A point-line approach to the nonexistence of buildings of types \,H₃\, and \,H₄\,

We present a novel proof for the fact that thick spherical buildings of types H₃ H 3 and H₄ H 4 do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type H₃ H 3 . This way, we can write our arguments purely in the language of point-line geometries, not needing the theory of buildings. Assuming the existence of thick spherical buildings of type H₃ H 3 , we construct nontrivial root elations of generalized pentagons contained within them. This leads to a contradiction with Tits’s result on the nonexistence of Moufang generalized pentagons. Consequently, we obtain a new, direct, and geometric proof for the nonexistence of thick spherical buildings of types H₃ H 3 and H₄ H 4 , without invoking Tits’s extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types Bₙ B n and Cₙ C n , and known constructions for buildings of type Aₙ A n , this yields an alternative, elementary proof for the fact that all thick irreducible spherical buildings of rank 3 have the Moufang property, not using Tits’s extension theorem.