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Ladders — page 7

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.
The basins of Newton's method on z³ = 1. The complex plane coloured by which cube root of one Newton's method converges to from each starting point.

Newton basins

  1. 1 Where Newton's method goes instead
1 rung · dynamics A set, its negation, and its double negation. Four bars on one number line showing an open set, its negation, their union, and the double negation.

Non classical logic

  1. 1 The middle that is not excluded
1 rung · 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
1 rung · geometry Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind.

Pell

  1. 1 One solution that makes all the others
1 rung · number 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
1 rung · number 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
1 rung · 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
1 rung · logic 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
1 rung · logic 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
1 rung · algebra The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Sums of two squares

  1. 1 Two squares, and a lattice
1 rung · number The four ways to glue a square's edges in pairs. Squares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.

Surface classification

  1. 1 Every surface is a sphere with handles
1 rung · topology 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
1 rung · dynamics Nine points of a triangle, on one circle. A triangle with the midpoints of its sides, the feet of its three altitudes and the midpoints from each corner to the orthocentre marked; all nine lie on a single circle of half the circumradius.

Triangle centres

  1. 1 Nine points on one circle
1 rung · geometry xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

Uniform convergence

  1. 1 A limit that forgets to be continuous
1 rung · analysis

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