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Concept

Mobius transformation

A map of the plane, extended by one point at infinity, given by a quotient of two linear expressions in a complex variable. Such maps carry circles and lines to circles and lines, preserve angles, and are exactly the compositions of an even number of inversions in circles.

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

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.

Conformal mapFixed pointInversionCircle packingComplex numbersCross ratioGraphGroupInvariantIterationOrientationPlanarity

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