Matpic
mathematics, in pictures
Concept

Conformal map

A map that keeps the angle at which two curves cross, while changing lengths and areas freely. Any hypothesis about tangency or perpendicularity survives one, which is what lets a configuration be moved to an easier position and moved back.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

A sphere is a plane plus one point

Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.

topology · stereographic projection 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.

geometry · inversion 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.

geometry · inversion 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.

geometry · inversion 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.

geometry · inversion

Named alongside it

The objects these essays reach for when they reach for this one.

InversionCircleCircle preservingConstructionFixed pointMobius transformationTangencyApollonius problemAstrolabeBijectionCircle packingComplex numbers

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