plot of the day: Jeff Tupper's Self-Referential Formula

Jan 18, 2007 00:38

Graph the set of points (x, y) such that


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radium January 18 2007, 10:21:00 UTC
No kidding. And not only is he a joker, but he's masquerading as a (possibly former) graduate student at the University of Toronto!

So I played around with the plot a little (I've cooked up a Mathematica 4-compatible version of the notebook if anyone's interested), here's the plot in context of a hundred vertical cells:


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langostino January 18 2007, 13:31:14 UTC
Occam's razor demands testing for Mathematica programmers' easter eggs first!

I'd say one could just as easily see it as Euclid's or Descartes (or any one of a number of people's) egg. In order for this to work, you need precisely the standard mathematical notation (< and the floor notation in particular are susceptible to being represented in other ways, but the numerals themselves could be changed; or switch to slash fractions or negative exponents), a particular choice of coordinates (doesn't work in polar!), and to represent graphs as subsets of a Euclidean space.

I think the Principle of Covariance should apply to proposed evidence of God.

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stereotype441 January 18 2007, 15:48:24 UTC
This reminds me of Fourteen Fruitful Fractions, a crazy mathematical trick invented by John Conway. You multiply integers by fractions according to some simple rules, interpret the resulting numbers in a particular way, and the sequence of all prime numbers seems to come out of nowhere.

At first glance it looks like an incredible mathematical coincidence. But it turns out that the "simple rules" are actually powerful enough to constitute a programming language (called "fractran"), and the fourteen fractions are simply a computer program for finding prime numbers.

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lars_larsen January 18 2007, 16:02:57 UTC
It would have been cooler if he put it in some funky graffiti font.

I didn't calculate it but I have a feeling the information content is in that big ass number. You dont really need an equation if you've got a big number that works as a bitmap.

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radium January 18 2007, 17:05:39 UTC
Yeah! I think that's it.

Which begs more questions -- why this function? What's the class of functions that can generate all possible bitmaps of a certain finite size? If I want a function that generates all 451 x 13285 pixel bitmaps, how do I generate it? :)

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lars_larsen January 18 2007, 17:54:34 UTC
I didnt look into it that closely, it might not be worth looking because it could be obfuscated for no reason. You're on the right track by tweaking it to see how it works though.

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mauitian January 18 2007, 16:36:05 UTC
Hey, that's cute.

As long as you're playing with it -- see if you can write a Mathematica function that takes a matrix of binary numbers (like the plot) and converts it to its corresponding big integer. Then you can get the Tupper number for you name, etc.

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radium January 18 2007, 16:58:43 UTC
Oh dear, this looks like an awful lot of fun.

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mauitian January 18 2007, 22:59:28 UTC
If you want to do it the AI way, instead of just figuring out the inverse formula, you could have a neural net feed the formula numbers and learn what numbers, and changes, produces which patterns, until your net can pick a number to reproduce any given pattern.

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radium January 20 2007, 06:39:15 UTC
There's a great discussion of how to generate Tupper numbers for arbitrary-sized bitmaps over at http://reddit.com/info/yxxw/comments, especially raldi's explanation and code. It seems much simpler than I was expecting...

I guess everything's easy once you know how to do it! :)

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really? kutta January 18 2007, 19:20:28 UTC
Is this function really self-referential? It seems to me that it's actually a function for rasterizing a 1D bitstring into a 2D image... And that he chose the big number to be a serialization of the image of the formula. You can make it print anything you want, though.

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Re: really? radium January 18 2007, 19:32:55 UTC
So demanding! :)

Perhaps you'd be happier if the image also contained the necessary 1D bitstring used to create the image? I wonder how massive the shortest such bitstring would be...

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Re: really? kutta January 18 2007, 20:20:22 UTC
I guess you're thinking of a 'quine'. The shortest one is usually an emptry string, but it depends on the program that is interpreting the bitstring.

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