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I have recently discovered that I can create letters and any shape I want by hiding parts of curves by making them complex. To generalise if I want $x>a$ then I multiply my function by $\sqrt{\frac{|x-a|}{x-a}\,}$ and replacing $x-a$ with $b-x$ I can make $x<b$.

The challenge I put forward is this:

Create an equation or set of equations, which graph the equations themselves

Is this possible?

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The empty set. :) – Hurkyl Dec 17 '12 at 22:46
You could say the same for a computer program quine. An empty program will print itself out ie nothing. Clever thinking though – Jordan Brown Dec 17 '12 at 22:48
You'd pretty much have to give a complete language for curves and fonts before you could answer this question. I suppose there might be a general proof that there is always a quine, no matter what the font definitions and descriptive language... – Thomas Andrews Dec 17 '12 at 22:48
@Jordan: My answer was an homage to that one. (that was actually the submission to the obfuscated C contest that put an end to that aspect of the competition) – Hurkyl Dec 17 '12 at 22:49
You might be interested in Tupper's self-referential formula – Ross Millikan Dec 17 '12 at 22:50

As Ross Millikan says in the comments, Tupper's self-referential formula is a famous example.

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+1, but Tupper's "self-referential" formula (which he never named so, in the excellent paper where he used it as example) just takes a bitstring (the region) and forms a bitmap out of it. This is like a program that echoes its input: sure, when given its own code as input it does print it out, but that doesn't make it a quine. However, Jakub Trávník has written a true mathematical "quine", whose graph contains all the information necessary to recreate it: here. – ShreevatsaR Jun 24 '13 at 9:13

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