Tupper's formula | Hacker News Reader

Tupper's formula

en.wikipedia.org

92 pointsagumonkey24 comments

The problem with "Tupper's self-referential formula" is that it's not actually self-referential at all. It's just a method for encoding a bitmap that's encoded in a constant value. That constant value is external data.

That's stated in the article.

Here's an actually self-referential variant that does include its magic number: http://jtra.cz/stuff/essays/math-self-reference/index.html

The always excellent Numberphile's video about it: https://www.youtube.com/watch?v=_s5RFgd59ao

You use the words "always excellent" as though you've never had to explain to someone that no, the sum of all positive nonzero integers is not actually equal to -1/12.

That's how I found out actually. I don't like the bias into self reference too much, the formula being re-encoded into K (IIRC). It seems like an algebraic encoding of a bitmap. Anyway, pretty neat find.

At least it seems reasonably efficient. The bitmap is composed of 106 * 17 = 1802 bits, while the digits contain 543 * log2(10) = 1803.8 bits; not bad, disregarding redundancy in the image.

Great point.

And here is a link to the image generated from the number in his book: http://i.imgur.com/qczpXxp.png :)

He confirmed it on a reddit thread yesterday http://www.reddit.com/r/math/comments/32q0cx/the_everything_...

How would someone even go about figuring this out? Maybe Tupper was incredibly bored?

You don't discover this, you construct it this way. It's pretty basic math.

The formula simply interprets the big magic number as "pixels". Then you just pick your magic number to encode an image that looks like your formula.

I would have been more impressed if that number k was also depicted in the image.

I doubt there's enough space in that grid, but maybe you could find another similar formula with a bigger grid?

The 543 digit number encodes log(10^543, 2)~=1803.8 bits, and the image is 17x106=1802 bits. It's going to be hard to fit it in there.

You could do it if you modified the formula to decode a compressed bitmap rather than an uncompressed bitmap.

You could probably make the compression as simple as encoding a "font" in the number (or formula)... then it's just an extra indirection to lookup each digit's bitmap.

But some numbers will get larger when compressed. And some here means most of them.

You're not compressing just any number... you're compressing a number representing a bitmap that's mostly white. That that can be done is not just theoretical... it's done all time time. (Computer files are, after all, just large numbers.)

You are of course correct, somehow I tricked myself into believing this would require compressing the number for every image into its image. But on a second thought there might still be a tough problem - altering the number will alter the image which will in turn require altering the number again. So the (possibly) hard problem is finding a fixed point which may actually not even exist (for a given formula).

Ah, but there is a science behind writing such things. They're called quines. Here's one I wrote: http://chris.pacejo.net/postscript/quine.ps

The trick is that the part of the number (program) that encodes the image of the number doesn't need to encode the image of itself -- that can merely be created directly from that part of the number. See e.g. the source of (or output of!) the quine I linked. The first half of the program is the "image"; the second half basically just prints the "image" once as it is represented in the source, and again after interpreting it a bit. (In fact this way you don't even need to compress the "image"... the compression comes from the fact that your "program" is roughly just two copies of the same thing.)

The number k is depicted in the image - take the pixels of the image, interpret them as a binary number, multiply it by 17, and there you have it.

Someone did that too! http://jtra.cz/stuff/essays/math-self-reference/index.html

Wow, that is very cool!

Now that, is cool.

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