Euler’s Theorem as the Path towards Geometry

The roots of this article reside as much in curricular pragmatism as in programmatic, ideologic convictions. By curricular pragmatism we mean the anankian[1] drive for one single course in Mathematics for Architecture Students at the Technion – Israel Institute of Technology, course that would encapsulate as much formative knowledge as deemed feasible. The need for an “Object Oriented”, fast approach, compact course, arises from curricular constraints: once the second half of a tandem of three-hours per week, semestrial courses (the first one comprised elements of Matrix Algebra and Introduction to Calculus), it was reduced to a single semestrial course, of two weekly-hours (the Algebra-Calculus half being abandoned completely). Moreover, this reduction in scope was accompanied by an augmentation of the Syllabus: while the previous geometric course comprised only symmetry (albeit treated in some detail[2]), the new Course was envisioned as a comprehensive introduction to incidence and symmetry of geometrical objects, an approach that is commonly hold to represents the corner-stone, the main goal of a Course of this type. (The chosen basic reference text being [Baglivo and Graver, 1983]). Thus, apart and beyond the absolute importance of Euler’s Formula relative to the corpus of classical mathematics and the role it played in its development (Betti numbers[3] and Homology in general, on one hand and the Global Gauss-Bonnet Theorem[4] on the other hand being, not the least of its outshoots), its simplicity, yet potency (in the sense of representing an jumping board, an opening towards a variety of subjects belonging to the fields of Topology and Geometry) recommend Euler’s Theorem as natural candidate for a cornerstone, a red-thread running along and directing the whole Course. Since a second purpose of any such course is to help develop geometric intuition and spatial imaginative powers, Euler’s Theorem introduces one effortlessly in the realms of geometric creativity, by its natural generalizations into two directions: (a) high genus and non orientable surfaces s.a. the Klein Bottle and The Projective Plane (thus representing an excursum in Non-Euclidian geometries and also into patterns and tilings), and (b) Star and Uniform Polyhedra. (We shall indicate how and where these objects and ideas arise.)