Magic moments for late starters (Q6638624)

This nice and interesting article presents a selection of people who have achieved remarkable results in the field of mathematics out of a passion for mathematics, without having a specific academic mathematical background. These are:\N\begin{itemize}\N\item Lee Sallows with a beautiful geometry theorem concerning triangles [\textit{L. Sallows}, Math. Mag. 87, No. 5, 381 (2014; Zbl 1325.97009)].\N\item Edouard Zeckendorf and his famous theorem. The \textit{Zeckendorf's theorem} is about the representation of integers as sums of Fibonacci numbers; although this theorem dates back from the year 1939, Zeckendorf only published it a third of a century later [\textit{C. Kimberling}, Fibonacci Q. 36, No. 5, 416--418 (1998; Zbl 1016.01504); \textit{E. Zeckendorf}, Bull. Soc. R. Sci. Liège 41, 179--182 (1972; Zbl 0252.10011)].\N\item Andrew Beal with the \textit{Beal conjecture}. This is a conjecture in number theory, formulated in 1993 and still waiting for a peer-reviewed proof, although Beal has offered a prize money of incredible \$1 million US Dollars, see also [\textit{R. D. Mauldin}, Notices Am. Math. Soc. 44, No. 11, 1436--1437 (1997; Zbl 0924.11022)].\N\item Preda Mihăilescu who proved \textit{Catalan's conjecture} [\textit{P. Mihăilescu}, J. Reine Angew. Math. 572, 167--195 (2004; Zbl 1067.11017)] more than 150 years after it was formulated. (Mihăilescu is an exception: he has indeed studied mathematics, however, he was over forty years old before he turned to ``pure'' mathematics professionally.)\N\end{itemize}\NThe following part of the article deals with amateur mathematicians who have worked on the topic of tiling. Which tile shape can be used to tile a plane, i.e. to cover it without gaps or overlapping?\N\begin{itemize}\N\item Richard James III and Marjorie Rice worked independently of each other with convex pentagons, all of which allowed tiling. A photograph of the foyer of the headquarters of the Mathematical Association of America in Washington, D.C. shows the ceramic tiles of the floor which are based on one of the tiling pentagons discovered by Rice.\N\item Joan Taylor ``became hooked'' on aperiodic tiles. When her work was already well advanced she contacted the American physicist Joshua Socolar and together they presented the first single prototile that can tile a plane but not admit a periodic tiling [\textit{J. E. S. Socolar} and \textit{J. M. Taylor}, J. Comb. Theory, Ser. A 118, No. 8, 2207--2231 (2011; Zbl 1232.05052)]. However, this was not a simple coherent tile.\N\item David Smith, on the other hand, showed an astonishingly simple solution to an aperiodic monotile which can be easily assembled from eight sixths of a regular hexagon. He called it a ``hat'' [\textit{D. Smith} et al., Comb. Theory 4, No. 1, Paper No. 6, 91 p. (2024; Zbl 1547.05049)].\N\end{itemize}\NAs anyone can imagine, such magic moments -- as the title of the article calls them -- are rare, so it was a real pleasure for the reviewer to hear about them.