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, Euler’s 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.
Author(s): Kristopher Tapp
Series: Texts for Quantitative Critical Thinking
Edition: 2
Publisher: Springer
Year: 2021
Language: English
Pages: 271
Tags: analytical thinking; symmetry;
Preface
Intended Audience
Improvements in the Second Edition
Advice to Instructors
Acknowledgements
Contents
Introduction to Symmetry
1A Counting Symmetries
1B Objects
1C Rigid Motions
1D Bounded and Unbounded Objects (Revisited)
1E Elements of Mathematics: The Contrapositive
Exercises
The Algebra of Symmetry
2A Composition
2B Cayley Tables
2C Groups
2D Symmetry Groups
2E One Reflection Is Enough
2F An Improved Classification of Rigid Motions
2G Elements of Mathematics: The Counterexample
Exercises
The Classification Theorems
3A Rigid Equivalence
3B Bounded Objects
3C Border Patterns
3D Wallpaper Patterns (optional)
3E Elements of Mathematics: Equivalence (optional)
Exercises
Isomorphic Groups
4A The Definition of an Isomorphism
4B Isomorphism Examples
4C A Better Notation for Cyclic Groups
4D Elements of Mathematics: The Converse
Exercises
Subgroups and Product Groups
5A Subgroups
5B Generated Subgroups (optional)
5C Product Groups (optional)
5D Elements of Mathematics: Sets versus Lists
Exercises
Permutation Groups
6A Counting Permutations
6B Cycle Notation and Composition
6C Even and Odd Permutations
6D Symmetries and Permutations
6E Elements of Mathematics: Well-Defined
6F Elements of Mathematics: The Inductive Proof (optional)
Exercises
Symmetries of 3D Objects
7A Basics of 3D
7B Essentially Two-Dimensional Objects
7C A Trick for Counting Symmetries
7D The Tetrahedron
7E The Cube
7F The Dodecahedron
7G The Classification Theorem
7H Chirality
7I The Full Story (optional)
Exercises
The Five Platonic Solids
8A The Classification of Platonic Solids
8B Counting Their Parts
8C Duality
8D Euler´s Formula
8E The Euler Characteristic
8F The Platonic Solids Through the Ages
Exercises
Symmetry and Optimization
9A Minimal Surfaces
9B The Circle Wins
9C Elements of Mathematics: Proof by Contradiction
Exercises
What Is a Number?
10A Natural Numbers
10B Integers and Rational Numbers
10C Irrational Numbers
10D Real Numbers
10E Which Real Numbers Are Rational?
10F Real Numbers and Symmetry
10G Elements of Mathematics: Construction of Rational and Real Numbers
Exercises
Excursions in Numbers
11A How Many Prime Numbers Are There?
11B The Meaning of ``Same Size´´
11C Are the Rational Numbers Countable?
11D Cantor´s Theorem
Exercises
Rigid Motions as Functions
12A Measuring Distance in Euclidean Space
12B Naming the Points on the Unit Circle
12C The Dot Product and Perpendicularity
12D Using the Dot Product to Find a Friend
12E Rigid Motions Are Functions
Exercises
Rigid Motions as Matrices
13A Matrix Computations
13B Representing Rigid Motions as Matrices
13C Orthogonal Matrices
13D Concluding Remarks
Exercises
Image Credits
Index
