March 17, 2003

This Week's Finds in Mathematical Physics (Week 194)

John Baez

I recently flew from Sydney, Australia to Waterloo, Canada. All of a sudden day became night and steamy 30 Celsius summertime suddenly switched to a -15 Celsius blizzard. Unsurprisingly, I came down with a cold. Nonetheless, I'm very happy to be here. I'm visiting the Perimeter Institute of Theoretical Physics, seeing old friends like Louis Crane, Fotini Markopoulou and Lee Smolin, and newer ones like Laurent Freidel, Hendryk Pfeiffer and Olaf Dreyer. There's a lot of interesting gossip about quantum gravity, string theory....

But more about that later! Now I want to talk about Conway's new book.

Last week I described what I thought were the Cayley integral octonions. But then Dan Piponi showed me that I had screwed up: they aren't closed under multiplication. I was very confused until I arrived here.

On the day I showed up, I got a packet of mail containing this book:

1) John H. Conway and Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Ltd., Natick, Massachusetts, 2003.

Conway and Smith sent it to me because they quoted my history of the octonions. And in this book, there is a description of my mistake and how to fix it! They attribute it to someone named J. Kirmse, and write:

Other people have made this very natural assumption, so it is convenient that it has a standard name: "Kirmse's Mistake." The product of two Kirmse integers happens to be a Kirmse integer rather more than one third of the time.

There's nothing like getting your mistake corrected by a book in the mail from Conway! Many of you probably know him for his work on surreal numbers and the game of Life. Among mathematicians he's famous for his work on game theory, the Leech lattice and finite simple groups. He's also famous for acting like he just quantum-tunnelled out of a Lewis Carroll novel. If you don't know what I mean, you're missing out on a lot of fun... so you should immediately read this:

2) Charles Seife, Mathemagician (impressions of Conway), The Sciences (May/June 1994), 12-15. Available at

Just to entice you, I'll quote the beginning:

"Have I done this to you yet?" He grabbed my hand and held it out in front of him, palm down. Before I could react, he pulled a rubber stamp out of his pocket, and my hand suddenly was emblazoned with big red letters. "John H. Conway's Seal of Grudging Approval." Within seconds, it had smeared to three red lines that wouldn't wash off for several days. Still grasping my hand, he pulled me toward his office. Brightly colored polyhedra hung in disarray from a network of strings dangling from the ceiling. The dim outline of a computer terminal was visible through a pile of Rubik's cubes and wooden toroids. "We'll be better off in the undergraduate lounge. The doctor says I should rest, and I can lie down over there."

Anyway, he's been busy writing books lately. Not too long ago, he finished one about the classification of quadratic forms:

3) John H. Conway and Francis Fung, The Sensual (Quadratic) Form, Mathematical Association of America, Washington DC, 1997.

and before that, a very fun elementary one about numbers:

4) John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus, New York, 1996.

Now he's into quaternions and octonions. But his new book with Derek Smith starts by talking about the real numbers and 1-dimensional geometry. Then it turns to complex numbers and 2-dimensional geometry, including the Gaussian and Eisenstein integers and the 17 "space groups" in 2 dimensions.

Perhaps I should say what these things are. The Gaussian integers are complex numbers of the form

a + bi
where a and b are integers. They form a square lattice:

             *     *     *     *

             *     *     *     *

             *     *     *     *

You can uniquely factor any Gaussian integer into primes - at least if you count differently ordered factorizations as the same, and ignore the ambiguity due to "units" - the invertible Gaussian integers 1, i, -1, and -i. You can prove this using the geometry of the square lattice... for details, read the book!

The Eisenstein integers are complex numbers of the form

a + bw
where a and b are integers and w is a nontrivial cube root of -1. These are closed under addition and multiplication, and they form a lattice with hexagonal symmetry:

           *       *      *      *

               *       *      *                

           *       *      *      *

Again you can use geometry to prove unique factorization up to reordering and units.

