What is lost at eight

The tower runs one dimension, then two, then four, then eight. At each step something is given up: the ordering, then the commuting, then the associating. The last step is the interesting one, because associativity is a law nobody expects to be able to do without.

Eight dimensions, a multiplicative norm, an inverse for every non-zero element, and a multiplication in which $(xy)z$ and $x(yz)$ can differ. That is the octonions, found by Graves in December 1843 — two months after Hamilton’s quaternions — and independently by Cayley in 1845.

Seven lines are the whole table #

The multiplication is usually written as a table of sixty-four entries, and it is much better read off a picture.

Take seven points and seven lines arranged so that every line holds three points, every point lies on three lines, and every pair of points lies on exactly one line. That configuration exists, is unique, and is the Fano plane — the smallest projective plane, with two points per line’s worth of everything.

Label the points $e_1$ through $e_7$ and orient each line, so that going round it in the arrow’s direction gives a cyclic triple. Then declare that each such triple multiplies exactly as $i, j, k$ do: the product of two consecutive units is the third, and reversing the order changes the sign. Every unit squares to $-1$.

That is the complete definition. Any two of the seven imaginary units lie on exactly one line, so their product is determined; the rest follows by linearity. Sixty-four table entries, produced by seven arrows.

Why the arrows have to be drawn carefully #

There is a trap in the picture that the generator here was written to avoid, and it is worth naming because most drawings of the Fano plane fall into it.

A line of the plane is three collinear points, and the multiplication on it is a cycle — $e_1 e_2 = e_3$, $e_2 e_3 = e_1$, $e_3 e_1 = e_2$. Drawing that cycle as three arrows on the line puts all three on top of one another, two of them pointing backwards, and the picture becomes unreadable.

The fix is to choose the labelling so that on every line the cyclic order matches the order the eye meets the points. Then one arrowhead per line says which way the cycle goes and the reader can compute any product. The generator searches for such a labelling over all seven-factorial assignments and asserts it found one, rather than taking a labelling on trust.

That is a small thing and it is the difference between a diagram that defines the octonions and a diagram that decorates them.

What fails, and by how much #

Associativity fails, and the figure counts the failures rather than asserting them. Of the $343$ ordered triples of imaginary units, $168$ have $(xy)z \neq x(yz)$ — just under half.

The smallest example is easy to see on the plane. Take $e_1$, $e_2$ and $e_4$: no line holds all three, since three points of the Fano plane are collinear only if the configuration says so, and this triple is not. Then $(e_1e_2)e_4$ and $e_1(e_2e_4)$ come out as negatives of one another.

The pattern is exact: a triple associates when its three units are collinear and fails when they are not. Three collinear units generate a copy of the quaternions inside the octonions, and inside a copy of the quaternions everything associates. Three non-collinear ones generate the whole thing, and the whole thing does not.

So the Fano plane is not only the multiplication table; it is also the complete answer to which triples behave.

Reading a product off the picture #

Since the figure is the definition, it is worth using it once, slowly, as a reader would.

To find $e_3 e_7$: locate the unique line through $e_3$ and $e_7$. Follow its arrow. If the arrow runs from $e_3$ to $e_7$, the product is the third point of that line; if it runs the other way, the product is minus the third point. That is the entire procedure and it settles any of the forty-two products of distinct imaginary units.

For the products involving $1$ there is nothing to do, since $1$ commutes with everything and changes nothing. For a unit times itself the answer is $-1$, by the same convention that governs $i$, $j$ and $k$. Forty-two plus seven plus fifteen accounts for all sixty-four entries.

What makes this genuinely a picture rather than a mnemonic is that the incidence structure carries the algebra. Which triples associate, which subalgebras are copies of the quaternions, and which products are which are all readable off the seven lines; nothing has to be memorised beyond the orientation convention.

The law that does not fail #

Losing associativity outright would make the algebra useless, and it is not lost outright. What holds is alternativity: any two elements generate an associative subalgebra, so $(xx)y = x(xy)$ and $(xy)y = x(yy)$ always.

