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Ladders — page 4

A field says what an essay is about. A ladder says what else there is to say about it — the distinct arguments that stand against one idea, from the one 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
4 rungs · 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
4 rungs · 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
4 rungs · 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
4 rungs · applied The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

Compass-only

  1. 1 The straightedge buys nothing
  2. 2 One circle, and a straightedge
  3. 3 The compass that will not open
3 rungs · computation 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
3 rungs · algebra 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
3 rungs · logic 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
3 rungs · applied the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linking number

  1. 1 Two loops and one number
  2. 2 Zero can mean two different things
  3. 3 Linked, and no two of them are
3 rungs · topology 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
3 rungs · discrete The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

Polynomial roots

  1. 1 A loop that cannot miss the middle
  2. 2 What the coefficients already know
  3. 3 A shared root, found without finding it
3 rungs · algebra One matching that is not stable, and all 24 counted by blocking pairs. An unstable matching with its blocking pair ringed and both members' rankings marked, above an exhaustive census of every matching of the instance by how many blocking pairs it has.

Stable matching

  1. 1 Nobody has a reason to run away
  2. 2 The side that proposes wins
  3. 3 No stable rule is safe from a lie
3 rungs · applied 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
3 rungs · algebra (p ∨ q) ∧ ¬r, drawn on the cube of 8 assignments. The assignments as corners of a cube, joined when they differ in one variable, with the satisfying corners filled.

Truth functions

  1. 1 A formula is a corner of a cube
  2. 2 One connective is enough
  3. 3 The map that puts neighbours side by side
3 rungs · logic Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.

Bayes

  1. 1 Bayes' theorem is a picture of a square
  2. 2 The door that was not opened
2 rungs · probability Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above it.

Cellular automata

  1. 1 Eight rules and a triangle
  2. 2 The rule that computes
2 rungs · dynamics 4 circles, and the 14 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

Class diagrams

  1. 1 Four circles cannot do it
  2. 2 Twenty-four out of two hundred and fifty-six
2 rungs · 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
2 rungs · applied Euclid's algorithm on a 34 by 13 rectangle. The rectangle is tiled by peeling off the largest square that fits, again and again, until nothing is left.

Euclidean algorithm

  1. 1 The oldest algorithm, drawn as a tiling
  2. 2 A fraction that never closes
2 rungs · geometry The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

Finite fields

  1. 1 The field with four elements
  2. 2 Every element is a power of one of them
2 rungs · computation

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