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Ladders — page 3

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.
Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit.

Random walk

  1. 1 A walk that always comes home, until it does not
  2. 2 The path folded at its first touch
  3. 3 Half the time is the rarest answer
  4. 4 Two barriers and a fair game
  5. 5 The walk that becomes a curve
5 rungs · probability The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale.

Regular polyhedra

  1. 1 Why the list of perfect solids stops at five
  2. 2 Thirteen more when one word is dropped
  3. 3 The four that are allowed to cross themselves
  4. 4 Six in four dimensions, and three forever after
  5. 5 The five solids as three groups
5 rungs · geometry Every order of arrival for three partners, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

Shapley value

  1. 1 The order everybody arrives in
  2. 2 None of the four conditions is spare
  3. 3 A share of the votes is not a share of the power
  4. 4 Too many orders to list
  5. 5 Sharing a cost that is not the sum of its parts
5 rungs · applied 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 rungs · topology The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates.

Strange attractor

  1. 1 Two lobes and no cycle
  2. 2 The flow that is really a map
  3. 3 A closer start buys only time
  4. 4 Stretch, fold, and what is left
  5. 5 Neither a surface nor a solid
5 rungs · dynamics Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

The derivative

  1. 1 The slope of a single point
  2. 2 A curve with a corner at every point
  3. 3 A rectangle grown on two sides
  4. 4 The flat map that fits closest
  5. 5 The slope of the mirror image
5 rungs · analysis eˣ and its tangent lines. The exponential curve with tangent lines at several points; at each point the slope equals the height.

The exponential

  1. 1 The curve that is its own slope
  2. 2 The area that names the number
  3. 3 The equation with only one answer
  4. 4 The exponential of a square
  5. 5 The constant that counts what does not happen
5 rungs · analysis The plane divided by nearest neighbour. 10 sites, and every point of the rectangle shaded by which site is closest to it. The boundaries are the places where two sites tie.

Voronoi

  1. 1 The plane, divided by whoever is nearest
  2. 2 One dimension up, and the circles disappear
  3. 3 Every site in the middle of its own cell
  4. 4 When the sites are not the same size
  5. 5 The tree inside the triangulation
5 rungs · geometry 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 rungs · geometry V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.

Euler characteristic

  1. 1 Every corner pays for itself
  2. 2 Two trees, and every edge in exactly one of them
  3. 3 Seven hundred and twenty degrees of gap
  4. 4 The solid where the answer is not two
4 rungs · topology Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

Fourier series

  1. 1 A square wave built entirely out of round ones
  2. 2 Where the coefficients come from
  3. 3 When the period grows without bound
  4. 4 The corners go first
4 rungs · analysis the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

Fractal dimension

  1. 1 A dimension that is not a whole number
  2. 2 Infinite on one side and nought on the other
  3. 3 A carpet with two dimensions
  4. 4 A dimension from the stretching rates
4 rungs · dynamics A wheel of 5 rim regions needs 4 colours. A hub touching 5 rim regions arranged in a ring. The rim is odd, so the whole map needs 4 colours and no fewer.

Graph colouring

  1. 1 Four colours, and a proof nobody can read
  2. 2 Five colours, and a chain that can be followed
  3. 3 Counting the colourings
  4. 4 Seven regions on a doughnut
4 rungs · discrete The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000.

Inner product

  1. 1 The dot product is a shadow
  2. 2 The square that cannot be negative
  3. 3 One subtraction clears a direction
  4. 4 The nearest point of a flat thing
4 rungs · algebra 3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

Judgement aggregation

  1. 1 The court that contradicts itself
  2. 2 No rule escapes the doctrinal paradox
  3. 3 Deciding the premises or the conclusion
  4. 4 Four ways out, and what each costs
4 rungs · applied 5,040 orders, 7 thresholds, one best rule. For each number of candidates passed over, the share of the 5,040 possible arrival orders in which the rule ends up with the best of the 7. The count is exhaustive.

Optimal stopping

  1. 1 When to stop looking
  2. 2 When the numbers are shown
  3. 3 Giving up on the best
  4. 4 Half of what an oracle takes
4 rungs · probability Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle.

Pascals triangle

  1. 1 Pascal's triangle, in two colours
  2. 2 The run that lands one place along
  3. 3 Every entry counts the routes to it
  4. 4 The carries decide the divisibility
4 rungs · discrete The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

Permutation parity

  1. 1 The crossings that will not come out even
  2. 2 The only bit that survives
  3. 3 The puzzle that is exactly half solvable
  4. 4 When the label may be a matrix
4 rungs · algebra 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 rungs · discrete A tableau for ((p → q) ∧ (q → r)) → (p → r). A branching tree of formulas, each branch ending in a contradiction or in a description of a counterexample.

Proof systems

  1. 1 The tree that closes
  2. 2 The assumption a proof pays back
  3. 3 A lemma, and the proof that never mentions one
  4. 4 The instance that has to be guessed
4 rungs · logic

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