The integers among the quaternions

Every ring of numbers has an obvious set of integers inside it, and for the quaternions the obvious choice is wrong. Correcting it is the point of this rung, and the correction brings a regular solid with it.

The obvious ring is the Lipschitz quaternions: those with all four coordinates whole. It is closed under addition and multiplication, its norm is a sum of four squares, and everything about it looks right until an attempt is made to divide with remainder.

What a division algorithm is for #

In the ordinary integers, dividing $a$ by $b$ leaves a remainder smaller than $b$, and that single fact carries the whole Euclidean algorithm and, through it, unique factorisation.

The same statement in a ring with a norm reads: for any $a$ and any non-zero $b$ there is a $q$ in the ring with $N(a - qb) < N(b)$. Since $N(a - qb) = N(b)\,N(a b^{-1} - q)$, that is the same as asking for a $q$ in the ring within distance strictly less than one of the arbitrary quaternion $ab^{-1}$.

So the question is purely geometric: how far can a point of four-space be from the nearest point of the ring? If that distance is always under one, division with remainder works and everything follows.

Where the Lipschitz quaternions fail #

For quaternions with whole coordinates, the worst case is the centre of a unit cube: the point with every coordinate a half.

Its distance to each of the sixteen nearest whole-coordinate quaternions is $\sqrt{4 \times (1/2)^2} = 1$ exactly. Not less than one — exactly one — and the algorithm needs strictly less.

That is a near miss and it is fatal. The remainder can be made equal in size to the divisor and no smaller, so the descent that the Euclidean algorithm depends on has nowhere to go. The hero figure exhibits the failure with a specific pair: dividing $1 + i + j + k$ by $2$ gives a quotient that is exactly a half in every coordinate, a full unit away from every whole-coordinate quaternion.

In two dimensions the same computation succeeds, which is why nobody meets this problem in the Gaussian integers. The centre of a unit square is at distance $\sqrt{2}/2 \approx 0.707$ from its corners, comfortably under one, and the Gaussian integers therefore have unique factorisation. Four dimensions is where the count $\sqrt{n}/2$ reaches one.

The sixteen that fix it #

Hurwitz’s repair is to add the quaternions all of whose coordinates are half an odd integer — the sixteen points $(\pm 1 \pm i \pm j \pm k)/2$ and their translates.

Two things have to be checked. First, the enlarged set is still a ring: the sum of two half-integer quaternions is whole, the sum of a whole and a half is a half, and the product of two halves is again in the set — which is not obvious and is exactly what the hero figure verifies over all 576 products of the units.

Second, the covering radius has come down. The worst-covered point is no longer the cube’s centre, because the cube’s centre is now itself in the ring. The new worst case is at distance $\sqrt{2}/2$, which is under one, so division with remainder works.

The consequence is immediate: the Hurwitz quaternions have a Euclidean algorithm, greatest common divisors, and a factorisation theory. Not quite unique factorisation, because the ring is not commutative and factors can be shuffled by units, but unique up to that — which is all the applications need.

The near miss, and why a half is exactly the wrong number #

The failure of the Lipschitz quaternions is worth staring at, because the margin is zero rather than large and that is unusual.

In $n$ dimensions the centre of a unit cube is at distance $\sqrt{n}/2$ from its corners. For $n = 1$ that is $0.5$, for $n = 2$ it is $0.707$, for $n = 3$ it is $0.866$, and for $n = 4$ it is exactly $1$. So the ordinary integers and the Gaussian integers have room to spare, the three-dimensional lattice would still have some, and the four-dimensional case fails by nothing at all.

Above four dimensions the miss becomes wide — $\sqrt{5}/2 \approx 1.118$ and growing — so there is no question of a small repair. Four is the last dimension in which adding points can fix it, and adding the cube centres is exactly the fix.

A quantity that hits a threshold precisely rather than crossing it is usually a sign that the right object is one step away. Here the right object is the lattice with the cube centres included, and the tell was that the failing distance was $1$ and not $1.4$.

Twenty-four units, and what they are #

A unit is an element of norm one, and the Lipschitz quaternions have eight of them: $\pm 1, \pm i, \pm j, \pm k$. That is the quaternion group the first rung ends on.

The Hurwitz quaternions have twenty-four, the extra sixteen being the half-integer points, each of which has norm $4 \times (1/2)^2 = 1$. They form a group under multiplication, and it is not the quaternion group with extras bolted on: it is the binary tetrahedral group, of order twenty-four, whose centre is $\{\pm 1\}$ and whose quotient by that centre is the rotation group of the tetrahedron.

