Geometry From Euclid to Knots

Organization The main purpose of this book is to provide prospective high school mathematics teachers with the geometric background they need. Its core, consisting of Chapters 2 to 5, is therefore devoted to a fairly formal (that is, axiomatic) development of Euclidean geometry. Chapters 6, 7, and 8 complement this with an exposition of transformation geometry. The first chapter, which introduces teachers-to-be to non-Euclidean geometries, provides them with a perspective meant to enhance their appreciation of axiomatic systems. The much more informal Chapters 9 through 12 are meant to give students a taste of some more recent geometric discoveries. The development of synthetic Euclidean geometry begins by following Euclid'sElementsvery closely. This has the advantage of convincing students that they are learning "the real thing." It also happens to be an excellent organization of the subject matter. Witness the well-known fact that the first 28 propositions are all neutral. These subtleties might be lost on the typical high school student, but familiarity with Euclid's classic text must surely add to the teacher's confidence and effectiveness in the classroom. I am also in complete agreement with the sentiments Todhunter displayed in the previous excerpt: No other system of teaching geometry is better than Euclid's, provided, of course that his list of propositions is supplemented with a sufficient number of exercises. Occasionally, though, because some things have changed over two millennia, or else because of errors in theElements,it was found advantageous either to expound both the modern and ancient versions in parallel or else to part ways with Euclid. In order to convince prospective teachers of the need to prove "obvious" propositions, the axiomatic development of Euclidean geometry is preceded by the informal description of both spherical and hyperbolic geometry. The trigonometric formulas of these geometries are included in order to lend numerical substance to these alternate and unfamiliar systems. The part of the course dedicated to synthetic geometry covers the standard material about triangles, parallelism, circles, ratios, and similarity; it concludes with the classic theorems that lead to projective geometry. These lead naturally to a discussion of ideal points and lines in the extended plane. Experience indicates that the nonoptional portions of the first five chapters can be completed in about three quarters of a one semester course. During that time the typical weekly homework assignment called for about a dozen proofs. Chapters 6 and 7 are concerned mostly with transformation geometry and symmetry. The planar rigid motions are completely and rigorously classified. This is followed by an informal discussion of frieze patterns and wallpaper designs. Inversions are developed formally and their utility for both Euclidean and hyperbolic geometry is explained. The exposition in Chapters 8 through 12 is informal in the sense that few proofs are either offered or required. Their purpose is to acquaint students with some of the geometry that was developed in the last two centuries. Care has been taken to supply a great number of calculational exercises that will provide students with hands-on experience in these advanced topics. The purpose of Chapter 8 is threefold. First there is an exposition of some interesting facts, such as Euler's equation and the Platonic and Archimedian solids. This is followed by a representation of the rotational symmetries of the regular solids by means of permutations, a discussion of their symmetry groups, and a visual definition of isomorphism. Both of these discussions aim to develop the prospective teacher's ability to visualize three dimensional phenomena. Finally comes the tale of Monstrous Moonshine. Chapter 9 consists of a short introduction to the notions of homeomorphism and isotopy. Chapter 10 acquaints students withStahl, Saul is the author of 'Geometry From Euclid to Knots', published 2002 under ISBN 9780130329271 and ISBN 0130329274.