Why these two functions, and what we are about to fix
The previous guide gave us the grand idea: when a function refuses to be single-valued, stop blaming the function and rebuild its domain. Glue together enough copies of the plane that the function has exactly one value over each point of the new, larger ground, and the multivaluedness simply evaporates. That idea is gorgeous in the abstract, but it only truly clicks once you have built a surface or two with your hands. So this closing guide does the two canonical constructions in full, the two everyone learns first: the Riemann surface of the square root and of the logarithm.
These two are the right place to start because their failure to be single-valued is honest and visible, not pathological. The square root z^(1/2) takes two values at each nonzero z, because two numbers square to the same thing: if w^2 = z then (-w)^2 = z too. The logarithm log z = ln|z| + i(arg z + 2 pi k) takes infinitely many values, one for each integer k, because the exponential is periodic with period 2 pi i. In both cases the multivaluedness is not a mistake — it is faithful bookkeeping of a many-to-one map being undone. Our job is to give that bookkeeping a geometry.
The square root: two sheets, one twist
Start with the square root, because its surface is small enough to picture completely. The trouble concentrates at a single point. The origin is a branch point: a small loop around z = 0 cannot be made without the square root's value flipping sign when you return, because going once around the origin adds 2 pi to arg z, and the square root halves the argument, so it adds pi — and e^(i pi) = -1. The point at infinity is the only other branch point. Everywhere else the two values w and -w sit comfortably apart and you can follow either one smoothly.
Here is the construction. Take two copies of the complex plane — call them sheet A and sheet B — and on each one slit open a branch cut from 0 out to infinity, say along the negative real axis. On sheet A we carry the branch of the square root whose values have argument between -pi/2 and pi/2 (the 'right-hand' values); on sheet B we carry the other branch, the negatives of those. Each slit plane is now an honest single-valued home for its own branch. The magic is in how we sew them: cross the cut on sheet A and you do not loop back onto sheet A — you step onto sheet B, and crossing the cut on sheet B steps you back onto A.
TWO-SHEET SURFACE OF w = z^(1/2)
sheet A (branch with arg w in (-pi/2, pi/2))
sheet B (branch with arg w in ( pi/2, 3pi/2))
cut along negative real axis on each sheet
cross the cut on A --> arrive on B
cross the cut on B --> arrive on A
one full loop around z = 0:
A --(cross cut)--> B --(cross cut)--> A
value: w --> -w --> w
TWO turns of the plane = ONE turn of the surface
branch points: z = 0 and z = infinity (where the two sheets pinch together)Walk it once to feel why this works. Start on sheet A near z = 1 holding w = +1. Circle the origin once: you cross the cut, land on sheet B, and your value has continuously slid to -1, exactly as monodromy promised. But you are no longer where you started — you are on B. Circle once more: you cross the cut again, return to sheet A, and your value slides back to +1. The path that used to change the value now takes two full turns to close, and over every point of this two-sheeted surface the square root has exactly one value. The function is single-valued at last; it simply needed a domain twice as large.
What that surface really looks like
The honest picture has a subtlety worth confronting. If you try to glue the two slit planes in flat three-dimensional space, the second seam has to pass through the first, and you draw an apparent self-intersection along the cut. That crossing is an artifact of the drawing, not a feature of the surface — exactly like the apparent self-crossing when you flatten a Klein bottle onto paper. Abstractly the surface is perfectly smooth, with no edge and no self-intersection: the two sheets join cleanly at the cut, which stops being a wall and becomes an ordinary interior seam you can stroll across.
What is the global shape? Over each ordinary point of the plane there are exactly two sheets, and at the two branch points (0 and infinity) the sheets pinch together into a single point — these are the sheets and branch points of the construction. If you compactify by including the point at infinity and zip everything up, the resulting closed surface is, topologically, a sphere. Two sheets, two pinch points, sphere: that is the entire square-root surface. It is the simplest interesting Riemann surface there is, and its genus — its number of handles — is zero.
The logarithm: a ramp that never closes
Now the logarithm, where the same construction goes to infinity. Recall log z = ln|z| + i(arg z + 2 pi k): the real part ln|z| is single-valued and harmless, but the imaginary part is the argument, and the argument is only defined up to adding multiples of 2 pi. Encircle the origin once and arg z increases by exactly 2 pi, so the logarithm increases by 2 pi i — and it never comes back. There is no n for which n loops restore the original value, the way two loops did for the square root. The logarithm has infinitely many branches, indexed by every integer k, and they march off forever.
So instead of two sheets we need a sheet for every integer k, stacked one above another and glued in an endless staircase. Take a slit plane for each k, lay them in a tower, and glue the top edge of each cut to the bottom edge of the cut on the sheet above. Crossing the cut now always carries you up to the next sheet, k to k+1, and you can keep climbing — or descending — forever without ever returning to where you began. This is the famous Riemann surface of the logarithm, and the picture everyone reaches for is a spiral parking ramp, or an infinite helical staircase winding around the central axis at z = 0.
On this ramp the logarithm becomes beautifully single-valued and even simple. Height on the ramp tracks the argument: as you spiral once around, you rise to the next floor and the imaginary part of the logarithm goes up by 2 pi i — the floors are the 2 pi i jumps, made into honest geometry. Radial distance from the axis tracks the modulus, with the real part ln|z| running from minus infinity near the axis out toward plus infinity. The origin itself is missing — it is a branch point that the ramp winds around but never touches, which is why there is a single central axis and no ground floor.
The deeper truth: these are covers of the punctured plane
There is a unifying idea sitting underneath both pictures, and it pays to name it. Each surface comes with a natural downward map: stand at any point on the square-root surface or the logarithm ramp, and you can always read off the underlying z it sits over (just forget which sheet you are on). This makes each surface a covering space of the punctured plane — the plane with the origin removed, since the branch point is the one place where the sheets misbehave. The square-root surface is a two-fold cover; the logarithm ramp is an infinite-fold cover. 'Sheets' is exactly the informal word for the layers of a covering.
The logarithm's surface is not just a cover, it is the most spread-out one possible: the universal cover of the punctured plane. It is the covering that is itself simply connected — every loop on the ramp can be shrunk to a point, because the troublesome loop around the origin has been unrolled into the spiral that never closes. That is the cleanest statement of what 'taming the multivaluedness' means: you keep unrolling the loops that cause trouble until none are left, and the logarithm, which had infinitely many values downstairs, has exactly one value at each point of its own universal cover upstairs.
Step back and the whole rung lands in one frame. Continuation along a path can change a branch; the monodromy theorem says it depends only on the homotopy class of the path; a loop around a branch point is the homotopy class that bites; and the Riemann surface is the new domain on which all those branches become the single values of one honest, single-valued holomorphic function. The square root and the logarithm are the two pictures to keep forever in your head — one finite and spherical, one infinite and spiral — because almost every more elaborate Riemann surface is built by gluing copies of exactly these two moves.
One honest closing caveat, so the picture does not overclaim. Drawing these surfaces in three dimensions forces apparent self-crossings and edges that the true, abstract surface does not have — the picture is a faithful guide to the topology, not a literal embedding. And the grand classification of all simply connected Riemann surfaces — that each is conformally the disk, the plane, or the sphere — is the uniformization theorem, a deep result we only name here. The square root and the logarithm are not the end of the story; they are the two clean doorways through which the whole theory of Riemann surfaces is most gently entered.