The figure verifies both facts by computation. The centre is found by testing which units commute with all the others, and it comes out as exactly two elements. The quotient’s size is found by computing the rotation each unit performs and counting the distinct ones, and it comes out as twelve — two units per rotation, which is the double cover again.

So the units of this ring are a double cover of a polyhedron’s symmetries, and that is the first hint that the twenty-four points are a shape rather than a list.

The solid they make #

Twenty-four points in four-space, every one at distance one from the origin, every one at distance one from exactly eight others. That is a regular polytope: the 24-cell.

It has twenty-four vertices, ninety-six edges, ninety-six triangular faces and twenty-four octahedral cells, and it is self-dual — its faces are arranged like its vertices. Four dimensions has six regular polytopes where three has five, and the 24-cell is the extra one, with no analogue above or below.

That last fact deserves emphasis. The simplex, the cube and its dual exist in every dimension; the icosahedron and dodecahedron have four-dimensional cousins and then stop. The 24-cell exists in four dimensions only, and the reason is the reason this whole ladder exists: it is the unit group of a division algebra that exists in four dimensions only.

Lagrange’s theorem, proved properly #

The reason to do any of this is that it gives the cleanest proof of the four-square theorem, and the proof is short enough to sketch.

Every positive integer is a product of primes, and the identity of the previous rung reduces the theorem to primes. So take an odd prime $p$ and show it is a sum of four squares.

In the Hurwitz quaternions, $p$ is not irreducible. The reason is a counting argument: the residues modulo $p$ contain $a$ and $b$ with $a^2 + b^2 + 1 \equiv 0$, because the squares and the negatives of one-plus-squares are each $(p+1)/2$ sets of residues and two sets of that size in a set of $p$ must meet. Then $p$ divides $N(a + bi + j) = a^2 + b^2 + 1$, so $p$ and $a + bi + j$ have a common divisor that is not a unit.

Take a greatest common divisor $d$, which exists because the ring is Euclidean. Its norm divides $p^2$ and is neither $1$ nor $p^2$, so it is $p$ — and the norm of a Hurwitz quaternion is a sum of four squares, possibly of halves. Doubling appropriately clears the halves, and $p$ is a sum of four whole squares.

The load-bearing step is the existence of a greatest common divisor, which needs the Euclidean algorithm, which needs the sixteen extra units. Without them the proof does not run.

The lattice, and its density #

There is a second reason the Hurwitz quaternions are the right ring, and it is about packing.

The Lipschitz quaternions are the ordinary four-dimensional integer lattice. Its closest pairs are at distance one, so spheres of radius $1/2$ centred at its points just touch, and since a four-dimensional ball of radius $r$ has volume $\pi^2 r^4/2$, those spheres fill $\pi^2/32$ of space — about $30.8\%$.

Now add the half-integer points. Each is at distance $\sqrt{4 \times (1/2)^2} = 1$ from the origin, so the closest pairs are still at distance one and the spheres are still of radius $1/2$ — but there are now twice as many of them in the same volume. The packing density doubles to $\pi^2/16$, about $61.7\%$, for no change in sphere size at all.

That lattice has a name outside this subject: it is $D_4$, and it is the densest known sphere packing in four dimensions, proved optimal among lattices by Korkine and Zolotarev in 1872. Each sphere touches twenty-four others, which is the largest number possible in four dimensions and is again the twenty-four units.

Three questions with one answer. Which quaternions should be called integers, which four-dimensional lattice packs spheres best, and which regular polytopes exist in four dimensions — and the same twenty-four points settle all three.

Factorisation, and what non-commutativity costs #

Having a division algorithm gives greatest common divisors and a factorisation theory, but the theory is not quite the familiar one and the difference is worth stating.

In the ordinary integers a number factors into primes uniquely up to order and sign. In the Hurwitz quaternions a number of norm $pq$ factors into two elements of norms $p$ and $q$, and the factorisation is unique only up to unit migration: replacing a factorisation $ab$ by $(au)(u^{-1}b)$ for any of the twenty-four units gives another one, and reordering is generally not allowed at all, since $ab$ and $ba$ differ.

So the statement is: every Hurwitz quaternion of norm $n$ factors into elements whose norms are the prime factors of $n$, in an order matching any chosen ordering of those primes, and the factorisation is unique once the order is fixed and units are taken into account. That is Lipschitz’s theorem in the form Hurwitz repaired it.

