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Library of Congress Cataloging-in-Publication Data | ISBN 0-8160-4875-4 | Pages: 289 | English | PDF | Size: 1.37 MB
INTRODUCTION
Geometry is the study of shape, form, and space, and is of interest for its own sake as well as for its many applications in the sciences and the arts. Geometry is an integral part of mathematics, drawing upon all other areas in its development, and in turn contributing to the development of other parts of mathematics. The roots of geometry lie in many different cultures, including the ancient Vedic, Egyptian, Babylonian, Chinese, and Greek civilizations. The development of geometry as a deductive science began with Thales and reached maturity with the Elements of Euclid around 300 B.C.E. In the Elements, Euclid sets out a list of statements called postulates and common notions, which are fundamental, self-evident truths of geometry. These statements express common experiences and intuitions about space: It is flat, it extends infinitely in all directions, and it has everywhere the same structure. From the postulates and common notions, Euclid derives 465 propositions that form the body of Euclidean geometry. Euclid’s presentation is systematic, beginning with simple statements that depend only on the postulates and common notions and concluding with intricate and exquisite propositions. Each proposition is supported by a logical deduction or proof, based only on the postulates, the common notions, and other propositions that have already been proved. Because of its clarity and logical rigor, the axiomatic method of Euclidean geometry soon became the model for how a discipline of knowledge should be organized and developed. The theorems of Euclidean geometry came to be regarded as absolute truth and assumed a special role as the crowning achievement of human thought. Trying to emulate the success of Euclid, philosophers and scientists alike sought to find the self-evident principles from which their disciplines could be derived. During the 19th century, however, the discovery of non-Euclidean geometries brought an end to the special and unique role of Euclidean geometry. Nicolai Lobachevsky, János Bolyai, and Bernhard Riemann showed that there are other geometries, just as systematic and just as rigorous as Euclidean geometry, but with surprisingly different conclusions. These new geometries revolutionized the way mathematicians viewed mathematics and the nature of mathematical truth. A mathematical theory must be based on a collection of postulates or axioms, which are not self-evident truths, but instead are rules assumed to be true only in the context of a specific theory. Thus, the axioms of Euclidean geometry are different from those of elliptic geometry or hyperbolic geometry. A mathematical theorem, proven by logical deduction, is not true, but rather only valid, and even then valid only in the realm governed by the axioms from which it was derived. Different axioms give rise to different, even contradictory, theorems. The discovery of non-Euclidean geometries strengthened the study of geometry, giving it new vigor and vitality and opening new avenues of investigation. However, non-Euclidean geometries were
only one aspect of the development of geometry in the 19th century, which was truly a golden age for geometry, with projective geometry, affine geometry, vector spaces, and topology all emerging as important and independent branches of geometry. Today, after more gradual growth during the first part of the 20th century, geometry is once again flourishing in a new golden age. There are many factors supporting this renewed interest in geometry. Computers have become both an inspiration and tool for geometry, responsible for computational geometry, computer graphics, and robotics. Many newer areas of geometry, such as combinatorial geometry, discrete geometry, differential geometry, algebraic topology, dynamical systems and fractals, graph theory, and knot theory are the natural unfoldment of discoveries of earlier centuries. Other areas of current interest have their origin in applications: crystallography, frameworks, minimal surfaces, sphere packings, and the string theories of modern physics. All these areas and more are included in this handbook, which will serve students of geometry, beginner or advanced, and all those who encounter geometric ideas in their pursuit of the sciences, arts, technology, or other areas of mathematics.
Euclidean geometry, trigonometry, projective geometry, analytic geometry, non-Euclidean geometry, vectors, differential geometry, topology, computational geometry, combinatorial geometry, knot theory, and graph theory are just some of the branches of geometry to be found here. The selection of topics has focused on what a student would first encounter in any of these areas. Terms from other parts of mathematics that are needed to understand geometric terms are included in the glossary, but the reader is referred to The Facts On File Algebra Handbook and The Facts On File Calculus Handbook for a more comprehensive discussion of these topics. Nowhere is it assumed here that the reader has studied calculus.
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