Matpic
mathematics, in pictures
Ladder

Inversion — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    The map that trades circles for lines

    Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.

    rung 1 · geometry
  2. Eight circles touching three. Three given circles and the eight circles tangent to all of them, each labelled by which of the three it contains and which it lies outside.

    Eight circles touching three

    Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.

    rung 2 · geometry
  3. An Apollonian gasket, 125 circles in. The Apollonian gasket generated from four mutually tangent circles of curvature −1, 2, 2 and 3, drawn to 4 generations; every curvature in it is a whole number.

    Curvatures that stay whole

    Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.

    rung 3 · geometry
  4. Two inversions, and the number four points agree on. Four points, their images after one inversion and after a second in a different circle, with the cross-ratio computed at each stage; it is conjugated once and restored twice.

    The number four points agree on

    One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.

    rung 4 · geometry
  5. Nineteen circles, one per vertex. A circle packing of a triangulation with seven interior vertices and twelve on the boundary: two circles touch exactly when their vertices are joined, and the radii were solved for rather than chosen.

    Every flat graph is a pile of circles

    A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.

    rung 5 · geometry

All ladders