Twenty-four units instead of two is where the extra freedom comes from, and it is also where the count in Jacobi’s formula comes from. How many units a ring has controls how many representations an integer gets, and twenty-four is a large number of units for a ring to have.

What “integer” ought to mean #

The episode is worth generalising, because the same correction is needed elsewhere and it is easy to get wrong twice.

In the field generated by $\sqrt{5}$, the naive integers are $a + b\sqrt{5}$ with $a, b$ whole, and the correct ones include $(1 + \sqrt{5})/2$ — the golden ratio, which satisfies $x^2 = x + 1$ and so is an algebraic integer by the only reasonable definition. The naive ring is not the right one there for the same reason it is not right here.

The definition that survives is: an element is an integer when it satisfies a monic polynomial equation with ordinary integer coefficients. Checking the half-integer quaternions, $(1+i+j+k)/2$ satisfies $x^2 - x + 1 = 0$, which is monic with whole coefficients, so it is an integer by that test and its exclusion was an accident of writing coordinates down.

The tower of field extensions is where this definition belongs, and the lesson transfers: the right ring of integers is almost never the one obtained by putting whole numbers in coordinates.

The maximal orders, and how many there are #

One more subtlety, since the Hurwitz ring is not merely the best of two.

A ring of integers inside a quaternion algebra is called an order, and one that is not contained in a larger one is maximal. The Lipschitz quaternions are not maximal — the Hurwitz quaternions contain them — and the Hurwitz quaternions are.

There are other maximal orders, obtained by conjugating this one, and for the quaternions over the rationals they are all equivalent. That equivalence is a class-number-one statement of exactly the kind the number ladder ends on: for some algebras all maximal orders are conjugate and for others they fall into several classes, and the count is a finite invariant of the algebra.

So the tidiness here is not automatic. It is the four-dimensional case being the good case, one more time.

The 24-cell’s self-duality, and why it is not a coincidence #

One property of the solid deserves its own paragraph because it comes straight from the ring.

The 24-cell is self-dual: replacing each of its twenty-four octahedral cells by a point at that cell’s centre, and joining points whose cells shared a face, reproduces a 24-cell. Among the six four-dimensional regular polytopes it is the only one that is self-dual and not a simplex.

The reason is visible in the units. The twenty-four unit quaternions split into the eight Lipschitz ones and the sixteen halves, and each set is the vertex set of a four-dimensional cross-polytope or hypercube respectively — the two dual solids. The 24-cell is their union, and a union of a shape with its dual, when the two are placed correctly, is self-dual by construction.

That decomposition is also visible in the group. The eight Lipschitz units are the quaternion group, a normal subgroup of index three inside the twenty-four; the three cosets are the eight units and the two halves of the sixteen, and rotating between them is an order-three symmetry of the whole solid. The solid’s three-fold symmetry and the group’s index-three subgroup are the same fact, which is the kind of correspondence that makes the object worth calling a shape rather than a set of coordinates.

What the pictures cannot show #

A four-dimensional solid is drawn by projection and the projection distorts. The hero picks a viewpoint in general position so that no two of the twenty-four vertices land on one another, and asserts that no two do — an earlier viewpoint beyond a vertex sent the two points $\pm 1$ to the same place, and the picture came out a tangle for a reason nothing counted. Edge lengths and angles in the drawing mean nothing.

The group structure is checked and not drawn. That the twenty-four points are closed under multiplication is 576 products, verified in the generator; there is no arrangement of dots that shows it.

And the division algorithm has no picture at all. What it says is that every point of four-space is within distance one of the lattice, which is a statement about a covering radius in a space no drawing holds.

Where the ladder goes next #

The last rung takes the doubling one step further. The octonions have eight dimensions, a multiplicative norm, an eight-square identity, and no associativity — and the question of what survives that loss turns out to have a precise answer.

Sideways: unique factorisation and where it fails is the property this rung was constructed to keep, and the Euclidean algorithm is the machine the sixteen extra units were let in to run.

What is worth carrying away #

A definition that looks canonical is often just the one that is easiest to write down.

Integer quaternions are the ones with integer coordinates is a sentence anybody would write, and it produces a ring with no division algorithm, no greatest common divisors, and no proof of the four-square theorem. Adding sixteen points that look like fractions repairs all of it at once.

The test of a definition is what it makes provable. The half-integer quaternions have to be admitted not because a general theory demands it but because without them the theorem does not come out — and the general theory, when it arrived, agreed.