The secondary role of parallelism in geometry and the real difference between euclidean and hyperbolic geometry

Author

Panagiotis Pittas, New Jersey Institute of Technology

Document Type

Thesis

Date of Award

5-31-1989

Degree Name

Master of Science in Applied Mathematics - (M.S.)

Department

Mathematics

First Advisor

Denis L. Blackmore

Second Advisor

John Tavantzis

Third Advisor

Jacob Klapper

Abstract

After a brief survey of the foundations of absolute geometry, we state certain definitions (that of an angle and an ordering between them) and properties (Theorem 1, lemma 1) of the absolute plane. Then. by adding the Playfair axiom (Statement E₁) to the axioms of absolute geometry, we obtain the Euclidean plane. We introduce statement E₂ and then we prove that it is equivalent to E₁. We then reason that E₂ is superior to E₁ in terms of geometric content, that it expresses a characteristic property of the Euclidean plane, and we show that the parallel axiom problem is essentially an angle containment problem and suggest that E₂ should take the place of E₁ in the foundations of Euclidean geometry. We then add the Lobachevskian axiom (Statement H₁) to the axioms of absolute geometry and thus obtain the hyperbolic plane. We introduce statement H₂ (negation of E₂) equivalent to H₁, and suggest that in the hyperbolic case H₁ should be taken as an axiom. Then using H₂ we prove the characteristic theorem of hyperbolic geometry (Theorem 3), which if compared to statement E₂ defines the most striking difference between Euclidean and hyperbolic geometry.

Recommended Citation

Pittas, Panagiotis, "The secondary role of parallelism in geometry and the real difference between euclidean and hyperbolic geometry" (1989). Theses. 2869.
https://digitalcommons.njit.edu/theses/2869