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Concept

Cross ratio

A number computed from four points, as a ratio of two ratios of differences between them. It is unchanged by every map that carries circles to circles and preserves angles, and it is real exactly when the four points lie on one circle or one line.

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

The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

The straightedge buys nothing

Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.

computation · compass only 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

Named alongside it

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

InversionCompassComplex numbersConformal mapConstructible numberConstructionFixed pointGroupIncidenceInvariantMidpointMobius transformation

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