Concept

Riemann sphere

The complex numbers with a single point at infinity adjoined, which stereographic projection turns into an ordinary sphere. On it a rational function is a map with a degree and no exceptions, and division by zero becomes an ordinary value.

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

One point on the sphere for every point of the plane. 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

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

Named alongside it

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

Projection Astrolabe Bijection Circle preserving Complex numbers Conformal Conformal map Continuity Fixed point Mobius transformation Point at infinity Rotation

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