Geometry and Its Applications

Walter A. Meyer

• Publisher: Academic Press
• Year: 2006
• eBook ISBN: 9780080478036
• Edition: 2
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

Meyer's Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters.

The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry's usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.

• Realistic applications integrated throughout the text, including (but not limited to):
• • Symmetries of artistic patterns
  • Physics
  • Robotics
  • Computer vision
  • Computer graphics
  • Stability of architectural structures
  • Molecular biology
  • Medicine
  • Pattern recognition
• Historical notes included in many chapters

Chapter 1 The Axiomatic Method in Geometry

Prerequisites: High school mathematics

We human beings are at home in the physical world: We see it clearly and move about with ease, and we need no theory to do so. But our natural understanding of geometry does not serve all our needs. The axiomatic method is the most powerful method so far devised for a more complete understanding of our geometric enivronment.

The key idea of the axiomatic method is that we start with assumptions we have complete confidence in, and we reason our way to things we might not believe without a proof. Our objective in this chapter is to provide an overview of this axiomatic method, including some attention to how standards of proof have evolved over time. A short discussion of computer graphics provides an opportunity to see how the ideal concepts of geometry apply to the less ideal, real world. In the last section of this chapter we display the axiom set we will rely on for the part of Euclidean geometry we develop in this and the next chapter.

1.1 The Aims of Axiomatic Geometry

In Euclidean geometry, as in all of mathematics, we try to discover new knowledge by applying already-known principles. This style of thinking is called deduction, or the axiomatic method, and it can also be seen in many other fields, including fields outside the sciences, such as politics and religion. As an example, much of Western culture long ago selected the Ten Commandments as central moral principles, with the presumption that in the thousands of different moral dilemmas that come up in life, we can always deduce the right thing to do from the Ten Commandments. A little more recently, in the U.S. Declaration of Independence, certain truths are said to be self-evident, and the document goes on to justify the independence of the American colonies as a deduction from those truths. Likewise, in our legal system, deductions are drawn from laws and evidence in order to decide individual cases. However, the application of deduction to nonmathematical fields often seems to leave lots of room for differences of opinion, something that rarely occurs in mathematics. For this reason, mathematics remains the purest and most successful example of deduction at work — an ideal to which other fields often aspire.

The idea that some scientific truths can be deduced from other, more basic ones was surely known before the development of Euclidean geometry, but it certainly received its biggest boost in the sciences from its use in geometry. In this section we want to briefly ask “Why is this idea so appealing in geometry?”

Approaches to Geometry

We can discover a lot about the physical space we live in through simple visual observation. Fred Flintstone and his prehistoric buddies could tell if a ditch was too big to jump over without having gone to school. Likewise, even children who have not attended school take for granted their ability to move about without bumping into things, to judge whether they can squeeze through tight spaces or jump over ditches. (Ironically, researchers have found it a tremendous challenge to get robots to make these judgments even half as well as people.)

The eye and the brain are very capable, but there is much they can’t do. If Fred Flintstone wants to know exactly how wide the ditch is, he will have to measure it — say, with a tape measure. This is the second great way to learn about our physical world: Apply measuring instruments such as rulers and protractors to it. The deductive me...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• Introduction
• Chapter 1: The Axiomatic Method in Geometry
• Chapter 2: The Euclidean Heritage
• Chapter 3: Non-Euclidean Geometry
• Chapter 4: Transformation Geometry I: Isometries and Symmetries
• Chapter 5: Vectors in Geometry
• Chapter 6: Transformation Geometry II: Isometries and Matrices
• Chapter 7: Transformation Geometry III: Similarity, Inversion, and Projection
• Chapter 8: Graphs, Maps, and Polyhedra
• Bibliography
• Answers to Odd-Numbered Exercises
• Index