Also available at http://math.ucr.edu/home/baez/week194.html

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 
http://www.users.cloud9.net/~cgseife/conway.html

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,
http://www.nist.gov/srd/webguide/nist42-3/appa.htm

6) Eric Weisstein, Wallpaper groups, 
http://mathworld.wolfram.com/WallpaperGroups.html

7) David Hestenes, Point groups and space groups in geometric algebra, 
modelingnts.la.asu.edu/pdf/crystalsymmetry.pdf

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"):

                       e6

                    e4    e1
                       e7

               e3      e2     e5

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 and 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.   

I'll finish off with a quote from a quaternion skeptic:

"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

-----------------------------------------------------------------------
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,

(ħ1ħieħjeħke)/2,

(ħeħiħjħk)/2,

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


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