Matpic
mathematics, in pictures
Concept

Fibration

A way of filling a space with copies of one smaller space, such as circles, so that every point lies on exactly one copy and nearby copies sit side by side. Hopf filled the three-sphere with circles this way, every two of them linked once.

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

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.

topology · Stereographic projection The three-sphere filled with trefoils. Seven fibres of the (2,3) Seifert fibration of S³, stereographically projected: five trefoils and two exceptional circles; pairwise linking numbers 6, 3, 2 and 1 verified.

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.

topology · Linking number

Named alongside it

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

Linking numberStereographic projectionTorusHopf fibrationKnotQuaternionsWinding number

All concepts