the Riemann surface of the logarithm
/ REE-mahn /
The logarithm is the most extreme multivalued function: at every nonzero z it has infinitely many values, log z = ln|z| + i(arg z + 2 pi k) for every integer k, differing by integer multiples of 2 pi i. Its Riemann surface is the geometric object on which all of these values coexist single-valuedly. The picture is an infinite spiral ramp — like an endless multistorey car park spiralling around the central column at z = 0. Each full turn around the column takes you up to the next level, where log has gained another 2 pi i, and the ramp never closes back on itself: there are infinitely many sheets, one for each integer, stacked in an unending helix.
Build it by gluing. Take a copy of the plane slit along the negative real axis for each integer k; on sheet k the logarithm is the principal value plus 2 pi i k. Now glue the top edge of sheet k's slit to the bottom edge of sheet (k+1)'s slit, all the way up and down the integers. The result is a single connected ramp with no top and no bottom, joined only at the central point z = 0, which is the single branch point — of infinite order, since you must spiral forever to exhaust the sheets and you never return to your starting level. On this ramp, log is a perfectly ordinary single-valued holomorphic function: walking once around the central axis moves you to the next floor and increases log by exactly 2 pi i, with no jump or ambiguity anywhere.
This surface makes vivid why the principal logarithm needs a branch cut and why arg z is only defined up to 2 pi: choosing a branch means choosing ONE floor of the car park and refusing to walk off its edge, while the branch cut is the slit you agree not to cross. Topologically the logarithm's surface is simple — it is just the spiral ramp, which is contractible (it can be squashed to a point), and in fact it is the universal cover of the punctured plane, the simplest covering space from which every branch can be read off. So the wildest multivalued elementary function has, geometrically, one of the cleanest possible surfaces.
Stand at z = 1 on floor 0, where log z = 0. Walk counterclockwise once around the origin: you ascend smoothly to floor 1 and arrive back over z = 1 with log z = 2 pi i. Keep circling: floor 2 gives 4 pi i, floor -1 (clockwise) gives -2 pi i. You never come back to floor 0 by circling — the ramp has no repeats, which is exactly the logarithm's infinite multivaluedness made into stairs.
An endless spiral ramp around z = 0: each turn climbs one floor and adds 2 pi i to log. The ramp never closes.
The single branch point z = 0 has infinite order — unlike the square root's surface, the logarithm's ramp never closes back on itself, so there is no finite number of sheets. This surface is contractible and is the universal cover of the punctured plane.