Matpic
mathematics, in pictures
Depth

Ladders — page 6

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.
A resolution refutation of four clauses on three variables. A derivation tree: the given clauses at the top, each later clause obtained by cancelling one variable between two clauses above it, ending in the empty clause.

Resolution

  1. 1 A proof with one rule
1 rung · logic The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

Roots of unity

  1. 1 The polygon an equation forces
1 rung · algebra Stereographic projection. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane.

Stereographic projection

  1. 1 A sphere is a plane plus one point
1 rung · topology 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
1 rung · 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
1 rung · dynamics The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Sums of two squares

  1. 1 Two squares, and a lattice
1 rung · number The four ways to glue a square's edges in pairs. Squares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.

Surface classification

  1. 1 Every surface is a sphere with handles
1 rung · topology 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
1 rung · analysis 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
1 rung · 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
1 rung · 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
1 rung · geometry

All essays