With g in SO(8) the product

g

defines a simple change of basis for the underlying octonion algebra

(the simplicity of the thing was pointed out to me by Shahn Majid).

In fact, given any n-dimensional algebra, and g in SO(n), this will

result in a change of basis.

What's interesting in the octonion case is that the X,Y-product also

gives a change of basis, and if we insist on restricting the change

in such a way that

e

then really interesting lattices start popping up, in particular the

laminated lattices of dimension 8 and 16. So what?

Well, my interest in in the division algebras arose from what seemed

obvious mathematical connections to physics at the Standard Model

level, and the fact that the division algebras were examples of

mathematical specialness, or, as I like to think of it,

When I finished my book I decided that mathematical resonance was

a good thing, and the more one knew about it, the better prepared one

would be to see how it connected to our physical reality. So, right

off the bat, yes, I have a strong faith or intuition that mathematical

resonance is intimately tied to the design of our physical reality.

To me this seems very pragmatic and down to earth. For others

pragmatism lies more with letting our physical reality itself tell

us the nature of its design. (Well, sure. That's reasonable. What it

has told me is that mathematical exceptionalness is of key importance

in that design; it requires rich mathematical structures to give rise

to a structure as rich as this universe.)

Ok, so...

Through browsing Conway and Sloane's addictive "Sphere Packings..."

book I became aware of the Leech lattice, which is the laminated lattice

of dimension 24. It is intimately related to the octonion algebra,

as are the laminated lattices of dimensions 8 and 16, but the Leech

lattice seems to be about the most exceptional, resonant lattice

there is, and I wish I could tell you why with some authority, but

it's been a while since I last looked at the book. (Indeed, I shouldn't

even be writing this; I should be working on a real life, but things

keep cropping up that get me thinking, and that together with an

inherent irresponsibility...).

Anyway, as mentioned, and as published in various places, I've used the

X,Y-product to generate really nice copies of the 8 and 16-dimensional

laminated lattices, and I hungered for the Leech, the 24-dimensional

laminated lattice. I felt and feel convinced that once understood,

it will provide some stunning insights into the exceptional nature

of octonion mathematics (see John Baez's mathematical weeklies, #104 I believe

for what provoked this new page).

Well, avoiding all coyness, my most recent idea was to see if this

algebraic change of basis,

g

could somehow be exploited to go beyond the 16-dimensional laminated

lattice up to the 24-dimensional case. The two thoughts I'd had

revolved around the octonion triple product (see Okubo's book), and

the exceptional Jordan algebra of 3x3 Hermitian matrices over the

octonions. I have lots of ideas, and very little time. I throw this into

the wind for what it's worth, and what better wind than the world wide wind.

Cheers

(The start of all this work lies in physical mathematics with my demonstration

that the mathematics of the division algebras comforms in many remarkable

ways with the mathematics of the standard model of quarks and leptons.

An introduction to that work can be found on my U(1)xSU(2)xSU(3) page.)

Note: that there is no page 13 is an accident.

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