The lemniscatic chessboard (Q2874412)

The authors derive properties mainly of the lemniscatic sine function \(\text{sl}:{\mathbb C}\to{\mathbb C}\cup\{\infty\}\) from two applications:NEWLINENEWLINE(1) a conformal mapping from an open square \(\mathcal S\) to a unit disc \(D\);NEWLINENEWLINE(2) a parameterization of an elliptic curve.NEWLINENEWLINEThe article provides an arclength parameterization of the lemniscate wherefrom the authors deduce among others: ``The family of lemniscate parallels is self-orthogonal.'' Moreover, the lemniscate is parametrizable by a simple rational expression in the lemniscatic sine function. ``The beautiful structure of this parametrization becomes fully visible only when complex values of the arclength parameter are allowed and the lemniscate is viewed as complex curve. To visualize such hidden structure'', the authors show how ordinary chessboards are turned to lemniscatic chessboards.