A rotation of four-space takes two of them

A rotation of the plane has an angle. A rotation of space has an angle and an axis. The natural guess for four dimensions is an angle and something bigger to spin about, and the natural guess is wrong in an interesting way: a rotation of four-space has two angles and no axis whatever.

The quaternions describe this exactly. Conjugating by a unit quaternion rotates the three-dimensional space of pure quaternions; multiplying by unit quaternions on both sides rotates the whole four-dimensional space, and every rotation of four-space arises that way.

The map, and why it is a rotation #

Fix unit quaternions $p$ and $q$ and define $R(x) = pxq^{*}$, where the star is conjugation.

It is linear in $x$, because quaternion multiplication is. It preserves length, because the norm is multiplicative: $|pxq^{*}| = |p||x||q^{*}| = |x|$. A linear map of four-space preserving all lengths preserves all angles too, since the inner product is recoverable from lengths, so $R$ is orthogonal.

That leaves two possibilities, a rotation or a reflection, and the determinant decides. It is $+1$: the map depends continuously on $p$ and $q$, both of which live on a connected sphere, and at $p = q = 1$ the map is the identity with determinant $+1$. A continuous determinant taking values in $\{+1, -1\}$ on a connected set is constant.

So every such map is a rotation, and the argument used nothing but multiplicativity of the norm and connectedness of a sphere. That is a very small set of ingredients for a description of a six-dimensional group.

Every rotation, and exactly twice #

The harder half is that nothing is missed. Every rotation of four-space is $x \mapsto pxq^{*}$ for some pair.

The counting argument is the quickest way to believe it. Pairs $(p, q)$ of unit quaternions form a product of two three-spheres, which is a six-dimensional object; rotations of four-space form a group of dimension $\binom{4}{2} = 6$. The map from pairs to rotations is a group homomorphism between two six-dimensional connected groups, and its kernel turns out to be finite, so it is onto.

The kernel is small and worth knowing: $R$ is the identity exactly when $pxq^{*} = x$ for all $x$. Taking $x = 1$ gives $p = q$, and then $pxp^{*} = x$ for all $x$ means $p$ commutes with everything, which forces $p = \pm 1$. So the kernel is $\{(1,1), (-1,-1)\}$ and the map is two-to-one.

Negating both quaternions gives the same rotation; negating one gives a different one. The hero figure checks both halves of that, since the first is what makes the description consistent and the second is what makes it two-to-one rather than four.

The two angles, and where the axis went #

A rotation of three-space has an axis because $3$ is odd: the characteristic polynomial of an orthogonal matrix of odd size and determinant one must have $+1$ as a root, so some direction is fixed.

In four dimensions there is no such obligation. The eigenvalues of a rotation come in conjugate pairs on the unit circle, and four of them pair up as $e^{\pm i\theta_1}, e^{\pm i\theta_2}$ with no real eigenvalue required. Geometrically the space splits into two orthogonal planes, and the rotation spins each by its own angle.

The angles come straight from $p$ and $q$. Writing $p = \cos(\alpha/2) + \ldots$ and $q = \cos(\beta/2) + \ldots$, the two rotation angles are $(\alpha - \beta)/2$ and $(\alpha + \beta)/2$. The figure computes them that way and then again from the trace of the matrix and of its square, which is a genuinely independent route: the trace of a rotation is twice the sum of the cosines of its angles, and the trace of its square gives the doubled angles, so two numbers determine the pair.

Counting the dimensions, which is the whole surprise #

It is worth doing the dimension count slowly, because it is where the difference between three and four dimensions is arithmetic rather than intuition.

A rotation of $n$-space is fixed by choosing a plane to spin in and an angle, and more generally by a skew-symmetric matrix, of which there are $\binom{n}{2}$ independent entries. So the rotation group has dimension one in the plane, three in space, six in four-space, ten in five-space.

Now compare with the number of unit quaternions needed. One unit quaternion is a point of the three-sphere: three parameters, matching the three of space’s rotation group with the two-to-one map absorbing nothing. Two unit quaternions: six parameters, matching four-space exactly.

Six equals three plus three, and that is the coincidence. In three dimensions the count $3 = 3$ makes conjugation by one quaternion enough; in four the count $6 = 3 + 3$ makes a pair enough. In five dimensions the rotation group has dimension ten and there is no supply of tens lying around, which is the arithmetic behind the failure the last section describes.

