occasional meanderings in physics' brave new world

Name:
Location: New Zealand

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.