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Exploration in Modern Mathematics
Mike Janssen
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Front Matter
Colophon
Introduction
References
A Note to Students
A Note to Instructors
Versions of this text
Acknowledgments
1
The Cube
1.1
Introduction and Background
1.1.1
Thinking about Play
1.1.2
Exploring the Cube
1.1.3
Exercises
1.2
Describing the Cube
1.2.1
Faces and Layers
1.2.2
The Need for Notation
1.2.3
Order
1.2.4
Exercises
1.3
Moving Cubies
1.3
Exercises
1.4
Reorienting Cubies
1.4
Exercises
2
Inductive and Deductive Reasoning
2.1
Inductive Reasoning
2.1
Exercises
2.2
Deductive Reasoning
2.2
Exercises
3
Modern Mathematical Logic
3.1
Logical Connectives and Rules of Inference
3.1
Exercises
3.2
Formal Systems and Incompleteness
3.2.1
Formal Systems
3.2.2
Infinity and Incompleteness
3.2.3
Exercises
4
Dynamical Systems
4.1
Sequences
4.1
Exercises
4.2
Discrete Dynamical Systems
4.2.1
Exponential Growth
4.2.2
A discrete predator-prey model
4.2.3
The discrete SIR epidemiological model
4.2.4
Exercises
5
Graphs
5.1
Intro to Graph Theory
5.1.1
The Bridges of Konigsberg
5.1.2
Background and terminology
5.1.3
Exercises
5.2
Planar Graphs and Graph Coloring
5.2.1
Planar Graphs
5.2.2
Graph Coloring
5.2.3
Exercises
5.3
Trees
5.3
Exercises
6
Modular Arithmetic and Coding Theory
6.1
Adding in Circles
6.1
Exercises
6.2
Coding Theory
6.2
Exercises
7
Voting
7.1
Elections with Two Candidates
7.2
Plurality and the Borda Count
7.2
Worksheet
7.2.1.1
Describing the Problem
7.2.1.2
A solution
7.2.1.3
The Borda Count
7.2
Exercises
7.3
Pairwise Comparisons and Instant Runoff Voting
7.3
Worksheet
7.3.1.1
Pairwise Comparisons
7.3.1.2
Instant Runoff Voting and Arrow’s Theorem
7.3
Exercises
8
Fractals and Geometry
8.1
Introduction to Fractals
8.2
Complex Numbers and Fractals
8.2.1
The Arithmetic and Geometry of Complex Numbers
8.2.2
Mandelbrot and Julia Sets
8.3
Chaos Theory
8.4
The Pythagorean Theorem
8.5
Tessellations and Symmetry
Backmatter
Index
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
