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Series — page 2

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.
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
  5. 5 A dimension for every rate of crowding
  6. +1 more
6 essays · dynamics 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
  5. 5 Moving a map across a product
  6. +1 more
6 essays · 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
  5. 5 The nearest consistent verdict
  6. +1 more
6 essays · applied Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders.

Latin squares

  1. 1 The thirty-six officers
  2. 2 A field's worth of squares
  3. 3 The plane hiding in the squares
  4. 4 Nine thousand four hundred and eight
  5. 5 Sixteen of five hundred and seventy-six
  6. +1 more
6 essays · computation A rule for moving between 3 states. 3 states drawn as circles with an arrow for every move the rule allows, labelled with its chance; a dashed loop is the chance of staying put.

Markov chains

  1. 1 The rule that forgets where it came from
  2. 2 The chain that stops
  3. 3 The chain that runs the same backwards
  4. 4 The time spent and the share held
  5. 5 Where the shares have nowhere to go
  6. +1 more
6 essays · probability Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Measure

  1. 1 Almost none of it left, and still uncountably many
  2. 2 No interval in it, and length to spare
  3. 3 Covering a set from outside
  4. 4 Which functions can be added up
  5. 5 A set that has no size at all
  6. +1 more
6 essays · analysis Which axioms hold on which frames. A table of frames against modal axioms, each cell decided by checking the axiom under every valuation.

Modal logic

  1. 1 The axiom is the shape of the graph
  2. 2 Two diagrams the language cannot tell apart
  3. 3 Worlds built out of sentences
  4. 4 The axiom with no property of the arrows
  5. 5 Necessity that means provable
  6. +1 more
6 essays · logic 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
  5. 5 The thresholds that nest
  6. +1 more
6 essays · 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
  5. 5 A remainder read two digits at a time
  6. +1 more
6 essays · 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
  5. 5 Every derivation is a term
  6. +1 more
6 essays · logic The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

Quaternions

  1. 1 A multiplication that remembers the order
  2. 2 A rotation of four-space takes two of them
  3. 3 The identity that multiplies sums of squares
  4. 4 The integers among the quaternions
  5. 5 What is lost at eight
  6. +1 more
6 essays · algebra The chance of being connected, against the chance of an edge. Curves of the exact probability that a random graph on three to six labelled points is connected, plotted against the probability of each individual edge.

Random graphs

  1. 1 The moment everything joins up
  2. 2 The moment a giant appears
  3. 3 Two thresholds, not one
  4. 4 Finding a threshold with two moments
  5. 5 The window where the giant is born
  6. +1 more
6 essays · probability A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Scissors congruence

  1. 1 Equal area is enough, and equal volume is not
  2. 2 A dissection that never comes apart
  3. 3 Slid, but never turned
  4. 4 Finitely many, and nobody says how many
  5. 5 The obstruction that was the only one
  6. +1 more
6 essays · computation 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. +1 more
6 essays · number Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

Taylor series

  1. 1 One point's worth of information
  2. 2 An error with an unknown in it
  3. 3 The centre is a choice
  4. 4 The points that ruin the fit
  5. 5 A denominator that reaches past the radius
  6. +1 more
6 essays · analysis 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
  2. 2 Which curves have a length at all
  3. 3 The length belongs to the journey
  4. 4 The length the derivative never sees
  5. 5 A length counted by the lines that cross it
5 essays · analysis A table of shares written as a lottery over 3 whole assignments. A doubly stochastic table of shares, and beneath it the permutation matrices and weights that add up to it exactly, each drawn as a grid with one marked cell per row.

Assignment

  1. 1 A lottery over whole assignments
  2. 2 The corners are whole assignments
  3. 3 A price for every person and task
  4. 4 One table, two lotteries
  5. 5 Where the corners stop being whole
5 essays · applied A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

Billiards

  1. 1 A bounce is a fold of the table
  2. 2 A table folded into a surface
  3. 3 A room that cannot be lit
  4. 4 The obstacle that makes a table chaotic
  5. 5 The triangle nobody can settle
5 essays · dynamics 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
  2. 2 Any unevenness brings the match sooner
  3. 3 A collision that finds a factor
  4. 4 Fourteen people within a day
  5. 5 A room where nobody is alone
5 essays · 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
  2. 2 Two opposite points that agree twice
  3. 3 As many cuts as colours
  4. 4 Opposite labels that have to meet
  5. 5 The colours a circle forces
5 essays · topology

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