Squigonometry by W. Wood (Disclaimer: I Do Not Own This Research Paper)

Squigonometry Author(s): William E. Wood Reviewed work(s): Source: Mathematics Magazine, Vol. 84, No. 4 (October 2011), pp. 257-265 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/10.4169/math.mag.84.4.257 . Accessed: 09/09/2012 06:26
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VOL. 84, NO. 4, OCTOBER 2011

257

Squigonometry
W I L L I A M E. W O O D
University of Northern Iowa Cedar Falls, IA 50614 [e-mail]

It is easy to take the circle for granted. In this paper, we look to enhance our appreciation of the circle by developing an analog of trigonometry—a subject built upon analysis of the circle—for something that is not quite a circle. Our primary model is the unit squircle, the superellipse defined as the set of points (x, y) in the plane satisfying x 4 + y 4 = 1, depicted in F IGURE 1. It is a closed curve about the origin, but while any line through the origin is a line of symmetry of the circle, there are only four lines of symmetry for the squircle. Many familiar notions from trigonometry have natural analogs and we will see some interesting behaviors and results, but we will also see where the lower degree of symmetry inconveniences our new theory of squigonometry. We only scratch the surface here, offering many opportunities for the reader to extend the theory into studies of

References: 1. J. Callahan, D. Cox, K. Hoffman, D. O’Shea, H. Pollatsek, and L. Senechal, Calculus in Context: The Five College Calculus Project, W.H. Freeman, 1995. 2. B. Cha, Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II, College Math. J. 38 (2007) 288–296. 3. W. Boyce and R. DiPrima, Elementary Differential Equations, 9th ed., Wiley, 2008. 4. E. F. Krause, Taxicab Geometry: An Adventure in Non-Euclidean Geometry, Dover, 1987. 5. C. C. Maican, Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering: A Self-Study Approach, International Press, 2005. 6. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1986. 7. R. M. Young, Execursions in Calculus, Mathematical Association of America, 1992. Summary Differential equations offers one approach to defining the classical trigonometric functions sine and cosine that parameterize the unit circle. In this article, we adapt this approach to develop analogous functions that parameterize the unit squircle defined by x 4 + y 4 = 1. As we develop our new theory of “squigonometry” using only elementary calculus, we will catch glimpses of some very interesting and deep ideas in elliptic integrals, non-euclidean geometry, number theory, and complex analysis. WILLIAM E. WOOD has recently joined the Mathematics Department at the University of Northern Iowa. He enjoys thinking about various problems across mathematics and somehow turning them all into geometry problems. He lives in Cedar Falls with his wife, cats, and board game collection.