Gronau, Detlef The spiral of Theodorus. (English) Zbl 1077.39002 Am. Math. Mon. 111, No. 3, 230-237 (2004). The notation of the spiral of Theodorus is referred to the representation by P. J. Davis [Spirals from Theodorus to chaos, A. K. Peters, Wellesley, MA (1993; Zbl 0940.00002)]. The underlying so-called Theodorus function \(T(x)\) is a particular solution of the difference equation \[ f(x+ 1)= \Biggl(1+{i\over\sqrt{n+1}}\Biggr)\cdot f(x)\;(- 1< x< \infty),\;f(0)= 1.\tag{\(*\)} \] The discrete points of this spiral are considered, which can be presented in the Gaussian plane recursively by \(z_0= 1\) and \(z_{n+1}= (1+ {i\over\sqrt{n+1}})\cdot z_n\) for \(n= 0,1,\dots\). There are infinitely many possibilities for connecting these points by a continuous curve. The question is: How can one characterize such a spiral-like curve through the “Theodorus points”? The author gives a characterisation of \(T(x)\) among the various solutions of the difference equation \((*)\) by way of monotonicity criteria. Finally, two further interesting examples of solutions are given. Reviewer: Erhard Quaisser (Potsdam) Cited in 1 ReviewCited in 3 Documents MSC: 39A10 Additive difference equations Keywords:spiral of Theodorus; iterative methods; spiral-like trajectories; difference equation; monotonicity criteria Citations:Zbl 0940.00002 PDFBibTeX XMLCite \textit{D. Gronau}, Am. Math. Mon. 111, No. 3, 230--237 (2004; Zbl 1077.39002) Full Text: DOI Online Encyclopedia of Integer Sequences: Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.