Two hundred and forty directions

The right ring of integers among the quaternions has twenty-four units, and those twenty-four points are the vertices of the twenty-four-cell: the most symmetric arrangement four dimensions has, the one with no analogue in three or in five, and the tightest way to place twenty-four spheres around one.

Eight dimensions has an arrangement of its own, and its count is two hundred and forty. The connection between the two numbers is not an analogy drawn afterwards. The quaternions are one of the ways the two hundred and forty are built, and this rung is that construction, made explicit enough to count.

What the collection is #

A root system is a finite set of vectors, closed under reflection in each of its own members, in which the vectors meet at very few angles. $E_8$ is one, it is the largest of the exceptional ones, and it is what the figure builds.

The construction is two rules. Take every vector with two coordinates $\pm 1$ and the rest zero — there are $\binom{8}{2} \times 4 = 112$ of those. Then take every vector with all eight coordinates $\pm \tfrac12$ and an even number of minus signs, of which there are $2^7 = 128$. Both kinds have squared length two, and $112 + 128 = 240$.

The half-integer vectors are the interesting half, and the parity condition on their signs is what makes the whole thing work. Without it there would be $256$ of them and the inner products would come out wrong; with it, every pair of the $240$ meets at an inner product of $0$, $\pm 1$ or $\pm 2$, which the figure checks over all $57{,}600$ pairs.

The definition has a second half, and it is the one the figure checks hardest. A root system must be closed under reflection in each of its own members: take any root $t$, reflect it in the hyperplane perpendicular to any root $r$, and the result must again be a root. For vectors of squared length two the reflection is $t - (t \cdot r)\,r$, which is why the inner products being whole numbers matters — a reflection is only guaranteed to land on a lattice point when the coefficient it uses is an integer. The figure applies that reflection at all $57{,}600$ ordered pairs and confirms every result is on the list, and a single failure would stop the picture being drawn.

That is a much stronger condition than it looks. Most sets of $240$ unit-length-squared-two vectors satisfy nothing of the kind, and the parity condition on the half-integer signs is exactly what buys it: drop the condition and reflections start landing on vectors with an odd number of minus signs, which are not in the collection.

That the profile is the same from every root is worth pausing on. It says the system has a symmetry group carrying any root to any other, which is what makes $240$ a meaningful count rather than a sum of unrelated pieces. The group is the Weyl group of $E_8$, and it has $696{,}729{,}600$ elements — a number nothing here computes and everything here is a shadow of.

What a root system is for #

A root system is not built for its own sake. It is the combinatorial shadow of a Lie algebra — a system of infinitesimal symmetries — and each root records one direction in which the algebra can be moved.

The arithmetic is exact and worth stating, because it is where the number $248$ comes from. The Lie algebra of $E_8$ has one dimension for each root and one for each dimension of the space the roots live in, so $240 + 8 = 248$. That algebra is the largest of the five exceptional ones, it is the reason $E_8$ appears in physics whenever anybody wants the biggest symmetry available, and every fact about it that a figure here could check is a fact about the $240$ vectors.

The reflections checked above generate the symmetry group of the root system, and its size is the count quoted at the end of the last section. So the collection carries the whole structure: reflect to get the group, count roots and dimensions to get the algebra, and the eight-dimensional arrangement of $240$ points is the object all of it is read off.

Two quaternion lattices inside it #

Split the eight coordinates into two blocks of four and sort the roots by where they live.

The twenty-four roots inside a block are the vectors with two coordinates $\pm 1$ among four, which is $\binom{4}{2} \times 4 = 24$. Those are the $D_4$ root system, and $D_4$ is the quaternions: the twenty-four Hurwitz units, scaled by $\sqrt 2$, are exactly those twenty-four vectors.

The scaling is worth making explicit, since it is where the identification lives. A Hurwitz unit is a quaternion of length one — the eight of the form $\pm 1, \pm i, \pm j, \pm k$ and the sixteen with all coordinates $\pm\tfrac12$. Multiply every one of them by $\sqrt 2$ and the lengths become $\sqrt 2$, so the squared lengths become two, which is what a root needs; and the twenty-four scaled points are precisely the twenty-four vectors with two coordinates $\pm 1$ in four dimensions. The two lists match point for point, and the rotations quaternions perform are the symmetries that carry the twenty-four-cell to itself.