The same coincidence has a name in the theory of Lie algebras: the algebra of four-dimensional rotations splits as a sum of two copies of the three-dimensional one, and it is the only rotation algebra that splits at all. Everything on this page is that one sentence made concrete.

Left and right are two separate rotations #

The clearest way to see the two planes is to take the two multiplications apart.

Left multiplication by $p$, with $q = 1$, is itself a rotation of four-space. So is right multiplication by $q^{*}$. And they commute — a left multiplication and a right multiplication always do, since $(px)q^{*} = p(xq^{*})$ is associativity and nothing else.

That commuting pair is unusual. In three dimensions two rotations about different axes never commute, and the failure is the whole reason the order of two turns matters. In four dimensions there are two whole three-dimensional families of rotations, each commuting with every member of the other, and the quaternions exhibit them as left and right multiplication.

Those families are what make four dimensions special. The rotation group of $n$-space is simple for every $n$ except $4$, and the exception is exactly this splitting into two commuting copies.

Isoclinic rotations, where every point moves the same amount #

When the two angles are equal the rotation is called isoclinic, and it is the case with no analogue at all in three dimensions.

Every point of the sphere is then moved through the same angle, whichever plane it started in. There is no equator that moves fastest and no pole that stays still: the motion is completely uniform over the sphere, which is impossible in odd dimensions where something always has to stay put.

Left multiplication by any unit quaternion is isoclinic, with the two angles equal in size; right multiplication is isoclinic the other way, with them equal and opposite. Every isoclinic rotation is one or the other, which is another way of saying that the two commuting families are exactly the two kinds of uniform motion the sphere admits.

The circles a uniform motion leaves behind #

A rotation’s orbits are the curves points travel along, and for an isoclinic rotation those orbits are the most interesting object in the picture.

Right multiplication by the unit complex numbers inside the quaternions moves each point round a circle, and the circles fill the three-sphere with none left over. There are as many of them as points of an ordinary two-sphere, and any two of them are linked — they pass through one another exactly once, like two rings of a chain, and cannot be pulled apart without breaking.

That is the Hopf fibration, and it is the first genuinely three-dimensional fact on this ladder. It says the three-sphere is not a product of a circle and a sphere: the circles are arranged over the sphere in a way that twists, and the linking is the measurement of the twist.

The figure computes each linking number from the drawn curves rather than quoting it, which is worth doing because linking is a property of the curves and not of the projection: the number does not change when the picture is deformed, so a number read off the projection is a number about the originals.

Why the fibration is the double cover in disguise #

There is a second reading of the same picture, and it explains where the sphere of circles comes from.

Sending a unit quaternion $q$ to the rotation it performs by conjugation is the two-to-one map onto rotations of three-space. Sending it instead to the image of $i$ under that rotation is a map onto the two-sphere, and its fibres are the circles above: two quaternions land on the same point exactly when they differ by right multiplication by a unit complex number.

So the circles are the sets of quaternions performing rotations that agree on where $i$ goes, and the sphere they sit over is the sphere of possible destinations. The double cover the first rung drew and the fibration here are two views of the same map, taken at different stages: one records the whole rotation, the other only what it does to one direction.

Two quaternions, and what each half does to the sphere #

Since the two multiplications commute, the general rotation can be understood by understanding each half, and the two halves treat the three-sphere differently in a way worth stating.

Left multiplication by $p$ moves every point of the three-sphere. So does right multiplication by $q$. Neither has a fixed point at all unless it is the identity, which is a sharp contrast with three dimensions where every rotation fixes two points of the two-sphere — the poles where its axis meets.

The reason is the group structure. Left multiplication by $p$ fixes $x$ exactly when $px = x$, which forces $p = 1$; there is nothing special about any point of the sphere from the point of view of multiplication, because the sphere is a group and translations of a group have no fixed points.

That is the deepest difference between the two-sphere and the three-sphere, and it is why the second is so much more tractable. The two-sphere is not a group — a theorem, and not an easy one — so its rotations must have poles. The three-sphere is a group, so it can move rigidly along itself, and the circles a uniform motion sweeps out are the orbits of that motion.

Written as matrices, and why nobody does #

Everything above can be done with four-by-four matrices, and it is instructive to see why the quaternion account is preferred.

