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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.
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
  6. +1 more
6 essays · geometry 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
  6. +1 more
6 essays · geometry A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

Voting rules

  1. 1 The majority that goes in a circle
  2. 2 Five rules and five winners
  3. 3 Four conditions, and no rule that has all of them
  4. 4 A lie that pays
  5. 5 How often the majority goes in a circle
  6. +1 more
6 essays · applied 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 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 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
5 essays · logic Every triangulation of a 6-gon. All 14 ways of cutting a convex 6-gon into triangles with non-crossing diagonals — the 4th Catalan number, counted by drawing them.

Catalan numbers

  1. 1 One sequence, counting everything
  2. 2 Counting the paths that go wrong
  3. 3 One word, and four objects
  4. 4 The equation a sequence satisfies
  5. 5 The solid whose corners are triangulations
5 essays · discrete 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
5 essays · 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
  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
5 essays · probability The de Bruijn graph on 2 letters and words of 3, and the cycle through it. A graph whose vertices are short words and whose arrows are words one letter longer, with a closed walk using every arrow exactly once marked, and the cyclic sequence it spells.

De bruijn

  1. 1 Every word once, around a cycle
  2. 2 Every necklace, in order
  3. 3 A memory of four bits
  4. 4 A page that knows where it is
  5. 5 A cycle for every pair
5 essays · computation 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
  2. 2 The one that hardly ever comes up
  3. 3 Two patterns, one chance, different waits
  4. 4 A coin that lets the first player win
  5. 5 Three patterns in a circle
5 essays · probability 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
5 essays · 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
  2. 2 Seven powers in a space of six
  3. 3 The integers a field contains
  4. 4 How a polynomial breaks modulo the primes
  5. 5 Units that form a lattice
5 essays · algebra 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
  2. 2 The series everything else is measured against
  3. 3 The repair at the boundary
  4. 4 A geometric series whose ratio is a matrix
  5. 5 A series that converges to minus one
5 essays · 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
  2. 2 What the search has when it fails
  3. 3 The piece that cannot pair off
  4. 4 Enough partners in every finite group
  5. 5 Matching as they arrive
5 essays · 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
  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
5 essays · probability Inversion in a circle of radius 1. Three points and their images under inversion in a circle: each image lies on the same ray from the centre, at the distance whose product with the original is the squared radius. Beside it, the tangent construction that finds the image with compass and straightedge.

Inversion

  1. 1 The map that trades circles for lines
  2. 2 Eight circles touching three
  3. 3 Curvatures that stay whole
  4. 4 The number four points agree on
  5. 5 Every flat graph is a pile of circles
5 essays · 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
  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
5 essays · geometry A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Jordan curve

  1. 1 Which side of the line is inside
  2. 2 A curve that has area
  3. 3 Every loop is a circle in disguise
  4. 4 Two pieces, in every dimension
  5. 5 A ball whose outside is not one
5 essays · topology the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linking number

  1. 1 Two loops and one number
  2. 2 Zero can mean two different things
  3. 3 Linked, and no two of them are
  4. 4 A whole number split into two that are not
  5. 5 Six points in space and a pair that must link
5 essays · topology

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