So $E_8 \supset D_4 \oplus D_4$, and the containment is proper by a wide margin: forty-eight roots out of two hundred and forty. The other hundred and ninety-two are the whole content of the statement that $E_8$ is more than two copies of the quaternions. Sixty-four of them mix the two blocks in the integer coordinates, and a hundred and twenty-eight are the half-integer vectors, which belong to neither block and are what glue the two lattices into one.

That is the honest version of the slogan that $E_8$ is quaternionic. Two copies of the quaternion lattice sit inside it orthogonally; four fifths of it is elsewhere.

One more count settles what the glue is doing. The sixty-four roots that straddle the halves have one non-zero coordinate in each block, which is $4 \times 4$ choices of position and four of sign. The hundred and twenty-eight half-integer roots have four half-coordinates in each block. So every root outside the two blocks touches both of them, and there is no third block hiding anywhere — the decomposition into $24 + 24 + 64 + 128$ is a partition and the figure checks that it sums to $240$ by a route independent of how the roots were built.

The construction that is genuinely quaternionic #

There is a second route, and this one produces all two hundred and forty from quaternions rather than a fifth of them. It needs the golden ratio.

Allow coordinates from $\mathbb{Z}[\varphi]$ — whole numbers together with the golden ratio — and the twenty-four units grow to a hundred and twenty. They are the binary icosahedral group, they are the vertices of the six-hundred-cell, and they contain the Hurwitz units exactly five times over.

The ring they generate is the icosian ring. It is a rank-four module over $\mathbb{Z}[\varphi]$, and $\mathbb{Z}[\varphi]$ is a rank-two module over $\mathbb{Z}$, so as a lattice over the ordinary whole numbers it has rank eight. Give it the norm that adds the quaternion norm to the conjugate of the quaternion norm — the golden ratio has a conjugate, $1 - \varphi$, and applying it to every coordinate gives a second copy of the same quaternion — and the result is an $E_8$ lattice, with the hundred and twenty icosians and their conjugates supplying the two hundred and forty shortest vectors.

So $240 = 2 \times 120$ is not a coincidence of counts. The eight dimensions are four quaternion dimensions doubled by the golden field, and the two hundred and forty directions are the hundred and twenty icosians, read twice.

The subgroup relation is worth reading as a statement about symmetry rather than about counts. The Hurwitz units perform the twelve rotations of a tetrahedron, two units to each; the icosians perform the sixty rotations of an icosahedron, again two to each. A tetrahedron sits inside an icosahedron five ways — which is the same five that the five solids’ symmetry groups are organised by — and those five ways are the five cosets the figure counts. Nothing about the quaternions is being used there except that they double the rotation groups faithfully, which is the two-to-one covering the first rung established.

Why the units, the kissing number and the lattice agree #

Three different questions have the same answer in each of these dimensions, and the agreement is what makes the objects exceptional rather than merely large.

How many units does the ring of integers have? Twenty-four for the Hurwitz quaternions, two hundred and forty for the integral octonions.

How many spheres can touch one sphere? Twenty-four in four dimensions, two hundred and forty in eight — both proved optimal, the second by an argument from 1979 and the first not until 2003.

How many shortest vectors does the densest lattice have? The same numbers again, in the same lattices, and the density itself was settled only in 2016 for eight dimensions — by an argument that constructs a single function with prescribed zeros, and which had been sought for a century.

And how many unit vectors form a group under multiplication? The same numbers a fourth time, which is the answer that explains the other three: the units are a group, a group acts transitively on itself, and an orbit of unit vectors is exactly the kind of arrangement a kissing configuration wants.

The multiplicative norm is the common cause. It makes the units a group, and a group of unit vectors is an arrangement of points on a sphere with far more symmetry than a random one can have — which is precisely what a kissing configuration and a dense lattice both want. The identity that multiplies sums of squares is the four-dimensional case, and the eight-square identity is the octonionic one, and there is no sixteen-square identity, which is why the pattern ends.

