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Ladders — page 5

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 Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Finite geometry

  1. 1 Seven points, seven lines
  2. 2 A schedule where every pair meets once
2 rungs · computation 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
2 rungs · algebra 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
2 rungs · geometry Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

Harmonic series

  1. 1 A sum whose terms vanish and whose total does not
  2. 2 The same terms, in a different order, adding to whatever is asked
2 rungs · analysis the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

Knots

  1. 1 Three moves, and what they cannot undo
  2. 2 Colours that count more than three
2 rungs · topology A linear map redrawing the plane. The integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.

Linear maps

  1. 1 A matrix is a picture of what happens to the grid
  2. 2 What a map throws away
2 rungs · algebra A line, a point, and many parallels. A disc whose lines are arcs meeting the boundary at right angles, showing several lines through one point that never meet a given line.

Models

  1. 1 Two worlds that both obey the rules
  2. 2 An infinite tree has an infinite path
2 rungs · logic 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
2 rungs · discrete 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
2 rungs · dynamics Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

Sensitive dependence

  1. 1 A difference too small to draw
  2. 2 How fast two orbits part
2 rungs · dynamics The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

The core

  1. 1 A split nobody can walk away from
  2. 2 The objection nobody can make louder
2 rungs · applied 8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

The integral

  1. 1 Adding up rectangles until they stop being rectangles
  2. 2 Area is the undoing of slope
2 rungs · analysis Two factor trees of 360. The same number split two different ways, both ending in the same primes.

Unique factorisation

  1. 1 One way to factor, and no other
  2. 2 The shape of a number's divisors
2 rungs · number Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

Arc length

  1. 1 The staircase that is not the diagonal
1 rung · analysis 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
1 rung · logic When a shared birthday becomes likely. The chance that some pair in a group shares a birthday, against group size. It passes a half at 23 people, where the probability is 50.7%.

Birthday problem

  1. 1 Twenty-three people
1 rung · probability One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

Borsuk ulam

  1. 1 One line that halves them both
1 rung · topology The dihedral group of a 4-sided shape, drawn as a map. A Cayley graph: one dot per motion of the shape, with one arrow per generator, so that multiplying by a generator is following an arrow of that colour.

Cayley graph

  1. 1 The group drawn as a map
1 rung · algebra 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
1 rung · geometry A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

Circular functions

  1. 1 A sine wave is a circle seen from the side
1 rung · analysis

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