The circles that fill a three-sphere

The three-sphere is the set of points at distance one from the origin in four-dimensional space. Nobody sees it, and stereographic projection from one of its points is what makes it drawable: the projection is faithful everywhere except at that one point, and the image is ordinary three-dimensional space.

What the projection makes visible is the reason to bother. The three-sphere is filled by circles. Every point lies on exactly one; no two meet; and every two are linked, exactly once, however far apart they seem.

Where the circles come from #

Write a point of the three-sphere as a pair of complex numbers $(z, w)$ with $|z|^2 + |w|^2 = 1$. That is four real coordinates with one condition, which is what a three-sphere is.

Now multiply both by the same complex number of modulus one: $(z, w) \mapsto (\lambda z, \lambda w)$ with $|\lambda| = 1$. The condition survives, so the new point is still on the three-sphere, and as $\lambda$ runs round the unit circle the point traces out a circle.

The circle is genuinely a circle rather than something circle-shaped: the map from $\lambda$ to $(\lambda z, \lambda w)$ is injective whenever $(z,w)$ is not the origin, which on the sphere it never is, so distinct phases give distinct points and the fibre is a faithful copy of the unit circle.

That circle is a fibre, and the fibres fill the three-sphere because every point is on one. Two points are on the same fibre exactly when they differ by such a multiplication, which is an equivalence relation — so the fibres partition the sphere, and no two of them meet.

The set of fibres is itself a space: the pairs $(z, w)$ up to a common phase, which is the complex projective line, which is the ordinary two-sphere of the rungs below. So the construction is a map from the three-sphere onto the two-sphere whose fibres are circles, and it is called the Hopf fibration.

Why the linking is the surprise #

Circles filling a space without meeting is not by itself remarkable — a stack of parallel circles fills a solid torus. What makes this one interesting is that no two of its circles can be pulled apart.

Linking number one, for every pair. That is checked in the hero figure by counting signed crossings in the projection, which is the definition the linking-number ladder gives, applied to curves computed in four dimensions and then flattened.

The consequence is that the fibration is not a product. If the three-sphere were the two-sphere multiplied by a circle — a stack of circles arranged over the base with no twisting — then two fibres over nearby base points would be unlinked, and they are not. The twisting is not removable, and the linking number is the measurement that says so.

Counting the crossings #

The linking number is the one quantity the figure computes rather than displays, and it is worth saying how, because the method is elementary and the result is not.

Project the two curves onto a plane. At every place where one crosses the other, decide which is in front and whether the crossing turns left or right, and assign $+1$ or $-1$ accordingly. Add the signs and halve. That is the linking number, and the rung on linking establishes that the answer does not depend on the projection or on how the curves are drawn.

Here the curves are polygons of 240 segments each, so the count is a loop over 57,600 pairs of segments, each tested for a crossing in the plane and compared for depth. The sum comes out $\pm 2$ for every pair of fibres, so the linking number is one, ten times over for five fibres.

What makes that a computation rather than a display is that it could have come out otherwise. Two curves drawn to look linked and not actually linked would give zero, and a bug in the four-dimensional formula would give an inconsistent set of answers across the pairs. All ten agreeing at exactly one is a check on the construction as much as a measurement of it.

The closest-approach figure is the other half of the check. Signed crossings are only meaningful for curves that do not intersect, and two fibres that touched would make the count undefined — so the minimum distance between every pair of sampled points is measured and reported, at 0.38.

What the projection does to the picture #

Projected into ordinary space, the fibres sit on a family of nested tori.

Take the fibres over a circle of latitude on the base two-sphere. Their union is a torus, and the fibres wind round it once each way — the standard $(1,1)$ curve. Different latitudes give tori of different fatness, nested inside one another, and the two poles of the base give the two degenerate cases: one fibre becomes the central circle of the whole nest, and the other becomes a straight line through the middle, since it passes through the projection point.

