Hexagon - How to draw
Discrete Mathematics 199 (1999) 161-l 71
How to draw a hexagon
hwtitut fi7r Analysis,
Andreas E. Schroth”
TU Braunschweiy, Pockelsstr. 14, D-38106
Received 13 October 1997; revised 16 March 1998; accepted 18 May 1998
Pictures of the Gz(2) hexagon and its dual are presented. A way to obtain these pictures is
discussed. @ 1999 Elsevier Science B.V. All rights reserved
Keywords: Generalised hexagon; Generalised polygon; Coordinatisation: Models of geometries
1. Introduction
This paper presents pictures of the two classical generalised hexagons over the field
with two elements. Information on these generalised hexagons is employed so that the
pictures make some properties of these hexagons obvious. Familiarity with the theory of
generalised polygons is not assumed. Perhaps this paper will convince the uninitiated
reader that even new and seemingly abstract geometrical structures sometimes have
nice visual presentations.
Generalised polygons were introduced by Tits in [7]. They serve, among other things,
as a geometric realisation of certain groups. Generalised triangles are essentially projec-
tive planes, generalised quadrangles are also known as polar spaces of rank 2. Classical
(thick) generalised n-gons, n >2, exist only for n E {3,4,6, S}. By [3] also thick finite
generalised n-gons, n > 2, exist only for n E { 3,4,6,8}. If the pictures and discussions
presented here whets your appetite for generalised polygons, I recommend [6,9] for
further reading.
There is a well-known picture of the smallest projective plane, also known as the
Fano plane (Fig. 1). A picture of the smallest generaiised quadrangle appeared first on
the cover of a book on generalised triangles [4] and reminds of an ordinary pentagon
(Fig. 2). The picture is due to S. Payne, who calls it doily for obvious reasons. More
stunning pictures of geometries may be found in Polster’s Geometrical Picture Book [5].
0012-365X/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved
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