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E-raamat: Geometry Through History: Euclidean, Hyperbolic, and Projective Geometries

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  • Formaat: PDF+DRM
  • Sari: Mathematics and Statistics
  • Ilmumisaeg: 21-Mar-2018
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319741352
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Geometry Through History: Euclidean, Hyperbolic, and Projective GeometriesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Mathematics and Statistics
  • Ilmumisaeg: 21-Mar-2018
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319741352
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Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises.

For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid’s work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the field.


Arvustused

While the book covers some advanced topics, it is also remarkably accessible: anyone with a year of college-level mathematics and a love of geometry will find value in this text, as Dillon (Kennesaw State Univ.) explores axioms, proofs, and an array of fascinating problems. One of the most appealing aspects of this book is the author's way of introducing projective geometry to pique the reader's interest in modern algebra. Summing Up: Recommended. Lower-division undergraduates and above. (J. A. Bakal, Choice, Vol. 56 (03), November, 2018)

It discusses a lot of interesting topics in geometry, it is clearly written and should be accessible to its target audience of students . it offers students an opportunity to see the relationship between geometry and other areas of mathematics, including abstract and linear algebra. If you are teaching a course in geometry at the college level and the topics discussed here are compatible with your course, this book certainly merits aserious look. (Mark Hunacek, MAA Reviews, October, 2018) This is a wonderful new book that could be used as a textbook for geometry, but would be of interest to any mathematician whose work includes some geometry or who is interested in the history of mathematical developments. The book offers a deep, thorough, and entertaining presentation of geometry through history. (Tara D. Taylor, Mathematical Reviews, January, 2019)

1 The Elements of Euclid
1(50)
1.1 Introduction
1(3)
1.2 Background on Euclid and The Elements
4(7)
1.3 Basic Constructions
11(3)
1.4 Triangles: Propositions I.4-26
14(16)
1.5 Parallels: Propositions I.27-32
30(7)
1.6 Areas: Propositions I.33--46
37(7)
1.7 The Pythagorean Theorem
44(4)
1.8 Euclid's Assumptions
48(3)
2 Neutral Geometry
51(26)
2.1 The Problem of the Parallels
51(3)
2.2 All For One and One For All
54(12)
2.3 Archimedes' Axiom and The Angle Sum in a Triangle
66(10)
2.4 A Word About the Geometry on a Sphere
76(1)
3 The Hyperbolic Plane
77(20)
3.1 Introduction
77(4)
3.2 Polygons, Perpendiculars, and Parallels
81(16)
4 Hilbert's Grundlagen
97(26)
4.1 Axioms of Incidence
98(3)
4.2 Axioms of Order
101(3)
4.3 Axioms of Congruence
104(12)
4.4 The Axiom of Parallels
116(4)
4.5 Axioms of Continuity
120(3)
5 More Euclidean Geometry
123(44)
5.1 Euclid Beyond Book I
123(13)
5.2 Similar Triangles and the Power of a Point
136(6)
5.3 Circular Inversion
142(12)
5.4 Triangles, Centers, and Circles
154(11)
5.5 Summary
165(2)
6 Models for the Hyperbolic Plane
167(20)
6.1 Introduction
167(1)
6.2 Beltrami
168(2)
6.3 The Poincare Disk
170(17)
7 Affine Geometry
187(54)
7.1 Introduction
187(1)
7.2 Some Linear Algebra
187(16)
7.3 Fields
203(4)
7.4 Linear Transformations
207(11)
7.5 Affine Geometry
218(14)
7.6 The Affine Group
232(9)
8 An Introduction to Projective Geometry
241(32)
8.1 Introduction
241(1)
8.2 Projective Space
242(12)
8.3 The Theorem of Desargues
254(4)
8.4 Harmonic Sequences
258(4)
8.5 Transformations and Pappus's Theorem
262(5)
8.6 Homogeneous Coordinates
267(6)
9 Algebraic Curves
273(40)
9.1 Introduction
273(1)
9.2 Polynomials, Rings, and Integral Domains
274(10)
9.3 Factoring and Division
284(7)
9.4 The Resultant
291(8)
9.5 Homogeneous Polynomials
299(4)
9.6 Bezout's Theorem
303(10)
10 Rotations and Quaternions
313(20)
10.1 Introduction
313(1)
10.2 Background on Rn
313(5)
10.3 Rotations in R2
318(3)
10.4 Rotations in R3
321(12)
List of Figures 333(8)
Bibliography 341(2)
Index 343
Meighan Dillon is a Professor of Mathematics at Kennesaw State University. She is a lifelong educator, whose research in infinite-dimensional Lie algebras is complemented by explorations in pedagogy and the use of projective geometry in art.
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