The nesting is worth following once with the arithmetic. A latitude at polar angle $a$ on the base gives a torus whose fibres wind on it; the projected radius of that torus grows as $a$ approaches the projection point’s own latitude, and shrinks to the central circle at the opposite pole. So the family runs from a circle, through tori of every size, out to the line — and the line is the fibre over the point that was projected from.

Every point of space is on exactly one torus and one curve on it, and space is filled with no gaps and no crossings. The straight line and the central circle are linked once, like every other pair, which is the case easiest to see and the one that makes the rest believable.

The map, and what it is a map of #

There are three descriptions of the same construction and each is the natural one somewhere.

As complex numbers. The map sends $(z, w)$ to $z/w$, read as a point of the Riemann sphere — with $w = 0$ going to infinity, which is the point the second chart of the rung below exists for. Two pairs give the same ratio exactly when they differ by a common phase.

As quaternions. The three-sphere is the unit quaternions, and the map sends a unit quaternion $q$ to $q\,i\,\bar{q}$, which is a unit imaginary quaternion and so a point of an ordinary two-sphere. The fibres are the left cosets of the circle group generated by $i$, so the fibration is a group divided by a subgroup — which is the cleanest of the three descriptions and the one that says why the fibres are all alike: multiplication by a fixed quaternion carries any fibre to any other.

As rotations. The unit quaternions are a double cover of the rotations of space, and the map records where a rotation sends one fixed axis. Two rotations sending the axis to the same place differ by a rotation about that axis, which is a circle’s worth — the fibre.

The third description is the one that makes the fibration inevitable rather than clever. Rotations of space form a three-dimensional family; specifying where one axis goes uses two of those dimensions; and the one left over is the spin about that axis. The circle is the freedom that remains after an axis is chosen, and the fibration is that sentence written as a map.

Why it cannot be untwisted #

A fibration whose fibres are circles over a two-sphere is either the product — a plain stack — or it is not, and the invariant deciding it is exactly the linking number of two fibres.

For the product, take the two-sphere and cross it with a circle. Two fibres over distinct points are unlinked: one can be pushed away from the other without crossing, because they sit over different base points and nothing connects them. For the Hopf fibration, they cannot.

The number is called the Hopf invariant, and Hopf’s discovery in 1931 was that it is not zero — which was the first example anybody had of a map from a three-sphere to a two-sphere that cannot be deformed to a constant. Before it, the expectation was that maps from a higher sphere to a lower one were always trivial, because that is what happens for maps from a lower sphere to a higher one.

So this picture is a counterexample to a reasonable guess, and the whole subject of higher homotopy groups began with it. The second homotopy group’s commutativity is on a neighbouring ladder and belongs to the same reconsideration.

The guess it refuted deserves stating properly, because it was not foolish. A map from a circle into a two-sphere can always be contracted — a loop on a sphere slides off. A map from a two-sphere into a three-sphere can be contracted too, and in general a map from a lower sphere into a higher one always can, by a dimension count: the image misses a point and a sphere minus a point contracts. The reasonable extrapolation is that a map from a higher sphere into a lower one is even more likely to contract, since there is more room to move about in the source.

It is false, and the reason is that the source’s extra dimension is what allows the twisting. More room in the domain is not more room to deform, and the three-sphere’s circles being linked is what a map with nowhere to go looks like.

The base sphere is a sphere of directions #

The third description — a rotation, recorded by where it sends one axis — is the one that says what the base sphere is, and it is worth following because it converts the fibration into a statement about an everyday object.

A rotation of ordinary space is determined by three numbers. Ask only where it sends the $x$-axis and two numbers are used: the answer is a point of the two-sphere of directions. The third number is unspoken, and it is how much the rotation additionally spins about that new axis — a full circle’s worth of freedom.

So the fibration is the statement that a rotation is a direction plus a spin, and the twisting is the fact that the spin cannot be measured against any fixed reference that varies continuously with the direction. If it could, the fibration would be a product.

