On the non-Paschian ordered planes

TL;DR

This paper introduces a non-Paschian ordered plane model based on betweenness that cannot be derived from a linear point ordering, exhibiting elliptic geometry properties such as triangles with angle sums exceeding π.

Contribution

It presents a novel non-Paschian plane model where betweenness is independent of point ordering, and defines angles and triangles with elliptic characteristics.

Findings

The plane cannot define segment congruence.

Angles and triangles are definable with elliptic properties.

The sum of triangle angles exceeds π in this model.

Abstract

We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle and angle measure, respectively. With respect to our definitions the plane has an elliptic character, meaning that the sum of the angles of a triangle is greater than $\pi$.