Number, Shape, & Symmetry An Introduction to Number Theory, Geometry, and Group Theory

Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors? successful work with undergraduate students at the University of Chicago, seventh to tenth grade mathematically talented students in the University of Chicago?s Young Scholars Program, and elementary public school teachers in the Seminars for Endorsement in Science and Mathematics Education (SESAME).

The first half of the book focuses on number theory, beginning with the rules of arithmetic (axioms for the integers). The authors then present all the basic ideas and applications of divisibility, primes, and modular arithmetic. They also introduce the abstract notion of a group and include numerous examples. The final topics on number theory consist of rational numbers, real numbers, and ideas about infinity.

Moving on to geometry, the text covers polygons and polyhedra, including the construction of regular polygons and regular polyhedra. It studies tessellation by looking at patterns in the plane, especially those made by regular polygons or sets of regular polygons. The text also determines the symmetry groups of these figures and patterns, demonstrating how groups arise in both geometry and number theory.

The book is suitable for pre-service or in-service training for elementary school teachers, general education mathematics or math for liberal arts undergraduate-level courses, and enrichment activities for high school students or math clubs.

The Triangle Game. The Beginnings of Number Theory. Axioms in Number Theory. Divisibility and Primes. The Division and Euclidean Algorithms. Variations on a Theme. Congruences and Groups. Applications of Congruences. Rational Numbers and Real Numbers. Introduction to Geometry and Symmetry. Polygons and Their Construction. Symmetry Groups. Permutations. Polyhedra. Graph Theory. Tessellations. Connections. Appendix. Glossary. Bibliography. Index.

Diane L. Herrmann is a senior lecturer and associate director of undergraduate studies in mathematics at the University of Chicago. Dr. Herrmann is a member of the American Mathematical Society, Mathematical Association of America, Association for Women in Mathematics, Physical Sciences Collegiate Division Governing Committee, and Society for Values in Higher Education. She is also involved with the University of Chicago’s Young Scholars Program, Summer Research Opportunity Program (SROP), and Seminars for Elementary Specialists and Mathematics Educators (SESAME).

Paul J. Sally, Jr. is a professor and director of undergraduate studies in mathematics at the University of Chicago, where he has directed the Young Scholars Program for mathematically talented 7-12 grade students. Dr. Sally also founded SESAME, a staff development program for elementary public school teachers in Chicago. He is a member of the U.S. Steering Committee for the Third International Mathematics and Science Study (TIMSS) and has served as Chairman of the Board of Trustees for the American Mathematical Society.