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Ladders — page 6

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.
Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square

  1. 1 Completing the square, by completing a square
1 rung · algebra five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

Concentration

  1. 1 How far from the average a thing can be
1 rung · probability 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
1 rung · discrete 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
1 rung · 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
1 rung · discrete Necklaces of 5 beads in 2 colours. Every string of beads, grouped by the rotations that carry one onto another.

Fermats little theorem

  1. 1 Necklaces that prove a theorem
1 rung · number 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
1 rung · algebra Odd numbers as square shells. Nested L-shaped shells of 1, 3, 5 … 11 cells stack into a 6 by 6 square.

Figurate numbers

  1. 1 Every square is a stack of odd numbers
1 rung · geometry A product of 3 polynomials, and what its coefficients count. The coefficients of a product of small polynomials, with the combinations of choices that reach one marked total written out beneath it.

Generating functions

  1. 1 A polynomial that counts
1 rung · discrete A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

Geometric series

  1. 1 The sum that fits in one square
1 rung · analysis A matching that covers all 5 of one side. A bipartite graph with every possible pairing drawn thin and one complete matching drawn thick, so that each vertex on the left is joined to a distinct vertex on the right.

Halls theorem

  1. 1 One bottleneck and nothing else
1 rung · discrete A closed walk on the 3-cube changing one place at a time. The corners of a 3-dimensional cube with a path through every one of them exactly once, each step moving along an edge, and the last corner one step from the first.

Hamiltonian cycles

  1. 1 A walk that changes one thing at a time
1 rung · discrete All 24 arrangements of 4 objects, and the 9 that move every one. Every permutation of 4 objects drawn as a grid of cells, with the diagonal — where an object stays where it began — shaded, and the arrangements that avoid it entirely marked.

Inclusion exclusion

  1. 1 Nobody gets their own hat
1 rung · probability An angle standing on a chord. A circle with a fixed chord and a movable apex on the major arc. The angle at the apex is 60 degrees wherever the apex is put, and the angle the same chord subtends at the centre is 120 degrees.

Inscribed angle

  1. 1 An angle that does not care where it stands
1 rung · geometry One perimeter of 300, spent five ways. Regular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².

Isoperimetric

  1. 1 The most area a fence can hold
1 rung · geometry All 16 trees on 4 labelled points. Every tree on 4 labelled points, drawn one by one. There are 16 of them, which is 4 to the power 2.

Labelled trees

  1. 1 Sixteen trees on four points
1 rung · discrete 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
1 rung · dynamics 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
1 rung · geometry A largest flow of 5 through two wide ends and a narrow middle. A network with a capacity on every road, the amount a largest flow sends along each, and the cut whose capacity equals that flow's value drawn as a line separating the places.

Network flow

  1. 1 The bottleneck is the whole story
1 rung · discrete 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
1 rung · computation

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