That last impossibility has a familiar cousin, and the relationship is close without being identity. A continuous choice of reference direction at every point of the two-sphere would be a non-vanishing tangent vector field, and a sphere cannot be combed flat. Both facts are obstructions of the same kind — an integer attached to a circle’s worth of freedom over a sphere, which vanishes exactly when the freedom can be labelled consistently — and the two integers are different: the Hopf fibration’s is one and the sphere’s tangent directions give two, which is the Euler characteristic.

Two circle bundles over the same sphere, twisted by different amounts, and neither untwistable. The Hopf fibration is the one twisted least, which is why it is the smallest example of the phenomenon and why Hopf found it first.

What the drawing is and is not #

The picture is a projection, so it is faithful about some things and not about others, and the division is worth stating.

Faithful: which curves are linked, and how many times. Linking is a topological property and the projection is a homeomorphism away from one point, so the counts read off the drawing are the counts in the sphere.

Faithful: which curves lie on a common torus, and the order of the nesting. Those are statements about which points are near which, and a homeomorphism preserves them.

Not faithful: the sizes. The tori appear to grow without bound as the latitude approaches the projection point, and on the three-sphere they do not — they are all the same size, in the sense that the fibration is symmetric under rotations carrying any point to any other. The nest of larger and larger tori is the projection’s distortion, exactly as the growing patches on the two-dimensional rung below are.

Not faithful either: the symmetry. On the three-sphere every fibre is equivalent to every other, by a rotation of the whole sphere; in the projection one of them is a straight line and the rest are circles of assorted sizes, so the picture has almost no symmetry left. Anything a reader concludes from a fibre looking special is a conclusion about the projection.

And a matter of taste: which fibre becomes the line. The projection point is a choice, and choosing a different one gives a picture in which a different fibre is the axis and the rest are rearranged. Nothing about the fibration prefers any of them.

What the pictures cannot show #

The fibres are computed in four dimensions and projected, so what is drawn is a faithful shadow rather than the object. The three-sphere itself is not drawn, cannot be, and every statement about it here is a statement about the shadow plus an argument that the shadow preserves it.

The linking numbers are counted from polygonal approximations to the curves, at 240 segments each. That is a genuine count of signed crossings, and it would fail if two segments crossed exactly, which is why the closest approach is measured and reported.

The tori are described and not drawn. Adding them would make the nesting obvious and would hide the curves, which are what the linking is counted from — so the figures draw the curves and leave the surfaces to the prose.

And the Hopf invariant is described and not computed. The linking of two fibres is what the figure counts, and the identification of that count with a homotopy invariant is a theorem no figure contains.

Where the ladder goes next #

This rung closes the ladder: the projection, the properties it preserves, the complex structure it produces, the atlas that makes it a manifold, and the object the atlas makes visible.

Named here as debts. The other Hopf fibrations, over the quaternions and the octonions, which fill a seven-sphere with three-spheres and a fifteen-sphere with seven-spheres and stop there for a reason worth its own rung. And the invariant computed as an integral, which is where the linking number and the homotopy class are identified.

Sideways, the two charts this is drawn in are the rung below, the base sphere is the Riemann sphere, the count of crossings is the linking number, and the quaternions the third description runs on are the ones with a lattice inside them.

What is worth carrying away #

When an object cannot be seen, the useful question is which of its properties survive a projection.

The three-sphere is out of reach and its fibration is not, because linking is topological and the projection is a homeomorphism away from one point. Everything the picture claims — no two fibres meet, every two are linked once — is of that kind, and everything it appears to claim about size is not.

The habit worth taking is to sort a picture’s content into what the map preserves and what it distorts. A projection is a tool with a specification, and reading a projected picture without the specification is how the tori’s growing sizes become a false belief about the sphere.

The corollary is about the value of a counterexample of this shape. Hopf’s map was the first thing anybody had that could not be deformed away, and the reason it was found is that somebody drew it. An object visible only through a projection is still visible, and a great deal of what is known about high-dimensional spheres started with a picture of circles on nested tori.