Foundations of Geometry

Description

Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach.

 

This text’s coverage begins with Euclid’s Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.

 

This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.

• Earlier presentation of neutral geometry has been achieved by shifting some topics to appendices and covering others more efficiently. For example, material on set theory and the real numbers was moved to an appendix because much of it is review for most students.
• The review of proof writing has been incorporated into the chapter on Axiomatic Systems.
• Description of different axiom systems for elementary geometry has been moved to its own appendix (previously found at the beginning of the chapter on The Axioms of Plane Geometry). Instructors can now cover this material at any time they choose during the course.
• New topics/content include:
• More exercises in elementary Euclidean geometry in Chapter 5
• A new section on ideal triangles and the classification of parallels in Chapter 6
• Reworking of Chapter 7 (Area) to include a relatively elementary proof of the Euclidean case of the dissection theorem, giving instructors the option of omitting all the non-Euclidean material in that chapter
• A section on similarity transformations in Euclidean geometry has been added to the chapter on Transformations (Chapter 10)
• Technology material facilitates the use of the open-source software Geogebra.
• A new Appendix B (Systems of Axioms for Geometry) explores in depth the range of options available in the choice of axioms and explains the rationale for the choices made in this book.

 

Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach. This text’s coverage begins with Euclid’s Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.

This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.

