Author: Harold E. Wolfe
Publisher: Courier Corporation
Release Date: 2013-09-26
College-level text for elementary courses covers the fifth postulate, hyperbolic plane geometry and trigonometry, and elliptic plane geometry and trigonometry. Appendixes offer background on Euclidean geometry. Numerous exercises. 1945 edition.
Author: Evan Chen
Publisher: The Mathematical Association of America
Release Date: 2016-05-02
This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class.
This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
Foundations of Three-Dimensional Euclidean Geometry provides a modern axiomatic construction of three-dimensional geometry, in an accessible form. The method of this book is a graduated formulation of axioms, such that, by determining all the geometric spaces which satisfy the considered axioms, one may characterize the Euclidean space up to an isomorphism. A special feature of Foundations of Three-Dimensional Euclidean Geometry is the introduction of the parallel axiom at an early stage of the discussion, so that the reader can see what results may be obtained both with and without this important axiom. The many theorems, drawings, exercises, and problems richly enhance the presentation of material. Foundations of Three-Dimensional Euclidean Geometry is suitable as a textbook for a one- or two-semester course on geometry or foundations of geometry for undergraduate and beginning graduate students. Mathematics majors in M.A.T. programs will find that this exposition of a classical subject will contribute greatly to their ability to teach geometry at all levels; and logicians, philosophers, and engineers will benefit from this book's applications to their own interests. Book jacket.
Author: Sotirios E. Louridas
Publisher: Springer Science & Business Media
Release Date: 2014-07-08
"Problem-Solving and Selected Topics in Euclidean Geometry: in the Spirit of the Mathematical Olympiads" contains theorems which are of particular value for the solution of geometrical problems. Emphasis is given in the discussion of a variety of methods, which play a significant role for the solution of problems in Euclidean Geometry. Before the complete solution of every problem, a key idea is presented so that the reader will be able to provide the solution. Applications of the basic geometrical methods which include analysis, synthesis, construction and proof are given. Selected problems which have been given in mathematical olympiads or proposed in short lists in IMO's are discussed. In addition, a number of problems proposed by leading mathematicians in the subject are included here. The book also contains new problems with their solutions. The scope of the publication of the present book is to teach mathematical thinking through Geometry and to provide inspiration for both students and teachers to formulate "positive" conjectures and provide solutions.
This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound “geometrical roots” and numerous applications to modern nonlinear problems, it is treated as a universal “object” of investigation, connecting many of the problems discussed. The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry.
This unique book overturns our ideas about non-Euclidean geometry and the fine-structure constant, and attempts to solve long-standing mathematical problems. It describes a general theory of "recursive" hyperbolic functions based on the "Mathematics of Harmony," and the "golden," "silver," and other "metallic" proportions. Then, these theories are used to derive an original solution to Hilbert's Fourth Problem for hyperbolic and spherical geometries. On this journey, the book describes the "golden" qualitative theory of dynamical systems based on "metallic" proportions. Finally, it presents a solution to a Millennium Problem by developing the Fibonacci special theory of relativity as an original physical-mathematical solution for the fine-structure constant. It is intended for a wide audience who are interested in the history of mathematics, non-Euclidean geometry, Hilbert's mathematical problems, dynamical systems, and Millennium Problems. Contents:The Golden Ratio, Fibonacci Numbers, and the "Golden" Hyperbolic Fibonacci and Lucas FunctionsThe Mathematics of Harmony and General Theory of Recursive Hyperbolic FunctionsHyperbolic and Spherical Solutions of Hilbert's Fourth Problem: The Way to the Recursive Non-Euclidean GeometriesIntroduction to the "Golden" Qualitative Theory of Dynamical Systems Based on the Mathematics of HarmonyThe Basic Stages of the Mathematical Solution to the Fine-Structure Constant Problem as a Physical Millennium ProblemAppendix: From the "Golden" Geometry to the Multiverse Readership: Advanced undergraduate and graduate students in mathematics and theoretical physics, mathematicians and scientists of different specializations interested in history of mathematics and new mathematical ideas.
Wie bewegt sich ein Massenpunkt in einem Gebiet, an dessen Rand er elastisch zurückprallt? Welchen Weg nimmt ein Lichtstrahl in einem Gebiet mit ideal reflektierenden Rändern? Anhand dieser und ähnlicher Fragen stellt das vorliegende Buch Zusammenhänge zwischen Billard und Differentialgeometrie, klassischer Mechanik sowie geometrischer Optik her. Dabei beschäftigt sich das Buch unter anderem mit dem Variationsprinzip beim mathematischen Billard, der symplektischen Geometrie von Lichtstrahlen, der Existenz oder Nichtexistenz von Kaustiken, periodischen Billardtrajektorien und dem Mechanismus für Chaos bei der Billarddynamik. Ergänzend wartet dieses Buch mit einer beachtlichen Anzahl von Exkursen auf, die sich verwandten Themen widmen, darunter der Vierfarbensatz, die mathematisch-physikalische Beschreibung von Regenbögen, der poincaresche Wiederkehrsatz, Hilberts viertes Problem oder der Schließungssatz von Poncelet.
Author: George Jennings
Publisher: Springer Science & Business Media
Release Date: 1994
This introduction to modern geometry differs from other books in the field due to its emphasis on applications and its discussion of special relativity as a major example of a non-Euclidean geometry. Additionally, it covers the two important areas of non-Euclidean geometry, spherical geometry and projective geometry, as well as emphasising transformations, and conics and planetary orbits. Much emphasis is placed on applications throughout the book, which motivate the topics, and many additional applications are given in the exercises. It makes an excellent introduction for those who need to know how geometry is used in addition to its formal theory.