Riemann Sphere to Torus

Revolve a circle of radius r about an axis lying at distance R from the circle’s centre. Four shapes come out:

R = 0

a sphere, traced twice, because the circle spins about its own diameter

R < r

a spindle torus: the circle crosses the axis, giving an apple outside and a lemon inside

R = r

a horn torus: the circle just touches the axis, and the hole has closed to a point

R > r

a ring torus, the ordinary doughnut

Your poles touch exactly when R = r, so the moment of contact is a horn torus. On the way there the outer skin is an apple. Keep the whole revolving circle and the shape is a spindle torus, which carries a second sphere inside it: the lemon, joined to the apple at 0 and ∞. The lemon shrinks into the touching point at the instant of contact. Switch on Spindle lemon to see it.

On the Riemann sphere, +∞, −∞, +i∞ and −i∞ are already one point, the north pole. So bringing the poles together glues exactly two points, 0 and ∞. Mathematicians know the result well. It is called the nodal curve, or pinched torus, and it appears in three guises:

• An equation. It is the nodal cubic y² = x²(x + 1). Lines through the node parametrise it by a sphere, and the two points of that sphere that land on the node can be placed at 0 and ∞.
• A group. Its smooth points form ℂ* = ℂ ∖ {0} under multiplication. The node ties together the two ends of multiplication, the infinitely small and the infinitely large.
• A limit. It is what any torus becomes when one of its circles pinches to a point. Among all tori it is the single extra point at the boundary: the cusp j = ∞ of the Deligne–Mumford compactification.

Why 0 and ∞, rather than any other pair? The map z ↦ 1/z swaps them, with the unit circle |z| = 1 as the mirror between them. In logarithmic coordinates w = log z they are the two ends of one infinite cylinder, and gluing them closes the direction of scale, log |z|, into a loop.

The animation shows one more thing. While the poles approach, the geometry changes smoothly but the conformal shape does not change at all: the apple is still the Riemann sphere, and the chart stays put. The topology changes all at once, at contact.

Close to the node the surface looks like two discs touching at a point: one around 0 with coordinate u = z, and one around ∞ with coordinate v = 1/z. The node is the equation uv = 0. Opening it replaces this with uv = q for a small number q, so the two discs are joined by a thin neck. This is called smoothing the node, or plumbing. Done to the whole sphere, it gives the Tate torus

E_q = ℂ* / ⟨ z ↦ qz ⟩

Take the annulus |q|^½ ≤ |z| ≤ |q|^−½ and glue its inner circle, which came from 0, to its outer circle, which came from ∞, using z ↦ z/q. The glued circle is the violet throat. It is called the vanishing cycle, because it vanishes back into the node as q → 0. Each point of the throat is a direction: arg z = α heading into 0, glued to arg z = α heading out to ∞. A complex q would add a twist of arg q to that gluing, which a torus of revolution cannot show.

The number q, called the nome, measures how far the node has opened. Every modular form has a q-expansion f = Σ a_n qⁿ, which is a series centred on exactly this pinched torus.

A torus of revolution is conformally a flat rectangle, and one number fixes its shape:

τ = i · r / √(R² − r²)q = e^2πiτ = exp(−2πr / √(R² − r²))

The surface metric (R + r cos v)² du² + r² dv² equals (R + r cos v)² (du² + dσ²) with dσ = r dv / (R + r cos v), and one trip around the tube gives ∮ dσ = 2πr / √(R² − r²). The grid on the model is drawn in exactly these coordinates, u = arg z and σ = log |z|, so its cells are conformal squares at every stage.

• R/r → 1: τ → i∞ and q → 0. The horn torus is the pinched limit.
• R/r = √2: τ = i, the square torus. The 12 × 12 grid closes exactly. This is also the Willmore torus, the stereographic image of the Clifford torus.
• R/r → ∞: τ → 0. A thin ring pinches the other circle, the one around the tube.

So the family runs from one pinch, through the most balanced torus, toward the other pinch.

On the sphere only the latitudes, |z| = constant, close up. The meridians run from 0 to ∞ and stop. A sphere has no room for two independent loops, since every loop on it can be shrunk to a point. Gluing 0 to ∞ closes the meridians, and the torus is the first surface with two independent circles:

• Phase, arg z, runs around the hole. The four quadrants become four slices of the ring.
• Scale, log |z|, runs through the hole: from the throat, where 0 ≡ ∞, out to the unit circle on the outer rim and back again.

That is why a mapping built on the quadrants of the plane and a mapping built on the axis from 0 to ∞ cannot both be circles on the sphere. On the torus they are its two independent cycles.

Each further hole is the same move: choose two more points, glue them, and open the node. A surface with g holes is a sphere with g pairs of discs removed and their rims glued by Möbius maps, which together form a Schottky group. For the torus the group is just z ↦ qz, and the set its orbits pile up on, the limit set, is exactly {0, ∞}. From two holes on, the limit set becomes a fractal dust, found inside circles nested within circles, as pictured in Mumford, Series and Wright’s Indra’s Pearls.