1. Prologue: Euclid’s Elements

1.1 Geometry before Euclid

1.2 The logical structure of Euclid’s Elements

1.3 The historical significance of Euclid’s Elements

1.4 A look at Book I of the Elements

1.5 A critique of Euclid’s Elements

1.6 Final observations about the Elements

2. Axiomatic Systems and Incidence Geometry

2.1 The structure of an axiomatic system

2.2 An example: Incidence geometry

2.3 The parallel postulates in incidence geometry

2.4 Axiomatic systems and the real world

2.5 Theorems, proofs, and logic

2.6 Some theorems from incidence geometry

3. Axioms for Plane Geometry

3.1 The undefined terms and two fundamental axioms

3.2 Distance and the Ruler Postulate

3.3 Plane separation

3.4 Angle measure and the Protractor Postulate

3.5 The Crossbar Theorem and the Linear Pair Theorem

3.6 The Side-Angle-Side Postulate

3.7 The parallel postulates and models

4. Neutral Geometry

4.1 The Exterior Angle Theorem and perpendiculars

4.2 Triangle congruence conditions

4.3 Three inequalities for triangles

4.4 The Alternate Interior Angles Theorem

4.5 The Saccheri-Legendre Theorem

4.6 Quadrilaterals

4.7 Statements equivalent to the Euclidean Parallel Postulate

4.8 Rectangles and defect

4.9 The Universal Hyperbolic Theorem

5. Euclidean Geometry

5.1 Basic theorems of Euclidean geometry

5.2 The Parallel Projection Theorem

5.3 Similar triangles

5.4 The Pythagorean Theorem

5.5 Trigonometry

5.6 Exploring the Euclidean geometry of the triangle

6. Hyperbolic Geometry

6.1 The discovery of hyperbolic geometry

6.2 Basic theorems of hyperbolic geometry

6.3 Common perpendiculars

6.4 Limiting parallel rays and asymptotically parallel lines

6.5 Properties of the critical function

6.6 The defect of a triangle

6.7 Is the real world hyperbolic?

7. Area

7.1 The Neutral Area Postulate

7.2 Area in Euclidean geometry

7.3 Dissection theory in neutral geometry

7.4 Dissection theory in Euclidean geometry

7.5 Area and defect in hyperbolic geometry

8. Circles

8.1 Basic definitions

8.2 Circles and lines

8.3 Circles and triangles

8.4 Circles in Euclidean geometry

8.5 Circular continuity

8.6 Circumference and area of Euclidean circles

8.7 Exploring Euclidean circles

9. Constructions

9.1 Compass and straightedge constructions

9.2 Neutral constructions

9.3 Euclidean constructions

9.4 Construction of regular polygons

9.5 Area constructions

9.6 Three impossible constructions

10. Transformations

10.1 The transformational perspective

10.2 Properties of isometries

10.3 Rotations, translations, and glide reflections

10.4 Classification of Euclidean motions

10.5 Classification of hyperbolic motions

10.6 Similarity transformations in Euclidean geometry

10.7 A transformational approach to the foundations

10.8 Euclidean inversions in circles

11. Models

11.1 The significance of models for hyperbolic geometry

11.2 The Cartesian model for Euclidean geometry

11.3 The Poincaré disk model for hyperbolic geometry

11.4 Other models for hyperbolic geometry

11.5 Models for elliptic geometry

11.6 Regular Tessellations

12. Polygonal Models and the Geometry of Space

12.1 Curved surfaces

12.2 Approximate models for the hyperbolic plane

12.3 Geometric surfaces

12.4 The geometry of the universe

12.5 Conclusion

12.6 Further study

12.7 Templates

APPENDICES

A. Euclid’s Book I

A.1 Definitions

A.2 Postulates

A.3 Common Notions

A.4 Propositions

B. Systems of Axioms for Geometry

B.1 Filling in Euclid’s gaps

B.2 Hilbert’s axioms

B.3 Birkhoff’s axioms

B.4 MacLane’s axioms

B.5 SMSG axioms

B.6 UCSMP axioms

C. The Postulates Used in this Book

C.1 The undefined terms

C.2 Neutral postulates

C.3 Parallel postulates

C.4 Area postulates

C.5 The reflection postulate

C.6 Logical relationships

D. Set Notation and the Real Numbers

D.1 Some elementary set theory

D.2 Properties of the real numbers

D.3 Functions

E. The van Hiele Model

F. Hints for Selected Exercises

Bibliography

Index

• Implementation of “The Mathematical Education of Teachers (MET)” and the Common Core State Standards for Mathematics (CCSM) achieves the goals of MET and CCSM within the context of a traditional axiomatic approach to geometry. Students are prepared to teach high school geometry by making connections that enable a deeper understanding of the subject.
• Comprehensive coverage of most of Euclid’s Elements includes almost all of the material in the first six books of that work.
• Careful statements of the axioms encourage students to understand how the theorems of geometry are built on the axioms.
• Coverage of transformations and the transformational approach to the foundations contains a complete classification of rigid motions of the plane and an explanation of how the Reflection Postulate can substitute for the Side-Angle-Side Postulate. This helps students understand the transformational perspective and how it can be incorporated into the axioms.
• A complete construction of the traditional models for Hyperbolic Geometry provides a model that helps students understand the relationships spelled out in the axioms.
• A study of geometry in the real world examines some non-traditional models and curved spaces, helping students arrive at answers to questions that arise naturally about the relationship between non-Euclidean geometry and the geometry of the real world.
• Careful attention to the historical development of geometry discusses certain cultural and philosophical issues, and demonstrates the changes that the foundations of geometry have gone through over time.
• An introduction to proof acts as a bridge between lower-level courses (in which technique is emphasized) to upper-level courses (in which proof and the understanding of concepts are emphasized). This enables students to experience the axiomatic method and study careful proofs that characterize more advanced mathematics.
• Emphasis on how to write proofs provides many model proofs as well as commentary on key proofs.
• Provides students with an excellent introductory chapter on how to write proofs, and provides more advanced techniques as they progress through the course.
• Includes many proofs that are exercises, with hints given so that students know roughly how the proof should be structured.
• Enables students to concentrate on how to organize and communicate the proof.

 

Gerard Venema earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education, he spent two years in a postdoctoral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin University, and has been a faculty member there ever since. While on the Calvin University faculty he also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He also spent two years as Program Director for Topology, Geometry, and Foundations in the Division of Mathematical Sciences at the National Science Foundation and nearly ten years as the Associate Secretary of the Mathematical Association of America.

Venema is a member of the American Mathematical Society and the Mathematical Association of America. He is the author of two other books. One is an undergraduate textbook, Exploring Advanced Euclidean Geometry with GeoGebra, published by the Mathematical Association of America. The other is a research monograph, Embeddings in Manifolds, coauthored by Robert J. Daverman, that was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition, Venema is author of over thirty research articles in geometric topology.