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The third image below was certainly unexpected for my soon-to-be-collaborators, Emmanuel Candes and Justin Romberg. They started with a standard image in signal processing, the Logan-Shepp phantom: …

This question was cross-posted on Math Stack Exchange. Here is a copy of my answer for it there. This is it.  The perfectly centered billiards break.  Behold. Setup This break was computed i …

The histogram of all OEIS sequences shows an unexpected gap known as Sloane's gap. The plot shows how cultural factors influence mathematics. (http://arxiv.org/abs/1101.4470v2)

The answer is 'no'. Making the substitution $$ x = \frac{(t-1)(t-5)(t^2+2t+5)}{16t^2}, $$ one finds $$ {\textstyle\sqrt{x+\sqrt{x+\sqrt{x+1}}}\,\mathrm{d}x} = \frac{(t^2-2t+5)(t^2-5)\sqrt{t^4{-}2t^2 …

Here is a simple story one can tell about the entropy $$H = -\sum_{i=1}^n p_i \log p_i$$ of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning …

                          This visual proof of $$\sum\limits_{n=1}^\infty \left (\frac{1}{2}\right)^{\,2n}=\frac{1}{3}$$ is from http://www.cecm.sfu.ca/~loki/Papers/Numbers/ (Visible Structures in N …

Some years ago I was pleasantly surprised when an idea of Jan Mycielski led me to find a very explicit Banach-Tarski paradox in the hyperbolic plane, $H^2.$ $H^2$ can be decomposed into three simple s …

One can obtain a nice picture showing somewhat unexpected patterns by marking all rational points on the unit sphere whose coordinates have denominator less than some upper bound, and projecting this …

I think one reason JSTOR doesn't have “loss of generality” before 1831 is that fewer scientists wrote in English. But one finds (with minor variants merged, translations *starred, and year first publi …

I'm adding a separate answer for the general question that the OP asked, which settles the question in the negative for all $n>2$ (and gives an alternate proof for $n=3$ to the one I gave above). Rec …

On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep. For example, one of the critical ideas in my paper Joel David Hamkins, Gap forcing, …

I was there. Arnol'd is one of my big mathematical heros, but I found the whole thing really sad. It was in French, but my French is decent. Arnol'd began his part with a phrase I've heard him say be …

Computer designers and programmers dreamed, from the earliest days of the computer, of a computer that could play chess and win. Even Alan Turing had that dream, and designed turochamp, the first ches …

I have proved this equality by means of Cauchy’s Theorem applied to an adequate function. Since my solution is too long to post it here, I posted it in arXiv: Juan Arias de Reyna, Computation of a De …

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other t …