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Coxeter versus Fano

by Steven H. Cullinane, July 8, 2010

The following excerpts from Coxeter's Projective Geometry
sketch his attitude toward geometry in characteristic two.
"... we develop a self-contained account... made
more 'modern' by allowing the field to be general
(though not of characteristic 2) instead of real or complex."

The "modern" in quotation marks may have been an oblique
reference to Segre's Lectures on Modern Geometry (1948, 1961).
(See Coxeter's reference 15 below.)

 Image-- Coxeter on the Fano Plane


"It is interesting to see what happens…."

Another thing that happens if 1 + 1 = 0 —

It is no longer true that every finite reflection group
is a Coxeter group (provided we use Chevalley's
fixed-hyperplane definition of "reflection").