Composing two rotations of four-space by matrices takes sixty-four multiplications. Composing them as pairs of quaternions takes two quaternion products, which is thirty-two multiplications, and the result is automatically a rotation with no re-orthogonalisation needed. The same advantage in three dimensions is the reason quaternions are used in graphics and in spacecraft attitude control, and it is sharper here.

More to the point, the pair $(p, q)$ is the decomposition into two commuting rotations, whereas the matrix hides it: extracting the two angles from a matrix takes an eigenvalue computation, and reading them off $p$ and $q$ takes an arccosine each. A better basis makes a map easy to read is the general lesson, and here the better basis is not a basis at all but a different algebra.

The finite rotation groups, and a solid nobody expected #

One consequence of the pair description is worth following, because it produces an object that has no three-dimensional analogue.

The finite subgroups of the unit quaternions are known and short: the cyclic groups, the binary dihedral groups, and three exceptional ones of orders $24$, $48$ and $120$ covering the rotation groups of the tetrahedron, the cube and the icosahedron. Each of them, acting on four-space by left multiplication, is a finite group of rotations — and the orbit of a point is a set of four-dimensional points arranged with that group’s symmetry.

Taking the group of order $24$ and the point $1$ gives twenty-four unit quaternions, and they turn out to be the vertices of a regular four-dimensional solid: the 24-cell, which has twenty-four octahedral faces and is its own dual. It exists in four dimensions and in no other, exactly as the pair description does, and it is not a coincidence that the two facts arrive together.

That solid is the subject of a later rung, where it arrives from arithmetic rather than from symmetry — the integers among the quaternions are exactly its vertices, and the two routes to the same twenty-four points are the kind of coincidence this ladder keeps producing.

What breaks in five dimensions #

The description stops here, and it is worth saying why rather than leaving the impression that quaternions describe rotations in general.

The argument needed a four-dimensional algebra with a multiplicative norm, so that multiplying preserved lengths. There are exactly four such algebras over the reals, in dimensions one, two, four and eight — which the first rung’s tower of doublings arrives at — so the trick is available in those dimensions and in no others.

Dimension eight is available, and the octonions do describe rotations of eight-space, in a way complicated by their failure to associate. Five, six and seven have no algebra to offer, and their rotation groups are described by other means entirely.

A construction that works in dimensions one, two, four and eight and nowhere else is a signature. Whenever that list appears the same four algebras are behind it, and this ladder’s last rung is about what the list is.

What the pictures cannot show #

Four dimensions are drawn by projection and the projection lies. The matrix figure shows sixteen numbers, which is honest and not a picture of anything moving. The Hopf circles are stereographic images of great circles in the three-sphere: the linking is faithful, and the sizes and shapes are not.

The two-to-one covering has no drawing. That $p$ and $-p$ give the same rotation is checked to the last decimal place in the generator and there is no figure of it, because a picture of two objects doing the same thing is a picture of one thing.

And an isoclinic rotation looks like a general one on paper. What distinguishes it is that every point moves through the same angle, and a still picture of a motion cannot show how far anything moved.

Where the ladder goes next #

The next rung takes the multiplicativity of the norm — used twice above without comment, to show the map preserves length — and reads it as a statement about integers. A product of two sums of four squares is a sum of four squares, and the four numbers are given by an explicit identity that is the quaternion product written out. That identity is what makes Lagrange’s four-square theorem need checking only at the primes.

Sideways: what a map does to a circle is the two-dimensional version of the angle extraction above, and the loop that needs two turns to close is the topology the double cover belongs to.

What is worth carrying away #

Adding a dimension can take structure away rather than add it.

Three-dimensional rotations have axes because three is odd, and the axis is such a convenient handle that it is easy to read it as part of what rotation means. In four dimensions it is gone, replaced by two planes and two angles, and every intuition trained on axes has to be retired.

The quaternions are the reason the four-dimensional case is nevertheless the easy one. The group that would take a page of linear algebra to describe is a pair of points on a sphere, multiplying — and the price, which is that the description is two-to-one, is the same price the three-dimensional case already paid.

The wider habit is to distrust a feature that depends on a parity. An axis exists because three is odd, a fixed point exists because a sphere of even dimension has one, a determinant is positive because a count came out even. Each of those looks like a property of rotation and is a property of a number, and the way to tell them apart is to go one dimension up and see which survives.