There is a fourth question with the same answer, and it is the one that makes the coincidence feel less like luck. How many ways can a lattice point have the minimum norm? In a generic lattice the answer is small — a few dozen, at most — because there is no reason for many vectors to share a length exactly. In these two the answer is enormous, and the reason is the group: the units act on the lattice by multiplication, transitively on the shortest vectors, so the count of shortest vectors is the size of the group. A large kissing number is what a group of unit vectors looks like from outside.

The other route, and why it is the same #

The integral octonions reach $E_8$ from the other side. Their unit group has two hundred and forty elements, and those elements are the same two hundred and forty vectors, in different coordinates.

Three constructions, then: a list of vectors with a parity condition; a ring of quaternions over the golden field; a ring of octonions. They produce the same object, and the fact that they do is the substance of calling $E_8$ exceptional. An object with one construction is a definition. An object with three unrelated ones is a fact about mathematics.

The three constructions are not equally informative, and it is worth saying which one explains what. The list of vectors explains nothing; it is a definition that happens to work, and its parity condition looks arbitrary until something else accounts for it. The octonionic construction explains the parity — it is the condition that makes the integral octonions closed under multiplication — and explains why eight is the last dimension where this happens. The icosian construction explains the number: $240$ is twice $120$, and $120$ is the order of a rotation group that exists because the icosahedron does.

Three constructions, three different things explained, and the object is the same in all three. That is the ordinary situation with the exceptional objects, and it is why they are studied by people who would not otherwise share a subject.

What the coincidence does not extend to #

Twenty-four dimensions has the Leech lattice, whose kissing number is $196{,}560$, and it is the other exceptional object of this kind. It is not built from a division algebra, because there is no division algebra in sixteen or twenty-four dimensions to build it from; it comes from an error-correcting code instead.

That is the point at which the quaternionic story stops rather than continuing. The pattern 1, 2, 4, 8 is a theorem about algebras and it has four terms. The pattern of exceptional lattices has more terms than that, and the extra ones arrive by different routes.

It is also worth being clear about which of the golden-ratio coordinates is doing the work. The icosians need $\mathbb{Z}[\varphi]$ because the icosahedral rotations do: a fivefold rotation has $\cos 72°$ in it, and that number is not rational. Four-dimensional space over a field of degree two is eight-dimensional space over the rationals, and the doubling is where the eight comes from — not from any property of eight, but from four times the degree of the field the icosahedron forced. The exceptional object in eight dimensions is the quaternions plus the icosahedron, and neither factor is optional.

Between and above these dimensions almost nothing is known. The densest packing in five dimensions is undetermined; the kissing number in dimension five is known only to lie between $40$ and $44$. The dimensions where the answer is exact are the dimensions where an algebra or a code supplies a group of unit vectors, and there are very few of those.

One consequence of that is worth recording, because it is the kind of thing this ladder keeps producing. The $E_8$ lattice has a multiplication on it — inherited from the icosians — and the ordinary integer lattice in eight dimensions does not, in any way compatible with its geometry. So the densest arrangement in eight dimensions is the one that is also an algebra, and the ordinary arrangement is neither dense nor multiplicative. Structure and density arrive together here, as they did at four dimensions, and as they stop doing above eight.

What this rung settles #

That the twenty-four of the last rung and the two hundred and forty of this one are the same phenomenon at two sizes, and that quaternions are how the second is built as well as the first.

Three counts carry it, and each is made rather than quoted. Two hundred and forty roots, with the neighbour profile checked from every one. Forty-eight of them in two orthogonal quaternionic blocks, with the remaining hundred and ninety-two counted and classified. A hundred and twenty icosians, closed under multiplication, containing the twenty-four Hurwitz units five times over.

What the figures cannot show is the eight-dimensionality itself — every picture here is a count, a grid or a histogram, because a set of vectors in eight dimensions has no honest projection that a reader could check. That is the trade this whole file’s rules were written for: a drawing that could be checked, in place of a drawing that could be admired. The famous circular portrait of $E_8$ is a projection onto a plane chosen to make the symmetry visible, and it is beautiful and it settles nothing; the grid and the bars above settle everything claimed here.

And the ladder closes where it began, with four characters written on a bridge in 1843. The multiplication that would not commute turned out to be the four-dimensional rotations, the integers with twenty-four units, the identity for sums of four squares, and — at one remove, through the golden ratio — the two hundred and forty directions of the largest exceptional root system there is.