Stereographic projection
Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.
Every triple, on one circle
Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.
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.
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.
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.
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.
Torus knots that fill the three-sphere
Hopf filled the three-sphere with circles, every two linked once. Let the circles turn at two different speeds instead — p turns one way while they make q the other — and the three-sphere is filled with (p, q) torus knots, trefoils when p and q are 2 and 3, every two of them linked exactly pq times. Two exceptional fibres remain plain circles; every knot links them q and p times, and they link each other once. Computed from the drawn curves for ten fibrations, every pair comes out at pq to six decimal places.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ProjectionConformalFibrationLinking numberMobius transformationRotationSphereTorusAngleAreaAstrolabeAtlas