We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{ R}^*$. The model enables one to represent the negation of equivalent versions of the parallel axiom, such as the existence of the circumcircle of a triangle, and Wallis' or Lagendre's axioms, as well as the difference between non-Euclidean and hyperbolic planes. The model has unique educational advantages as expounding its crucial ideas requires only the basics of Cartesian geometry and non-Archimedean fields.
Categories: math.HO