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2000 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Geometry: Euclid and Beyond

verfasst von: Robin Hartshorne

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
Alittle after the time of Plato, but before Archimedes, in ancient Greece, a man named Euclid wrote the Elements, gathering and improving the work of his predecessors Pythagoras, Theaetetus, and Eudoxus into one magnificent edifice. This book soon became the standard for geometry in the classical world. With the decline of the great civilizations of Athens and Rome, it moved eastward to the center of Arabic learning in the court of the caliphs at Baghdad.
Robin Hartshorne
Chapter 1. Euclid’s Geometry
Abstract
In this chapter we create a common experience by reading portions of Euclid’s Elements. We discuss the nature of proof in geometry. We introduce a particular way of recording ruler and compass constructions so that we can measure their complexity. We discuss what are presumably familiar notions from high school geometry as it is taught today. And then we present Euclid’s construction of the regular pentagon and discuss its proof.
Robin Hartshorne
Chapter 2. Hilbert’s Axioms
Abstract
Our purpose in this chapter is to present (with minor modifications) a set of axioms for geometry proposed by Hilbert in 1899. These axioms are sufficient by modern standards of rigor to supply the foundation for Euclid's geometry. This will mean also axiomatizing those arguments where he used intuition, or said nothing. In particular, the axioms for betweenness, based on the work of Pasch in the 1880s, are the most striking innovation in this set of axioms.
Robin Hartshorne
Chapter 3. Geometry over Fields
Abstract
Beginning with the familiar example of the real Cartesian plane, we show how to construct a geometry satisfying Hilbert’s axioms over an abstract field. The axioms of incidence are valid over any field (Section 14). For the notion of betweenness we need an ordered field (Section 15). For the axiom (C1) on transferring a line segment to a given ray, we need a property (*) on the existence of certain square roots in the field F. To carry out Euclidean constructions, we need a slightly stronger property (**)-see Section 16.
Robin Hartshorne
Chapter 4. Segment Arithmetic
Abstract
Segment arithmetic allows us to complete the chain of logical connections between an abstract geometry satisfying axioms studied in Chapter 2 with the geometries over fields studied in Chapter 3. We will show how to define addition and multiplication of line segments in a Hilbert plane satisfying the parallel axiom (P). In this way, the congruence equivalence classes of line segments become the positive elements of an ordered field F (Section 19). Using this field F we can recover the usual theory of similar triangles (Section 20).
Robin Hartshorne
Chapter 5. Area
Abstract
Looking at Euclid’s theory of area in Books I-IV, Hilbert saw how to give it a solid logical foundation. We define the notion of equal content by saying that two figures have equal content if we can transform one figure into the other by adding and subtracting congruent triangles (Section 22). We can prove all the properties of area that Euclid uses, except that “the whole is greater than the part.” This is established only when we relate the geometrical notion of equal content to the notion of a measure of area function (Section 23).
Robin Hartshorne
Chapter 6. Construction Problems and Field Extensions
Abstract
During the earlier parts of this book, we started always from Euclid’s geometry, developing and expanding it using our modern mathematical awareness. Because of the construction of the field of segment arithmetic, one could even argue that the use of fields in Chapter 4 arises naturally from the geometry. In this chapter, however, we will make use of modern algebra, the theory of equations and field extensions, and in particular the Galois theory, as it developed in the late nineteenth and early twentieth centuries.
Robin Hartshorne
Chapter 7. Non-Euclidean Geometry
Abstract
Certainly one of the greatest mathematical discoveries of the nineteenth century was that of non-Euclidean geometry: seen but not revealed by Gauss, and developed in all its glory by Bolyai and Lobachevsky. The purpose of this chapter is to give an account of this theory, but we do not always follow the historical development. Rather, with hindsight we use those methods that seem to shed the most light on the subject. For example, continuity arguments have been replaced by a more axiomatic treatment.
Robin Hartshorne
Chapter 8. Polyhedra
Abstract
Polyhedra are solid figures bounded by plane polygons. Most famous among these are the five regular, or Platonic, solids, identified classically with the four elements, earth, air, fire, water, and the whole universe. Euclid begins his Elements with the construction of an equilateral triangle (I.1) and ends in Book XIII with the construction of these regular solids. It has been suggested that Euclid’s purpose in writing the Elements was to fully elucidate the geometry behind these five figures.
Robin Hartshorne
Backmatter
Metadaten
Titel
Geometry: Euclid and Beyond
verfasst von
Robin Hartshorne
Copyright-Jahr
2000
Verlag
Springer New York
Electronic ISBN
978-0-387-22676-7
Print ISBN
978-1-4419-3145-0
DOI
https://doi.org/10.1007/978-0-387-22676-7