The valuations of the near hexagons related to the Witt designs S (5,6,12) and S (5,8,24)

Valuations of dense near polygons were introduced in 16. In the present paper, we classify all valuations of the near hexagons 1 and 2 , which are related to the respective Witt designs S (5,6,12) and S (5,8,24). Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to 1 or 2 , then H is classical in S . We will use this result to determine all dense near octagons that contain a hex isomorphic to 1 or 2 . As a by‐product, we obtain a purely geometrical proof for the nonexistence of regular near 2 d ‐gons, d  ≥ 4, whose parameters s , t , t i (0 ≤  i  ≤  d ) satisfy ( s , t 2 , t 3 ) = (2, 1, 11) or (2, 2, 14). The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 214–228, 2006