The Gaussian and Eisenstein integers are the most symmetrical lattices in 2 dimensions: they have 4-fold and 6-fold rotational symmetry, respectively. As I explained in "week124" and subsequent Weeks, this is related to the appearance of the number 24 in bosonic string theory. But these lattices also play a role in crystallography, in the classification of 2-dimensional "space groups".

I'm not sure what the definition of a "space group" is - the references I've seen are annoyingly reticent on this point - but it's something like a subgroup of the Euclidean group (the group generated by rotations, reflections and translations) that acts transitively on a lattice. There are 17 space groups in 2 dimensions, also called "wallpaper groups" since they give different symmetries of repetitive wallpaper patterns. Of these, 2 act on a lattice with no special symmetry:

         *          *         *         *

           *          *         *         *       

             *          *         *         *

7 act on a lattice with rectangular symmetry:

             *        *        *        *

             *        *        *        *

             *        *        *        *

or alternatively, on a lattice with rhombic symmetry:

                  *                 *

         *                 *                *

                  *                 *

3 act on a lattice with square symmetry, and 5 act on a lattice with hexagonal symmetry. For more details, with pictures, see:

5) NIST, The 17 two-dimensional space groups,

6) Eric Weisstein, Wallpaper groups,

7) David Hestenes, Point groups and space groups in geometric algebra,

After this low-dimensional warmup, Conway and Smith's book turns to the quaternions and their applications to 3-dimensional and 4-dimensional geometry. They classify the finite subgroups of the 3d rotation group SO(3), its double cover SU(2), and the 3d rotation/reflection group O(3). They also classify the finite subgroups of the 4d rotation group. They mention but do not study the 230 space groups in 3 dimensions.

Then they turn to quaternionic number theory! The "Lipschitz integral quaternions" are of the form

a + bi + cj + dk
where a,b,c,d are integers. But number theory works better for the "Hurwitz integral quaternions", which are of the form
a + bi + cj + dk
where a,b,c,d are either all integers or all half-integers. These are closed under addition and multiplication, and they form a lattice called the D4 lattice, which gives the densest lattice packing of spheres in 4 dimensions - each sphere has 24 nearest neighbors. They prove a version of unique prime factorization for Hurwitz integral quaternions. But the sense of "uniqueness" here is a lot more tricky, in part because the quaternions are noncommutative.

Finally, they study the octonions. They start with a truly excellent study of Moufang loops, isotopies and triality - three fairly esoteric subjects that are crucial for understanding octonions. Then they tackle octonionic number theory! The "Gravesian integral octonions" are octonions of the form

a0 + a1 e1 + a2 e2 + a3 e3 + a4 e4 + a5 e5 + a6 e6 + a7 e7 
where all the coefficients are integers. The "Kleinian integral octonions" are those where the coefficients are either all integers or all half-integers. Both these are closed under addition and multiplication. To get even denser lattices closed under multiplication, we need the octonion multiplication chart (see "week104"):

This has 7 lines in it, if we count the circle containing e1,e2,e4 as an honorary "line". To get the "double Hurwitzian integral octonions", first pick one of these lines. Then, take all integral linear combinations of Gravesian integral octonions, octonions of the form

(± 1 ± ei ± ej ± ek)/2
where ei, ej, ek lie on this line, and those of the form
(± ei ± ej ± ek ± el)/2
where ei, ej, ek, and el all lie off this line. We get 7 different versions of the double Hurwitzian integral octonions this way. Each is closed under addition and multiplication, and each is a copy of the lattice called D4 x D4.

To get an even denser lattice, we can take the union of all 7 different double Hurwitzian integral octonions. I talked about this last week. We get an E8 lattice, which gives the densest packing of spheres in 8 dimensions - each sphere has 240 nearest neighbors. I thought this lattice was closed under multiplication, but it's not! Conway and Smith mockingly call it the "Kirmse integral octonions".

