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Series — page 4

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
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
  2. 2 The sum of the parts, taken again
  3. 3 Why a quarter of numbers overshoot
  4. 4 Divisors that add to three times the number
  5. 5 A ratio nobody else has
  6. +4 more
9 essays · number 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
  4. 4 Every power sum, from the coefficients alone
  5. 5 The roots of the slope stay inside
  6. +4 more
9 essays · algebra 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
  5. 5 Every derivation is a term
  6. +4 more
9 essays · logic 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
  5. 5 The arcs a line crosses on its way to a number
  6. +4 more
9 essays · number 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
  6. +4 more
9 essays · dynamics 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
  2. 2 A matrix that counts the returns
  3. 3 The folds that measure chaos
  4. 4 Counting in a base that is not a whole number
  5. 5 Where a base-β orbit spends its time
  6. +4 more
9 essays · 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
  6. +4 more
9 essays · analysis 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
  3. 3 An endless region with a finite area
  4. 4 A rectangle cut by a curve
  5. 5 An area measured by walking round it
  6. +4 more
9 essays · analysis 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
  2. 2 One circle touching four
  3. 3 A centre is three weights
  4. 4 Every point has a partner across the bisectors
  5. 5 A triangle that fits once fits everywhere
  6. +4 more
9 essays · geometry Hamilton's method on 27 seats and 5 regions. A worksheet of populations, exact quotas, floors, remainders and the seats Hamilton's method awards to 5 regions.

Apportionment

  1. 1 The seat that vanishes when the house grows
  2. 2 Five rules and one dial
  3. 3 The rule with no favourites
  4. 4 Two out of three, and never all three
  5. 5 Choosing what unfair means
  6. +3 more
8 essays · applied A closed interval and an open one, matched point for point. Two number lines, one closed and one open, with arrows showing the countable sequence of points that has to move.

Cardinality

  1. 1 Two injections make a bijection
  2. 2 The arithmetic that loses subtraction
  3. 3 A line with as many points as a square
  4. 4 Countable, and everywhere
  5. 5 The size that cannot be pinned down
  6. +3 more
8 essays · logic 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
  2. 2 How fast the ball fills
  3. 3 The edge that is as big as the ball
  4. 4 What is left when the middle is taken out
  5. 5 The polygon a lattice becomes from far away
  6. +3 more
8 essays · algebra 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
  3. 3 One thing in each region is enough
  4. 4 The conclusion is what survives the erasing
  5. 5 A walk that beats trying everything
  6. +3 more
8 essays · logic The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

Collatz

  1. 1 The question nobody can answer
  2. 2 Almost every number comes down
  3. 3 The heuristic that cannot be a proof
  4. 4 How short a cycle could be
  5. 5 Every pattern happens exactly once
  6. +3 more
8 essays · dynamics 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
  2. 2 Where two roots run into each other
  3. 3 One number under every bell
  4. 4 The signs no completion can change
  5. 5 Regression is a square completed halfway
  6. +3 more
8 essays · algebra 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
  4. 4 The area of the Mandelbrot set, from two sides
  5. 5 The fraction written on every bulb
  6. +3 more
8 essays · algebra The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

Determinant

  1. 1 The number that says how much room is left
  2. 2 The only function that behaves like a volume
  3. 3 One point in every big enough shape
  4. 4 A determinant that counts trees
  5. 5 The same sum without its minus signs
  6. +3 more
8 essays · algebra 3 rounds on chains of 4 and 5. Two chains of dots with pebbles placed in turn, and the transcript of a play: Spoiler picks an element of one chain, Duplicator answers in the other, and the pebbles must keep the same order.

Ehrenfeucht–Fraïssé games

  1. 1 A game that decides what can be said
  2. 2 Nearly always, or nearly never
  3. 3 The distance a sentence can see
  4. 4 The game the algorithm was playing
  5. 5 A language that can name a set
  6. +3 more
8 essays · logic The directions the map leaves alone. Unit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.

Eigenvectors

  1. 1 The directions a map leaves alone
  2. 2 The polynomial whose roots are the stretches
  3. 3 The same map in a better basis
  4. 4 Symmetry forces a right angle
  5. 5 What a map does to a circle
  6. +3 more
8 essays · algebra 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
  3. 3 The player who meets the first long run
  4. 4 The remainders that count the roots
  5. 5 What the signs allow
  6. +3 more
8 essays · geometry

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