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<div style="text-align: center;"><big><big><big><span style="font-weight: bold;">Coxeter versus Fano</span></big></big></big><br />
<br style="font-weight: bold;" />
<span style="font-weight: bold;">by Steven H. Cullinane,
July 8, 2010<br />
</span></div>
<div style="text-align: center;"><br />The following excerpts
from Coxeter's <span style="font-style: italic;">Projective
Geometry</span><br />
sketch his attitude toward geometry in characteristic two.<br />
"... we develop a self-contained account... made<br />
more 'modern' by allowing&nbsp;the field to be general<br />
(though not of characteristic 2) instead of real or complex."<br />
<br />
The&nbsp;"modern" in quotation marks may have been an oblique<br />
reference to Segre's&nbsp;<span style="font-style: italic;">Lectures on 
Modern Geometry</span> (1948, 1961).<br />(See Coxeter's reference 15 
below.)<br />
<br />&nbsp;<a href="http://www.log24.com/log/pix10A/100708-CoxeterCharTwo.gif"><img alt="Image-- Coxeter on the Fano Plane" src="versus_files/100708-CoxeterCharTwo.gif" border="0" /></a></div>
<div style="text-align: center;">
<p style="text-align: center; font-weight: bold;"><big>POSTSCRIPT</big></p><p style="text-align: center; font-weight: bold;">"It
is interesting to see
what happens&#8230;."</p>
<p style="text-align: center; font-weight: bold;">Another
thing that happens
if 1 + 1 = 0 &#8212;</p>

<p style="text-align: center; font-weight: bold;">It
is no longer true that
every finite reflection group<br />
is a Coxeter group (provided we use Chevalley's<br />
<a href="refl168.html">fixed-hyperplane
definition</a> of "reflection").</p><p style="text-align: center; font-weight: bold;">&nbsp; &nbsp; </p>
</div>
</body></html> "reflection").</p><p style="text-align: center; font-weight: bold;">&nbsp; &nbsp; </p>
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