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Series — page 3

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.
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. +5 more
10 essays · analysis 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
  3. 3 Five weighings and the question is closed
  4. 4 An objection one player makes to another
  5. 5 The corners are the orders of arrival
  6. +5 more
10 essays · applied 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
  6. +5 more
10 essays · analysis 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
  2. 2 The angle that is really an area
  3. 3 When two circular motions come home
  4. 4 Three sines tie a knot
  5. 5 The sine of a figure eight
  6. +4 more
9 essays · analysis The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

Compass-only

  1. 1 The straightedge buys nothing
  2. 2 One circle, and a straightedge
  3. 3 The compass that will not open
  4. 4 The price of a construction
  5. 5 The centre a straightedge cannot find
  6. +4 more
9 essays · computation 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
  2. 2 The bound is the answer to a search
  3. 3 No single input can move it far
  4. 4 The median of many small averages
  5. 5 A sphere that is nearly all equator
  6. +4 more
9 essays · probability The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.

Diagonalisation

  1. 1 The row that is not on the list
  2. 2 A list that cannot contain itself
  3. 3 The sentence that says it has no proof
  4. 4 The word that cannot describe itself
  5. 5 The program that prints itself
  6. +4 more
9 essays · logic 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
  2. 2 One residue whose powers are all of them
  3. 3 The exponent that is smaller than Euler's
  4. 4 An order that proves a prime
  5. 5 Two primes where Fermat holds twice
  6. +4 more
9 essays · number 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. +4 more
9 essays · dynamics 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
  2. 2 A sum stopped early still says something
  3. 3 How many get their own hat
  4. 4 The cells a permutation must miss
  5. 5 A round table with no couple together
  6. +4 more
9 essays · probability Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

Irrationality

  1. 1 The square that cannot shrink
  2. 2 Which roots refuse to be fractions
  3. 3 A tail too small to be a whole number
  4. 4 An integral that cannot be a whole number
  5. 5 Approached too fast to be algebraic
  6. +4 more
9 essays · number 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
  2. 2 Half a circle against a wall
  3. 3 Nearly the most means nearly round
  4. 4 The least wall for equal rooms
  5. 5 Every chord slid to the middle
  6. +4 more
9 essays · geometry the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

Iteration

  1. 1 The staircase that shows the whole orbit
  2. 2 How fast the staircase arrives
  3. 3 The same map in different coordinates
  4. 4 The histogram an orbit leaves
  5. 5 The orbit a computer draws
  6. +4 more
9 essays · dynamics 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. +4 more
9 essays · analysis A Möbius band. A strip joined end to end after a half twist, so it has one side and one edge.

Orientability

  1. 1 The surface with one side, and what happens when it is cut
  2. 2 The bottle that needs a fourth dimension
  3. 3 A disc sewn to a Möbius band
  4. 4 Two sheets over a one-sided surface
  5. 5 Orientation is a sign
  6. +4 more
9 essays · topology 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 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

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