The Four Pillars of Geometry
(Sprache: Englisch)
This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the...
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Produktinformationen zu „The Four Pillars of Geometry “
This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises
Klappentext zu „The Four Pillars of Geometry “
Many people think there is only one "right" way to teach geometry. For two millennia, the "right" way was Euclid's way, and it is still good in many respects. But in the 1950s the cry "Down with triangles!" was heard in France and new geometry books appeared, packed with linear algebra but with no diagrams. Was this the new "right" way, or was the "right" way something else again, perhaps transformation groups? In this book, I wish to show that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. Two chapters are devoted to each approach: The ?rst is concrete and introductory, whereas the second is more abstract. Thus, the ?rst chapter on Euclid is about straightedge and compass constructions; the second is about axioms and theorems. The ?rst chapter on linear algebra is about coordinates; the second is about vector spaces and the inner product.
This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract.
Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter.
Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter.
Inhaltsverzeichnis zu „The Four Pillars of Geometry “
- Preface- Straightedge and compass
- Euclid's approach to geometry
- Coordinates
- Vectors and Euclidean spaces
- Perspective
- Projective planes
- Transformations
- Non-Euclidean geometry
- References
- Index.
Autoren-Porträt von John Stillwell
Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract. The book contains more than 100 figures, and a large selection of exercises in each chapter. Bibliographische Angaben
- Autor: John Stillwell
- 2005, XI, 229 Seiten, Masse: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 0387255303
- ISBN-13: 9780387255309
- Erscheinungsdatum: 09.08.2005
Sprache:
Englisch
Rezension zu „The Four Pillars of Geometry “
From the reviews: "This is an introductory book on geometry, easy to read, written in an engaging style. The authora (TM)s goal is a ] to increase onea (TM)s overall understanding and appreciation of the subject. a ] Along the way, he presents elegant proofs of well-known theorems a ] . The advantage of the authora (TM)s approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)
"The pillars of the title are a ] Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. a ] The writing style is both student-friendly and deeply informed. The pleasing brevity of the book a ] makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors a ] . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)
"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. a ] Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. a ] The book can be recommended to be used in undergraduate courses on geometry a ] ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)
"Any new mathematics textbook by John Stillwell isworth a serious look. Stillwell is the prolific author of more than half a dozen textbooks a ] . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an
... mehr
in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)
"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. a ] the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte fA1/4r Mathematik, Vol. 150 (3), 2007)
"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. a ] The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)
"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. a ] the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte fA1/4r Mathematik, Vol. 150 (3), 2007)
"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. a ] The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)
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Pressezitat
From the reviews:"This is an introductory book on geometry, easy to read, written in an engaging style. The author's goal is ... to increase one's overall understanding and appreciation of the subject. ... Along the way, he presents elegant proofs of well-known theorems ... . The advantage of the author's approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)
"The pillars of the title are ... Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. ... The writing style is both student-friendly and deeply informed. The pleasing brevity of the book ... makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors ... . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)
"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. ... Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. ... The book can be recommended to be used in undergraduate courses on geometry ... ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)
"Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks ... . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an
... mehr
in-depth,semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)
"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. ... the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)
"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. ... The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)
"The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. ... The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. ... The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view." (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)
"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. ... the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)
"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. ... The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)
"The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. ... The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. ... The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view." (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)
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