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Ladder

Stereographic projection — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · topology
  2. A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line.

    Angles survive and areas do not

    Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.

    rung 2 · topology
  3. 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.

    rung 3 · topology
  4. Two charts on one sphere, meeting by the reciprocal. A sphere with its two polar caps marked, each the part missed by one of the two stereographic charts, and the band where both charts are defined shaded between them.

    One chart is never enough

    Stereographic projection matches the sphere minus a point with the whole plane, and the missing point is not a blemish to be tidied away. It is a theorem — no single flat picture covers a sphere — and the repair is two pictures with a rule for passing between them.

    rung 4 · topology
  5. 5 circles filling a three-sphere, every pair linked once. Several closed curves in space, nested on tori of different sizes, each pair passing through the other exactly once and none of them touching.

    The circles that fill a three-sphere

    A three-sphere is filled by circles — one through every point, no two meeting, every two linked exactly once. Stereographic projection is the only way anybody sees it, and the projected picture is a nest of circles on tori whose linking can be counted off the drawing.

    rung 5 · topology

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