Geometry and Its Applications

Walter J. Meyer

• Publisher: Chapman and Hall/CRC
• Year: 2022
• ISBN:
  9780429582264
• Edition: 3
• Topic: Matemáticas
• Subtopic: Matemáticas general

This unique textbook combines traditional geometry presents a contemporary approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, introduces axiomatic, Euclidean and non-Euclidean, and transformational geometry. The text integrates applications and examples throughout. The Third Edition offers many updates, including expaning on historical notes, 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.

• The Third Edition streamlines the treatment from the previous two editions
• Treatment of axiomatic geometry has been expanded
• Nearly 300 applications from all fields are included
• An emphasis on computer science-related applications appeals to student interest
• Many new excercises keep the presentation fresh

1 The Axiomatic Method in Geometry

DOI: 10.1201/9780429198328-1

We human beings are at home with the physical world—our senses guide our movements and help us estimate sizes. But from earliest times, we have wanted to know things about our physical world that our senses and measuring instruments could not tell us—for example, what is the circumference of the earth? For this, we needed the kinds of geometry we will explore in this book: axiomatic geometry. This means that we start with assumptions people are willing to accept as true (perhaps “for the sake of argument”) and use logical arguments based on these agreed-upon principles, instead of our senses. The process is called deduction and the starting principles are called axioms.

Geometry, that of Euclid in particular (which we start with), has paid dividends for over 2,000 years, but there are still frontiers to explore. For example, the development of robots has led to the desire to mimic whatever mysterious processes our human minds do, through geometry, to move about in the world safely and effectively.

Prerequisites:

high school mathematics,

the notation of set theory (in just a few places)

Section 1. Axioms for Euclidean Geometry

In this section, we describe the basic principles from which we drive everything else in our study of Euclidean geometry. These basic principles are called axioms. We then carry out rigorous proofs of the first few deductions from this axiom set.

Our axiom set is a descendant of the five axioms provided by Euclid, but there are some differences. There are two main reasons why we do not use Euclid's axioms as he originally gave them:

1. Euclid phrases his axioms in a way which is hard for the modern reader to appreciate.
2. It has been necessary to add axioms to Euclid's set to be able to give rigorous proofs of many Euclidean theorems.

A number of individuals, and at least one committee, have taken turns in improving Euclid's axiom set: notably, David Hilbert in 1899, G. D. Birkhoff in 1932, and the School Mathematics Study Group (SMSG) during the 1960s. Even though these axiom sets differ from one another—and from Euclid's—they all lead to the well-known theorems in Euclid's Elements. Consequently, we say that they are all axiom sets for Euclidean geometry. The axioms we list below for our use are a minor rewording of the SMSG axiom set.

As we embark on our study of axiomatic Euclidean geometry, you will be asked to consider proofs of some statements you may have learned before. To enter the spirit of our study, you must put aside what you have learned before or find obvious. In earlier sections, we have relied on some geometry you have previously learned but in this section, we strive to construct proofs only from the axioms we are about to list, and any theorems we have previously proved from those axioms. Keep in mind that our objective in our axiomatic discussion of Euclidean geometry is not to learn facts about geometry, but to learn about the logical structure of geometry.

Axioms About Points on Lines

Axiom 1: The Point-Line Incidence Axiom.

Given any two different points, there is exactly one line which contains them.

We denote the line connecting A and B by $\overleftrightarrow{AB}$. Our first theorem about lines uses proof by contradiction or indirect proof. It is based on the idea that the truths of Euclidean geometry do not contradict one another; if you reason correctly on true statements, then you can never deduce a statement that contradicts another which is known to be true. If you do find a contradiction, then one of the statements you have been reasoning from must be false. In our proof, we will make a supposition and show that it leads t...