Matpic
mathematics, in pictures
Depth

Ladders

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.
The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Pythagoras

  1. 1 Two squares, four triangles, and no algebra
  2. 2 Euclid proves it without moving anything
  3. 3 Every triple, on one circle
  4. 4 The rope that squares a corner
  5. 5 Two right angles and the diagonal of a box
  6. +3 more
8 rungs · geometry A Galton board after 600 balls. 600 balls fall through 12 rows of pegs, each bouncing left or right at random, and pile up in a bell-shaped heap.

Central limit

  1. 1 A bell curve assembled out of coin flips
  2. 2 The average settles and the wobble does not
  3. 3 The shape that averaging leaves alone
  4. 4 An average that never settles
  5. 5 How fast the bell arrives
  6. +2 more
7 rungs · probability A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

Fixed points

  1. 1 Something always stays put
  2. 2 Nothing on a sphere can be combed flat
  3. 3 A point that pulls, and a point that pushes
  4. 4 Three colours force a triangle
  5. 5 A map that shrinks everything
  6. +2 more
7 rungs · topology 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. +1 more
6 rungs · applied Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

Constructible numbers

  1. 1 What two points can build
  2. 2 Every step is a square root
  3. 3 The cube that will not double
  4. 4 The angle that will not divide by three
  5. 5 Which polygons can be drawn
  6. +1 more
6 rungs · computation A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

Convexity

  1. 1 The curve of the average, and the average of the curve
  2. 2 A line under every point
  3. 3 The function seen from its tangents
  4. 4 Where the guarantee stops
  5. 5 Three points, however many there are
  6. +1 more
6 rungs · analysis A code on the 3-cube, and the balls around its words. The corners of a hypercube with the chosen codewords marked and the words within one error of each shaded.

Error-correcting codes

  1. 1 Distance is a picture
  2. 2 Sixteen spheres that fill a cube
  3. 3 Finding the error without reading the message
  4. 4 A polynomial through the gaps
  5. 5 The best a code can be
  6. +1 more
6 rungs · 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 rungs · 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 rungs · 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 rungs · 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 rungs · 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 rungs · probability 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 rungs · 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 rungs · dynamics 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 rungs · 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 rungs · discrete 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
5 rungs · dynamics Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

Conic sections

  1. 1 One cone, four curves
  2. 2 Every ray comes back to the other focus
  3. 3 Aimed at one focus, turned towards the other
  4. 4 Two families that cross at right angles
  5. 5 One sign decides which curve
5 rungs · geometry The line spiralling over the circle. A circle with a helix drawn above it: the helix is the real line, and the map that sends each of its points straight down onto the circle covers the circle once per turn. Above one marked point sits a column of points, one per turn.

Covering spaces

  1. 1 The same loop, unrolled
  2. 2 The subgroup that is freer than the group
  3. 3 The symmetries a cover has of its own
  4. 4 A covering is a permutation
  5. 5 Folding a graph until it decides
5 rungs · topology 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 rungs · computation

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