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Series — page 7

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 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 Two graphs that will not lie flat, and one that will. K4, K5 and K3,3 in the best straight-line drawings a search could find. K4 has no crossings; the other two have one each, and Euler's formula shows that none can have none.

Planarity

  1. 1 Two graphs that will not lie flat
  2. 2 Five spokes squeezed into K5
  3. 3 The few points that cut a flat graph
  4. 4 The crossings a graph cannot avoid
  5. 5 Every point at the average of its neighbours
5 essays · discrete The widest layer of the subsets of a set of 4. A Hasse diagram of a small order with the widest layer marked, and the largest set of mutually incomparable elements found by examining every subset.

Posets

  1. 1 The widest layer and the longest chain
  2. 2 Everybody's share of the chains
  3. 3 The cube cut into chains
  4. 4 The largest family that always meets
  5. 5 How many ways to sort it
5 essays · discrete Counting a 5 by 3 rectangle two ways. Lattice points in a rectangle cut by a diagonal of slope q over p, coloured by which side they fall.

Quadratic reciprocity

  1. 1 Counting one rectangle, twice
  2. 2 The two supplements, and where the eight comes from
  3. 3 The symbol is the sign of a shuffle
  4. 4 One sum, squared two ways
  5. 5 Which primes a form takes
5 essays · number A resolution refutation of four clauses on three variables. A derivation tree: the given clauses at the top, each later clause obtained by cancelling one variable between two clauses above it, ending in the empty clause.

Resolution

  1. 1 A proof with one rule
  2. 2 Two literals make an arrow
  3. 3 A failed search is a proof
  4. 4 A contradiction that is only a sum
  5. 5 Two terms made equal, and no more
5 essays · logic One point on the sphere for every point of the plane. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane.

Stereographic projection

  1. 1 A sphere is a plane plus one point
  2. 2 Angles survive and areas do not
  3. 3 The sphere that complex numbers live on
  4. 4 One chart is never enough
  5. 5 The circles that fill a three-sphere
5 essays · topology Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Complex numbers

  1. 1 Multiplying is turning
  2. 2 The shape in every picture of itself
  3. 3 One c, one picture
  4. 4 The area of the Mandelbrot set, from two sides
4 essays · algebra A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Constant width

  1. 1 Round is not the only way to be the same width
  2. 2 The shape described from outside
  3. 3 The least area a width can hold
  4. 4 The same question in space
4 essays · geometry Königsberg as a graph. The four landmasses as circles and the seven bridges as edges; every circle has an odd number of edges.

Eulerian paths

  1. 1 Seven bridges, and the invention of throwing things away
  2. 2 A walk that splices in its own detours
  3. 3 The streets a postman walks twice
  4. 4 Every edge on exactly two cycles
4 essays · discrete The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly.

Golden ratio

  1. 1 The rectangle that eats itself
  2. 2 Three gaps and no more
  3. 3 The diagonal no unit measures
  4. 4 Tiles that never repeat
4 essays · geometry Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Modular arithmetic

  1. 1 Numbers that wrap
  2. 2 Two dials at once
  3. 3 Multiplying every number on the dial at once
  4. 4 A root lifted one digit at a time
4 essays · discrete Trisect every angle, and an equilateral triangle appears. A triangle with angles 78°, 54°, 48°, its six angle trisectors, and the triangle whose corners are where the trisectors nearest each side meet. That inner triangle is equilateral, which is Morley's theorem.

Morley

  1. 1 Three trisectors and a triangle nobody expected
  2. 2 Seven pieces and an equilateral middle
  3. 3 Eighteen equilateral triangles
  4. 4 The trisectors on a sphere almost agree
4 essays · geometry A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Pick theorem

  1. 1 Area by counting dots
  2. 2 The theorem that has no version in space
  3. 3 Sixteen polygons with one dot inside
  4. 4 The dots a circle catches
4 essays · discrete 13 into 12. 13 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1.

Pigeonhole

  1. 1 More things than boxes
  2. 2 How close a fraction can get
  3. 3 The orbit that must come back
  4. 4 A sum that forbids half the pairings
4 essays · discrete

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