### 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.

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.

## 2 Comments:

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

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

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