# New model of non-Euclidean plane

## Piotr Błaszczyk

**Authors:** Piotr Błaszczyk and Anna Petiurenko

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 \(\mathbb{R}^*\), adheres to Hilbert’s axioms of absolute geometry, deviates from the parallel and Archimedean axioms, and is not hyperbolic. 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.
In our model, we present counterexamples to various versions of the parallel axiom, including those found in Euclid’s *Elements*, such as the circumcircle proposition. Additionally, we demonstrate the failure of the Wallis axiom and Legender’s axiom, showing discrepancies between their predictions and our model’s outcomes, thus challenging the equivalence of these axioms to the parallel axiom. We emphasize that besides the well-known non-Archimedean Pythagorean fields, there exist non-Archimedean Euclidean fields as well.
The model has unique educational advantages because, unlike standard models of non-Euclidean planes involving non-Euclidean representations of straight lines (Poincaré) or angles (Klein), the key idea of our model requires only the basics of Cartesian geometry and non-Archimedean fields.

## Bibliography

- Max Dehn,
*Legendre’schen Sätze über die Winkelsumme im Dreieck*,,vol. 53 (1900), no. 3, pp. 404–439.*Mathematische Annalen* - Piotr Bł,aszczyk, Anna Petiurenko,
*New model of non-Euclidean plane*,,2023.*arXiv:2302.12768 [math.HO]* - Piotr Bł,aszczyk, Anna Petiurenko,
*New model of non-Euclidean plane*,,2023.*arXiv:2302.12768 [math.HO]*