I've just seen a video on Tupper's self-referential formula. When I heard that this formula was not at all self-referential but merely a simple way to generate every possible $17\times 107$ dot matrix each on top of the other, I was very disappointed.
Thus I'm wondering whether there is a truly self-referential plotted formula $\phi$ containing two free variables $x$ and $y$ in the sense that, if you plot all points $(x,y)\in \Bbb Z^2$ such that $\phi(x,y)$ is true, you get a calculator style pixel display of your formula somewhere on your plane and (unlike the above-mentioned formula) nothing else.
Of course, the requirements are a bit imprecise, since I specified neither the precise nature of $\phi$ nor its encoding as a calculator display formula, but I think that the rule "I know it when I see it" should be sufficient.
I'd be satisfied with any of the following (but I would prefer to see both, if possible):
- An general argument showing that, independent of our precise display conventions, as long as we allow $\phi$ to be a formula in a sufficiently expressive language (e. g. at least as strong as Peano arithmetic), there will always be such a formula. (I suspect that there should be a neat diagonal argument, but cannot think of one right now.)
- An impressive concrete example of such a formula and its plot.
(P. S.: As a precaution against witty commenters, I'd better require $\phi$ to be nonempty)
It seems that someone has already constructed such a function: http://jtra.cz/stuff/essays/math-self-reference/.