Every Two Circles Linked

A Circle Over Every Point

The points with x² + y² = 1 make a circle in the plane. The points with a² + b² + c² + d² = 1 make the three-sphere S³, which lives in four dimensions. Write the four numbers as two complex numbers z₁, z₂, and the condition is |z₁|² + |z₂|² = 1.

The map h(z₁, z₂) = (|z₁|² − |z₂|², 2 z₁ z̄₂) gives three real numbers whose squares add up to exactly 1, so h sends every point of S³ to a point of the ordinary sphere S².

Multiply z₁ and z₂ both by e^(iθ): neither length changes, z₁ z̄₂ stays the same, and so does h. As θ goes round once it draws a circle in S³. What lands on each point of S² is exactly one such circle, the fibre over that point. Fibres over different points never meet.

The porcelain globe on the left is S², the red pin a point P on it. The red thread on the right is the fibre over P. A steel bead runs round it, and its image on S² stays at P.

Down to Three Dimensions

S³ cannot be drawn in three-dimensional space as it is; it has to be projected. Stereographic projection starts from one point N of S³ and sends every other point along a straight line into three-dimensional space. N itself goes off to infinity.

One dimension down it is easier to see: projected from its top point, a circle becomes a line; projected from its top, a sphere becomes a plane. The projection turns circles into circles; only circles through N become straight lines.

In S³ every fibre is a great circle, and after projection it is still a circle. The one through N becomes a straight line: the blued-steel rod in the middle of the model, over the globe’s top point. The fibre over the bottom point is the unit circle round the rod.

Call P’s height on the globe p₁: −1 at the bottom, 1 at the top. As P climbs, the red circle’s radius grows as √(2/(1 − p₁)), and at the top it is a straight line.

Linked in Pairs

The red circle and the blue one never touch, yet they are linked: they cannot be pulled apart without cutting one. Every two fibres are linked like this, exactly once.

How many times can be computed. The Gauss linking number is a double integral along the two curves, and it always comes out a whole number. The readout is its value for the red and blue circles on the model, computed live.

The line in the middle is a fibre too, so every circle is also threaded on the rod.

Latitudes Become Tori

Take a circle of latitude on S² and put together the fibres over all its points: they make a torus. The nearer the latitude is to the top, the larger the torus; the nearer the bottom, the thinner it gets, closing in on the unit circle in the middle.

Its circles are Villarceau circles: cut a torus with a slanted plane that touches it in two places, and the cut is two such circles. With A = √((1 + p₁)/2) and B = √((1 − p₁)/2), the torus has major radius 1/B and tube radius A/B, and each fibre’s radius is 1/B as well.

The three tori belong to the three latitudes inlaid in the globe, in copper, brass and nickel silver, at p₁ = −0.8, −0.42 and 0.1.

Filling Space

Latitudes cover S², so the tori, one inside the next, fill three-dimensional space. Every point of space lies on exactly one fibre: on one of the circles, or on the line in the middle.

Up close, S³ looks like a small patch of S² times a circle. As a whole it is not S² × S¹: in S² × S¹ a loop once round the circle cannot be shrunk to a point, while in S³ every loop can. The circles are put together with a twist, and their linking in pairs is what that twist looks like.

Rotations and Qubits

The four numbers can also be written as one quaternion, r = a + bi + cj + dk. The unit quaternions make up exactly S³, and each stands for a rotation of three-dimensional space.

Rotate a pointer by r, and its tip lands on a point of S²: that is h(r). Turn the pointer further about itself and its tip stays put, but the rotation is a different one. All the rotations that bring the pointer to the same point form one fibre.

The small vane on the pin stands for the turn about the pointer. One full turn of the vane moves the bead only halfway round the fibre; it takes two turns to come back. r and −r are the same rotation, so S³ counts every rotation twice.

A qubit’s state is a pair of complex numbers (α, β) with |α|² + |β|² = 1: a point of S³. Multiplying both by e^(iθ) makes no difference to any measurement, so the states that can be told apart are the fibres, one for each point of S², and that S² is called the Bloch sphere.

It Cannot Be Shrunk

In 1931 Heinz Hopf published this map and proved that it cannot be deformed continuously into a constant map, one that sends all of S³ to a single point. Among maps from a sphere to a sphere of lower dimension, it was the first shown to be impossible to shrink.

His tool was the linking. Take two points of S²: the linking number of the fibres over them does not change under continuous deformation. For a constant map it is 0; for h it is 1. The number came to be called the Hopf invariant.

The same construction works twice more: quaternions give S⁷ → S⁴ with fibre S³, octonions give S¹⁵ → S⁸ with fibre S⁷. In 1960 J. F. Adams proved that, apart from the trivial S¹ → S¹, maps of Hopf invariant one exist only in these three cases.