Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Friday, February 08, 2008

M Theory Lesson 155

It is often said that the Fano plane is the smallest projective geometry, because there is a silly axiom ruling out diagrams with fewer points. Well, this axiom has the reasonable motivation, arising from perspective drawing, that a third point on a line specifies its direction.

But if the Fano plane represents the units of the octonions, we should have a geometry that represents the units of the quaternions, not to mention the complex numbers. The triangle clearly fills this role: for any 2 points there is only one line running through them, and given any 2 lines there is only 1 point incident upon both. Similarly, a single point and line represents the complex number $i$. Since we allow a one point field $\mathbb{F}_{1}$, this should be a perfectly legitimate projective geometry. $PG(2,1)$ has only one element, because there is only one element in the field.


Blogger kneemo said...

Yup, you got it. The Fano plane can be reduced to the triangles corresponding to all the quaternionic subalgebras of the octonions.

February 09, 2008 4:19 PM  
Blogger Kea said...

Hi kneemo. Good to see you. Of course when I mention projective geometry, I'm thinking about Motives, as always!

February 09, 2008 5:48 PM  

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