascal's triangle, in two colours
  2. 2 The run that lands one place along
  3. 3 Every entry counts the routes to it
  4. 4 The carries decide the divisibility
  5. 5 A remainder read two digits at a time
  6. +1 more
6 essays · discrete 28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

Perfect numbers

  1. 1 Numbers that are their own parts
  2. 2 The sum of the parts, taken again
  3. 3 Why a quarter of numbers overshoot
  4. 4 Divisors that add to three times the number
  5. 5 A ratio nobody else has
  6. +1 more
6 essays · number The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

Period-doubling

  1. 1 The road paved with doublings
  2. 2 A constant that does not care which map
  3. 3 The dark lines are one point's orbit
  4. 4 The window that opens with a stutter
  5. 5 Every window is the whole diagram again
  6. +1 more
6 essays · dynamics 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 A tableau for ((p → q) ∧ (q → r)) → (p → r). A branching tree of formulas, each branch ending in a contradiction or in a description of a counterexample.

Proof systems

  1. 1 The tree that closes
  2. 2 The assumption a proof pays back
  3. 3 A lemma, and the proof that never mentions one
  4. 4 The instance that has to be guessed
  5. 5 Every derivation is a term
  6. +1 more
6 essays · logic 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 five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale.

Regular polyhedra

  1. 1 Why the list of perfect solids stops at five
  2. 2 Thirteen more when one word is dropped
  3. 3 The four that are allowed to cross themselves
  4. 4 Six in four dimensions, and three forever after
  5. 5 The five solids as three groups
  6. +1 more
6 essays · geometry 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 relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

Symmetry groups

  1. 1 Eight ways to leave a square alone
  2. 2 Colourings nobody can tell apart
  3. 3 The blocks a subgroup cuts out
  4. 4 Twenty-four ways to set a cube down
  5. 5 Necklaces made of symmetries
  6. +1 more
6 essays · algebra 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 A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

Circle area

  1. 1 A circle unrolled into a triangle
  2. 2 Pinned between two sequences
  3. 3 The slice that has to match
  4. 4 A band of a sphere is a band of its cylinder
  5. 5 The ball that is largest in five dimensions
5 essays · geometry 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 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

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