Thales Meets Poincaré

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According to [2, 39, p. 100], Euclidean geometry and (BolyaiLobachevskian) geometry are the only types of absolute (also known as neutral) plane geometry. These geometries are distinguished by the number of lines that pass through a given point and are parallel to a given line. This number may be 1 (the Euclidean case) or greater than 1 (the case): see axioms E and BL [2, p. 197] or postulates (I) and (II) [3, p. 317]. Euclidean and geometiy may also be distinguished by considering the sum of the (radian) measures of the (interior) angles any triangle. Indeed, Euclidean geometry, this sum is T, while geometry, this sum is less than v (and may vary from triangle to triangle): see [2, 2, p. 264 and 2, p. 278], [3, p. 118], [4, Theorems 10.1 and 10.3]. The main purpose of this paper is to see whether these two geometries may also be distinguished by Thales' [2, 13, p. 269], a classical result on triangles Euclidean geometry whose proof depends on the behavior of parallel lines and similar triangles the Euclidean setting. To study the possible validity of Thales' geometry, we shall work inside the Poincare half-plane model, whose salient features are reviewed Section 2. There is no loss of generality using this model, as geometry is categorical [2, Proposition 7, p. 345], the sense that all its models are isomorphic. One benefit of using the Poincare model is that the question of a possible hyperbolic Thales' Theorem comes down to asking whether two specific numbers are equal; the calculations the Example Section 2 give a negative answer. This accomplishes our main purpose: Thales' does distinguish Euclidean geometry from geometry. But we can say more. In Section 3, continuing to work the Poincare model, we further analyze the two numbers that need to be calculated and compared testing for a hyperbolic Thales' Theorem. By subjecting the triangular data to a limiting process that is designed to, so to speak, minimize the difference between the and Euclidean metrics, we show (see the Section 3) that the ratio of the two numbers question has limit 1. Thus, although Thales' is false geometry, we can say that it holds in the limit, for a suitable Euclidean-seeking limit process. We hope that this work finds use as enrichment material model-oriented courses on absolute geometry. However, the technical details Section 3 depend on a more central part of the curriculum, namely real-analytic functions, as studied advanced calculus. The material being reinforced includes the Binomial for power series, the calculations involved multiplying, dividing, or composing functions defined by power series, and the Maclaurin series for ln(1 + x). A suitable reference for this material is [1].