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An Alternative Way to Generalize the Pentagon


  • Contract grant sponsor: Spanish Ministry of Science and Education; Contract grant sponsor: Catalan Research Council (S. B.); Contract grant sponsor: Spanish MEC projects ARES; Contract grant number: CSD2007-00004; Contract grant sponsor: RIPUP; Contract grant number: TIN2009-11689; Contract grant sponsor: Government of Catalonia; Contract grant number: 2009 SGR 1135; Contract grant sponsor: FPU; Contract grant numbers: BOEs 17/11/2009 and 11/10/2010 (K. S.) .


We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.