Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 1, pp. 199-201 (2000)

Affine Super-Pythagorean Geometry

Victor Pambuccian

Department of Integrative Studies, ASU West, P. O. Box 37100, Phoenix AZ 85069-7100, USA. e-mail:

Abstract: Super-Pythagorean fields can be thought of as arising from a weakening of the requirement of Euclideanity, while preserving the requirement of Pythagoreanity. Given that Pythagorean and Euclidean fields arise naturally as coordinate fields of planes with free mobility, and of those satisfying the circle axiom, it is natural to ask whether super-Pythagorean fields have a natural geometric meaning. In [P] we have provided axiom systems for Euclidean planes with super-Pythagorean coordinate fields with or without order. Two axioms of those axiom systems, A7 and A8, are artificial, in the sense that they do not reflect simple geometric intentions. This shortcoming will be remedied in this paper, where we shall display two axioms with clear geometric content to replace the above two. Moreover, in their new formulations these axioms will be affine statements. [P] Pambuccian, V.: Euclidean superpythagorean geometry. Beitr. Algebra Geom. 39 (2), (1998), 255-258.

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