Overview
- Introduces generalized trigonometric functions through an expanded notion of unit circle
- Showcases new results alongside well-established theory
- Engages readers with illustrations, projects, and opportunities for research
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
Part of the book sub series: SUMS Readings (SUMSR)
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Table of contents (20 chapters)
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Front Matter
Keywords
- generalized trigonometry introduction
- generalized trigonometry textbook
- squigonometry
- squigonometric functions
- p-norm introduction
- p-circle
- p-circle parametrizations
- taxicab geometry
- elliptic integrals
- generalized exponential function
- alternative values of pi
- Piet Hein superegg
- geometry in the p-norm
- lemniscatic functions
- dual norm
- alternative pi days
- undergraduate research mathematics
- senior capstone mathematics
About this book
This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by |x|p + |y|p = 1 where p ≥ 1. Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be.
Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of π, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein’s work in design. From here, more technical pathways offer further exploration. Topicsinclude infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor.
Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.
Reviews
Authors and Affiliations
About the authors
Robert D. Poodiack is Professor of Mathematics at Norwich University, Vermont, the nation’s oldest private military college. He earned his PhD in mathematical sciences from the University of Vermont in 1999. He has supervised dozens of senior capstone projects in his 23 years at Norwich, many on squigonometry-related topics. He is very active in the Northeastern Section of the Mathematical Association of America. In his spare time, he enjoys running, baking, and cooking.
William E. Wood is an Associate Professor of Mathematics at the University of Northern Iowa. His mathematical pursuits tend toward geometry, including discrete conformal geometry, visualization, applications, and analysis, with particular interest in finding ways to bring geometric ideas to students in the classroom and through research projects. He earned his PhD from Florida State University.
Bibliographic Information
Book Title: Squigonometry: The Study of Imperfect Circles
Authors: Robert D. Poodiack, William E. Wood
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-3-031-13783-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Softcover ISBN: 978-3-031-13782-2Published: 16 December 2022
eBook ISBN: 978-3-031-13783-9Published: 15 December 2022
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XIX, 289
Number of Illustrations: 32 b/w illustrations, 95 illustrations in colour
Topics: Special Functions, Geometry, Functional Analysis