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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 5 essays across 2 fields — 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 A turn of the sphere, seen from the plane. A square grid in the plane and its image under the map obtained by lifting to the sphere, rotating by 62° about a tilted axis, and coming back down. The lines become arcs of circles and the crossings stay at right angles.

The sphere that complex numbers live on

Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.

topology · Stereographic projection The two-handled surface's octagon, eight to a corner. Regular hyperbolic octagon with angles π/4 and 64 neighbouring copies in the Poincaré disc.

An octagon only the hyperbolic plane can hold

Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.

topology · Surface classification An astrolabe is a stereographic projection in brass. Latitude 52°: 16 engraved curves, worst departure from a circle 8.88e-15; altitude-circle formulas within 2.2e-16; Cancer radius 0.6564, Capricorn 1.5235 of the equator's.

An instrument that computes with circles

An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.

topology · Stereographic projection

Named alongside it

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

Fixed pointComplex numbersConformalConformal mapInversionProjectionRotationSphereStereographic projectionAstrolabeCircle packingCircle-preserving

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