To fix this problem, you need to perform a slight trick. Pick a number i from 1 to 7. Then, take all the Kirmse integral octonions

a0 + a1 e1 + a2 e2 + a3 e3 + a4 e4 + a5 e5 + a6 e6 + a7 e7
and switch the coefficients a0 and ai. Bizarrely, the resulting "Cayley integral octonions" are closed under multiplication. But they are still an E8 lattice - just a rotated version of the Kirmse integral octonions.

Since this trick involved an arbitrary choice, there are 7 different copies of the Cayley integral octonions containing the Gravesian integral octonions. And this is as good as it gets: each one is maximal in a certain sense which Conway and Smith explain. They study prime factorization in the Cayley integral octonions, but it's very tricky, since the octonions are nonassociative.

I've got a bunch more to talk about, but I've probably scared away everybody except the octonion-heads, so I'll wait until next week. I'll just mention this review article, which octonion-heads should enjoy:

8) B. S. Acharya, M theory, G2 manifolds and four-dimensional physics, Class. Quant. Grav. 19 (2002), 5619-5653.

It's nice because it goes all the way from the definition of a G2 manifold to (sketchy but readable) physical considerations like the rate of proton decay.

Addendum: Tony Smith writes:

Thanks for mentioning the John Conway - Derek Smith book
in week 194. I have ordered it from Amazon.

BTW - ( and my apologies if you have already seen these details
if they are in the Conway-Smith book ) -
Kirmse's mistake is described in some detail in Coxeter's pper
Integral Cayley Numbers (Duke Math. J., v. 13, no. 4,
December 1946), in which Coxeter says:
"... Kirmse ... selects an eight-dimensional module ...
which is closed under subtraction and contains eight
linearly independent members. .. a module is called
an INTEGRAL DOMAIN if it is closed under multiplication.
A simple instance is the module Jo consisting of all
Cayley numbers ... [that are] integers. ...
[Kirmse] then defines a maximal ... integral domain over Jo
as an extension of Jo which cannot be further extended
without ceasing to be an integral domain.
He states that there are EIGHT such domains,
one of which he calls J1 and describes in detail.
Actually, there are only SEVEN, which presumably are
the remaining seven of his eight.
... J1 itself is not closed under multiplication. ...
Since the 168-group is doubly transitive on the
seven [imaginary octonions], ANY transposition [of the
imaginary octonions] will serve to rectify J1 in the desired manner.
But there are only seven such domains,
since the (7|2) = 21 possible transpositions fall
into 7 sets of 3, each set having the same effect.
In each of the seven domains, one of the [imaginary octonions]
plays a special role, viz., that one which is not affected
by any of the three transpositions.
Comparing Kirmse's multiplication table with Cayley's
... we see that  ... Kirmse's J1 could be used as it
stands if we replaced his multiplication table with Cayley's. ..."


These integral domains are also discussed
in Coxeter's paper Regular and Semi-Regular Polyotpes III
(Math. Z. 200, 3-45, 1988), where he describes the 240 units
of an E8 integral domain as

"... the 16 + 16 + 16 octaves

ħ1, ħi, ħj, ħk, ħe, ħie, ħje, ħke,



and the 192 others derived from the last two expressions by
cyclically permuting the 7 symbols [ i,j,k,e,ie,je,ke ]
in the peculiar order
     e, i, j, ie, ke, k, je
It seems somewhat paradoxical ... that the cyclic permutation
    ( e, i, j, ie, ke, k, je ),
which preserves the integral domain
(and the finite projective [Fano] plane ...)
is not an automorphism of the whole ring of octaves;
it transforms the associative triad ijk
into the anti-associative triad j ie je.

On the other hand, the permutation
    ( e ie je i k ke j ),
which IS an automorphism of the whole ring of octaves
(and of the finite [Fano] plane ...)
transforms this particular integral domain into another
one of R. H. Bruck's cyclic of seven such domains.  ...".

Tony          18 March 2003

"Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way." - Lord Kelvin

© 2003 John Baez