Spiraling the Earth with C

Elliptic functions have been known for about 150 years, mainly through the work of the mathematicians Legendre, Abel, and most of all, C. G. J. Jacobi ~1804–1851!. They are doubly periodic functions of a variable in the complex plane with the additional property that they are meromorphic, i.e., they have only poles as singularities. Among the elliptic functions, the so-called Jacobian elliptic functions are of second order, that is, they have two poles of first order in the period parallelogram. Most frequently, these latter functions are defined as the inverse functions of certain integrals that cannot be expressed in terms of those functions which, somewhat presumptuously, are called elementary. This definition usually forms the basis of the applications of the Jacobi elliptic functions in physics. In mechanics they appear in the theories of the simple pendulum and of the spinning top ~i.e., a rotating rigid body with one point fixed!; in superconductivity they appear as the solutions of the Ginzburg–Landau equations; in mathematical biology they occur as solutions of reactiondiffusion equations, etc. Here I wish to develop an approach to the Jacobian elliptic functions that will appeal to readers who like to think in terms of geometrical images. I believe that this approach will provide an insight into the behavior of these functions for different ranges of their parameters and some understanding of how these functions are interconnected. This approach was discovered by A. Seiffert and communicated in 1896 in a publication which may have escaped the attention of some of the readers of this Journal. It was briefly referred to by Whittaker and Watson. My contribution to the subject consists only in a few additions and in giving the elliptic functions a pictorial interpretation.