The clean way to state it uses the associator $[x, y, z] = (xy)z - x(yz)$, which measures the failure. In the octonions the associator is alternating: it changes sign when any two of its arguments are swapped, and therefore vanishes whenever two of them are equal. The figure checks both halves over all $343$ triples — that swapping two arguments negates it, and that none of the failures has a repeated argument.

Alternativity is enough for most of what associativity is used for. Inverses are unambiguous, powers are well defined without brackets, and the norm remains multiplicative. What is lost is the ability to write products of three or more distinct elements without brackets, and the ability to represent the algebra by matrices — since matrix multiplication associates and octonion multiplication does not.

The eight-square identity, still standing #

The norm survives the loss, and with it the arithmetic.

For octonions $|xy| = |x||y|$ still holds, so writing out coordinates gives an identity expressing a product of two sums of eight squares as a sum of eight squares. Degen found it in 1818, twenty-five years before there was an algebra to explain it, and the same thing happened one level down with Euler’s four-square identity and Hamilton.

The generator checks the identity in exact integers on a specific pair, and checks that every non-zero octonion has an inverse by producing one and multiplying. Both are computations rather than quotations, and both hold despite the associator being non-zero all around them — which is the point: the norm does not notice the failure.

Hurwitz’s theorem in 1898 closed the list. A bilinear identity of that kind exists only for one, two, four and eight squares. Not because nobody has looked, but because the linear algebra such an identity would require has no solution in any other dimension.

Why sixteen loses division #

The doubling construction that produced each level does not stop at eight. Applying it again gives the sedenions, sixteen-dimensional, and they are a genuine algebra — but they have zero divisors: two non-zero elements whose product is zero.

Once that happens division is finished, since dividing by either factor would give a contradiction. And the norm is no longer multiplicative, since a product of two things of positive length has length zero.

So the tower stops for a definite reason rather than from exhaustion. Each doubling costs something, and the fourth doubling costs the property the whole construction was for. The first rung’s table of the tower counts the failures at each level; this is where the count reaches the one that matters.

The symmetry group nobody expected #

The octonions have an automorphism group — the invertible linear maps preserving multiplication — and it is one of the five exceptional simple Lie groups.

For the complex numbers the group is of order two, generated by conjugation. For the quaternions it is the rotation group of three-space, since an automorphism must permute the imaginary units rigidly. For the octonions it is fourteen-dimensional and is called $G_2$, and it is not a rotation group or a matrix group of any familiar kind — it is one of five groups that exist for no reason expressible in a family.

That is worth stating plainly. The classification of simple Lie groups produces four infinite families and exactly five exceptions, and every one of the five is connected to the octonions. The largest, $E_8$, is $248$-dimensional and turns up in lattice packings and in physics; all of them trace back to this eight-dimensional algebra.

So the price of losing associativity buys the exceptional structures. An algebra that behaved would not have produced them.

Where the octonions are actually used #

The algebra spent a century as a curiosity, and it is worth saying where it stopped being one.

The exceptional Lie groups are the first place. All five of them — $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ — are constructed from the octonions in one way or another, and the classification of simple Lie groups would produce four infinite families and nothing else if the octonions did not exist. That is a strong statement: a whole class of exceptions in a classification theorem traces to one eight-dimensional algebra.

The second is geometry. There is a projective plane over the octonions, and there is no projective space of higher dimension over them, because the coordinate arguments that build one need associativity. So the octonionic plane exists and the octonionic three-space does not, and that asymmetry is the reason the classification of projective planes has an exceptional entry too.

The third is lattices and codes. The $E_8$ lattice, densest in eight dimensions and built from the octonion integers, is the ancestor of a family of extremely good error-correcting codes, and a lattice that packs well is exactly what a code that corrects well is.

None of that was visible in 1843. What Graves had was an eight-dimensional multiplication with a strange defect, and the defect turned out to be the source of every exceptional structure in two separate classification theorems.

Where the seven lines come from #

The Fano plane’s appearance is not decorative and there is a reason for exactly seven.

The seven imaginary units are the non-zero vectors of a three-dimensional space over the field with two elements: there are $2^3 - 1 = 7$ of them. Lines through the origin in that space, taken projectively, are the points of the Fano plane, and a line of the plane is a two-dimensional subspace minus the origin, which holds three non-zero vectors.

