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E-raamat: Symmetry: A Mathematical Exploration

  • Formaat: PDF+DRM
  • Ilmumisaeg: 02-Dec-2011
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461402992
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 02-Dec-2011
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461402992
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This textbook is perfect for a math course for non-math majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas.  It includes many topics commonly found in sampler courses, like Platonic solids, Eulers formula, irrational numbers, countable sets, permutations, and a proof of the Pythagorean Theorem.  All of these topics serve a single compelling goal: understanding the mathematical patterns underlying the symmetry that we observe in the physical world around us.

The exposition is engaging, precise and rigorous.  The theorems are visually motivated with intuitive proofs appropriate for the intended audience.  Students from all majors will enjoy the many beautiful topics herein, and will come to better appreciate the powerful cumulative nature of mathematics as these topics are woven together into a single fascinating story about the ways in which objects can be symmetric.

Arvustused

From the reviews:

The author starts with a very concrete introduction to symmetry in the plane, using rigid motions for a definition. The book can serve very well as an introduction to algebraic combinatorics for math students and also for many interested students of other fields, specially Computer Science and natural sciences. (Ulrich Knaner, Zentralblatt MATH, Vol. 1233, 2012)

In this book Tapp uses a large number of high-quality colored images to illustrate various properties derived from symmetry as well as to introduce students to the supporting mathematics . a vital resource for teachers of children as they try to get young minds interested in learning mathematics. an excellent text for math courses for non-math majors, as it will teach them math without seeming to do so. Artists, in particular, will find a great deal that they can apply in their work. (Charles Ashbacher, The Mathematical Association of America, February, 2012)

This textbook provides a great journey to the wonderful world of symmetry and shows how to understand and study it. The textbook is suitable for both mathematics and non-mathematics majors, as only high school algebra is needed to understand the material. All necessary definitions and notations are provided and highlighted, so it is not difficult to find them if needed. this textbook can also be beneficial for graduate students and undergraduates who take upper-level classes in mathematics. (Natali Hritonenko, Mathematical Reviews, January, 2013)

Preface v
Intended audience vi
Acknowledgments vii
Table of Notation
ix
1 Introduction to Symmetry
1(16)
A precise definition of "symmetry"
2(4)
Types of symmetries and types of objects
6(6)
The classification of plane rigid motions
12(2)
Exercises
14(3)
2 The Algebra of Symmetry
17(18)
Cayley tables
18(3)
Symmetry groups
21(4)
The power of inverses
25(3)
An improved classification of plane rigid motions
28(2)
Exercises
30(5)
3 Isomorphism
35(16)
What is an isomorphism?
36(3)
Isomorphism examples
39(4)
Rigid Equivalence
43(3)
A better notation for the cyclic groups
46(1)
Exercises
46(5)
4 The Classification Theorems
51(12)
Bounded objects
51(3)
Border patterns
54(2)
Wallpaper Patterns
56(3)
Summary
59(1)
Exercises
60(3)
5 Subgroups and Product Groups
63(12)
Subgroups
63(3)
Generating subgroups
66(3)
Product groups
69(4)
Exercises
73(2)
6 Permutations
75(12)
Permutation groups
76(4)
Even and odd permutations
80(4)
Exercises
84(3)
7 Symmetries of Solid Objects
87(28)
Rigid motions of space
87(6)
The symmetry group a tetrahedron
93(3)
The proper symmetry group a cube
96(1)
The proper symmetry group a dodecahedron
97(2)
Solid objects which are "essentially two-dimensional"
99(2)
The classification theorem for bounded objects
101(3)
Chirality
104(3)
Proper versus full symmetry groups
107(4)
Exercises
111(4)
8 The Five Platonic Solids
115(24)
Counting their parts
120(1)
Duality
121(3)
Euler's formula
124(5)
The Euler characteristic
129(2)
An algebraic proof that there are only five Platonic solids
131(3)
The Platonic solids through the ages
134(1)
Exercises
135(4)
9 Symmetry and Optimization
139(10)
Minimal surfaces
139(3)
The circle wins
142(5)
Exercises
147(2)
10 What is a Number?
149(18)
Natural numbers
149(2)
Rational numbers
151(2)
Real numbers
153(3)
Which real numbers are rational?
156(2)
How many primes are there?
158(4)
Exercises
162(5)
11 Cantor's Infinity
167(12)
The modern meaning of "same size"
168(4)
Are the rational numbers countable?
172(2)
Cantor's Theorem
174(3)
Exercises
177(2)
12 Euclidean Space
179(20)
The Pythagoran Theorem and distance formula
180(3)
Naming the points on the unit circle
183(2)
The dot product and perpendicularity
185(2)
Using the dot product to find a lover or a song
187(3)
What is a rigid motion?
190(3)
Two exotic examples
193(2)
Exercises
195(4)
13 Symmetry and Matrices
199(14)
Matrix computations
199(3)
Representing rigid motions as matrices
202(4)
Orthogonal matrices
206(2)
You finished the book. Now what?
208(1)
Exercises
209(4)
Index 213
Kristopher Tapp is currently a mathematics professor at Saint Joseph's University.  He is the author of 17 research papers and one well-reviewed undergraduate textbook, Matrix Groups for Undergraduates.  He has been awarded two National Science Foundation research grants and several teaching awards.  He lives in the Philadelphia area with his wife and daughter, whose pictures appear in this book.