John Donnelly,
A Model of Continuous Plane Geometry that is Nowhere Geodesic,
Forum Geometricorum, 18 (2018) 255-273.

Abstract. We construct a model M_1 of plane geometry that satisfies all of Hilbert's axioms for the euclidean plane (with the exception of Sided-Angle-Side), yet in which the geodesic line segment connecting any two points A and B is never the shortest path from A to B.  Moreover, the model M_1 is continuous in the sense that it satisfies both the Ruler Postulate and Protractor Postulate from Birkhoff's set of axioms for the euclidean plane.


 [ps file] [pdf]

Return to Forum Geom., 18 (2018) Table of Contents