Labelling the units by their coordinate triples, the product of two units is the unit labelled by the sum of their labels over the field with two elements. So the multiplication’s shape is addition in a vector space over two elements, and the only extra information is the signs — which is the orientation of each line.

The smallest projective plane turning up as a multiplication table is the sort of coincidence this subject specialises in, and here it has a one-line explanation.

The coincidence between those seven regions and these seven units is exact rather than numerical. In both cases the objects are the non-zero elements of a three-dimensional space over the two-element field, and in both cases the natural operation on them is addition in that space. What the octonions add is a sign on each line, and the sign is the only part of the multiplication that is not pure combinatorics.

Counting the sign choices #

There is one more question the picture raises: how much freedom is there in the orientations?

Each of the seven lines can be oriented two ways, giving $2^7 = 128$ candidate tables. Not all of them define an octonion algebra — the sign conventions must be consistent — and the ones that do are exactly $16$ of the $128$. Taken together with the $30$ ways of identifying the seven units with the points of the plane that respect its structure, there are $480$ different multiplication tables, all giving isomorphic algebras.

That $480$ is a number worth knowing because it is where confusion in the literature comes from: two textbooks writing down different tables are usually not disagreeing, and checking which of the $480$ each is using resolves it.

The corresponding count for the quaternions is $2$ — the choice of whether $ij$ is $k$ or $-k$ — which is why nobody ever has this problem one level down.

Integers, one level up #

The previous rung found that the quaternions have a natural ring of integers only after sixteen half-coordinate points are let in. The octonions have the same story with a larger repair.

The naive integer octonions — those with all eight coordinates whole — have $16$ units and no division algorithm, for exactly the reason the Lipschitz quaternions failed: the centre of a unit cube in eight dimensions is at distance $\sqrt{8}/2 = 1.414$ from its corners, and now the miss is wide rather than exact.

The repair is Coxeter’s, and it adds enough half-integer points to bring the covering radius down to $1/\sqrt{2}$. The resulting ring has $240$ units, and those $240$ points are the shortest vectors of the $E_8$ lattice — the eight-dimensional analogue of the twenty-four points of the last rung, with the same three-way coincidence between units, kissing number and lattice density.

Twenty-four in four dimensions and two hundred and forty in eight. Both are the largest kissing numbers their dimensions admit, both are the unit groups of the right ring of integers, and both are the vertex sets of the most symmetric object available. The pattern that began with the Gaussian integers has four terms and then stops, for the same reason everything else here stops.

What the picture cannot show #

A drawing of seven points does not show non-associativity. The failure is a statement about triples of elements, and the figure counts $168$ of them; a reader can verify any one by following arrows, and the count itself is arithmetic done in the generator.

The eight-square identity is not drawn at all. It is an identity in sixteen variables and it is checked at one point, in exact integers, which is a demonstration rather than a proof.

And the fourteen-dimensional automorphism group has no picture. Its existence is the most remarkable thing on this page and it is stated rather than shown, because nothing in fourteen dimensions fits on a page.

Where the ladder goes next #

This closes the ladder. It began with four characters — $i^2 = j^2 = k^2 = ijk = -1$ — and it ends two doublings later, at the point where dividing stops being possible and the reason is a theorem rather than a failure of ingenuity.

What is left unwritten from here belongs to other anchors rather than this one: the exceptional Lie groups as objects in their own right, the $E_8$ lattice as a sphere packing, and the projective plane over the octonions, which exists and has no analogue in higher dimensions for the same reason everything else here stops. Sideways: the smallest projective plane is the configuration that turned out to be the multiplication table, and what the tower costs at each step is where the ladder started.

What is worth carrying away #

A law can fail and leave something behind that is nearly as good, and the residue is often more interesting than the law.

Commutativity fails in the quaternions and what remains — that conjugation reverses products — is the structure rotations are built from. Associativity fails in the octonions and what remains is alternativity, which is enough for inverses, for powers, and for a multiplicative norm.

The general habit is to ask, when a law breaks, exactly which of its consequences broke with it. Usually fewer than expected. Here the answer is that brackets became necessary and matrices became unavailable, and everything else survived — including the arithmetic identity that was written down twenty-five years before anybody knew what algebra it belonged to.