Advertisement

Squigonometry: Trigonometry in the p-Norm

Chapter
  • 455 Downloads
Part of the Foundations for Undergraduate Research in Mathematics book series (FURM)

Abstract

We can define the traditional trigonometric functions in several different ways: via differential equations, via an arclength definition on the unit circle x2 + y2 = 1, or via an analytic approach. In this project, we adapt these approaches to define analogous functions for a unit squircle |x|p + |y|p = 1, p ≥ 1. As we develop these functions using only elementary calculus, we will ponder the importance and role of π, and glimpse some very deep ideas in elliptic integrals, special functions, non-Euclidean geometry, number theory, and complex analysis.

References

  1. 1.
    Adler, C.L., Tanton, J.: π is the minimum value of Pi. College Mathematics Journal 31(2), 102–106 (2000)Google Scholar
  2. 2.
    Andrews, G., Askey, R., Roy, R.: Special Functions (Encyclopedia of Mathematics and its Applications). Cambridge University Press (1999).  https://doi.org/10.1017/CBO9781107325937
  3. 3.
    Blåjö, V.: The Isoperimetric Problem. The American Mathematical Monthly 112 (6), 526–566 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borwein, J., Bailey, D.: Mathematics by Experiment: Plausible Reasoning in the 21st Century. A K Peters (2003)Google Scholar
  5. 5.
    Boyce, W., DiPrima, R.: Elementary Differential Equations, 9th edition. Wiley (2008)Google Scholar
  6. 6.
    Callahan, J., Cox, D., Hoffman, K., O’Shea, D., Pollatsek, H., Senechal, L.: Calculus in Context: The Five College Calculus Project. W.H. Freeman (1995)Google Scholar
  7. 7.
    Cha, B.: Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II. College Mathematics Journal 38(4), 288–296 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Euler, R., Sadek, J.: The π’s go full circle, Mathematics Magazine 72, 59–63 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Grammel: Eine Verallgemeinerung der Kreis- und Hyperbelfunktionen, (German) Ing.-Arch. 16, 188–200 (1948)Google Scholar
  10. 10.
    Hardy, G., Littlewood, J., Pólya, G., Inequalities, Cambridge Univ. Press (1934)zbMATHGoogle Scholar
  11. 11.
    Keller, J., Vakil, R.: Open image in new window, the Value of π in Open image in new window. The American Mathematical Monthly, 116(10), 931–935 (2009). https://doi.org/10.4169/000298909X477069
  12. 12.
    Krause, E.: Taxicab Geometry: Adventures in Non-Euclidean Geometry. Dover (1987)Google Scholar
  13. 13.
    Lang, J., Edmunds, D.: Eigenvalues, Embeddings and Generalised Trigonometric Functions, Springer Lecture Notes in Mathematics (2016)Google Scholar
  14. 14.
    Levin, A.: A Geometric Interpretation of an Infinite Product for the Lemniscate Constant. The American Mathematical Monthly, 113(6), 510–520 (2006). https://doi.org/10.2307/27641976 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lindqvist, P., Peetre, J.: p-arclength of the q-circle, Math. Student 72 (1–4) 139–145(2003)Google Scholar
  16. 16.
    Maican, C.C: Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering: A Self-Study Approach, International Press, 2005.Google Scholar
  17. 17.
    Markushevich, A.I.: The Remarkable Sine Functions, Elsevier (1966)Google Scholar
  18. 18.
    Poodiack, R.: Squigonometry, Hyperellipses, and Supereggs, Mathematics Magazine, 89(2) 92–102 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shelupsky, D.: A Generalization of the Trigonometric Functions, The American Mathematical Monthly, 66(10), 879-884 (1959). https://doi.org/10.1080/00029890.1959.11989425 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Singh, S.: Fermat’s last theorem: the story of a riddle that confounded the world’s greatest minds for 358 years. Harper Perennial (2007)Google Scholar
  21. 21.
    Weisstein, E.W.: Gamma Function. From Mathworld – A Wolfram Web Resource. http://mathworld.wolfram.com/GammaFunction.html
  22. 22.
    Weisstein, E.W.: Elliptic Integral Singular Value. From Mathworld – A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticIntegralSingularValue.html
  23. 23.
    Wood, W.: Squigonometry, Mathematics Magazine, 84(4): 257–265 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Young, R.M. Excursions in Calculus: An Interplay of the Continuous and the Discrete, Cambridge Univ. Press (1992)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Northern IowaCedar FallsUSA
  2. 2.Department of MathematicsNorwich UniversityNorthfieldUSA

Personalised recommendations