Automorphisms of Hilbert's non-desarguesian affine plane and its projective closure

  • 1 International University in Germany, Campus 1, 76646 Bruchsal, Germany.
  • 2 Institute of Geometry and Topology, University of Stuttgart, 70550 Stuttgart, Germany.


Hilbert constructed an early example of a non-desarguesian plane by modifying the lines of the real plane inside an ellipse. We employ methods from elementary algebraic geometry to show that the group of automorphisms of this plane leaves the ellipse invariant (unless the ellipse is a circle). Using this result, we determine the group of automorphisms of Hilbert's plane and some generalizations.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.