Tupper’s Self-Referential Formula is an equation that, when graphed, displays the formula itself. Here’s the equation (same as the image of the output shown at the beginning of this post):

Check out the link for the specifics of the functions and values of the variables.

(Via kottke)

## 12 thoughts on “Tupper’s Self-Referential Formula”

1. Wilberforce says:

To me, this is no less amazing than tuning your television to an empty channel and finding a schematic of how to build the television depicted in the static. I tried to explain this to my wife a few times, but I just get blank stares after a while.

Of course, I have no way to test if this is true, or if this is some joke that would make milk come out of mathematician’s nose, but I gathered from some other sources I found on the web that it is.

The fact that someone would find this odd equation and happen to graph it over such an odd range of numbers seems astronomically improbable. What are 105 and 9.609… *10^453 doing running around together? I’d say 105 is dating waaaaay out of his league.

Over the weekend I started thinking about this and figured that Tupper couldn’t have just stumbled across this; it has to be “manufactured.”

For one thing, 9.609*10^453 (which I’m going to call ‘n’ from now on) does not look like a random number to me (as ridiculous as that sounds. I mean, what number isn’t random at some point in time?). n seems to contain too many stretches of repeating digits for it to be just any old number. To me it just seems suspicious.

If you want to figure this out on your own, stop reading now.

To test my theory, I busted out the Java BigInteger class and converted n to binary. Sure enough, taken 17 bits at a time, n looks a lot like the graph of Tupper’s Self-Referential Formula. In fact 106*17 (the size of the graph) is 1,802, which is exactly the number of binary digits it takes to represent n. Dividing n by 17, converting it to binary, and arranging the bits in columns of 17 bits each yields Tupper’s formula exactly. What Tupper did was develop a formula that converts y into its binary representation and then constructed n such that it represented the formula itself – sort of like dot-matrix printing in reverse.

I still think this is pretty dang clever, but before I understood how it worked I thought it was nothing short of mystical.

2. James G. says:

Re: Comment By Wilberforce

Great explenation! Amazingly behind your sarcastic remark, the way you were able to explain how you came up with the way the equation works, to print itself, is great. Thanks for your wonderful interjection, hopefully you get to read my responce.

3. KPSS Matematik says:

This is great issue. thank you

4. Web Design Company USA says:

Nice insight :)

5. Web Design Company USA says:

Nice insight :)

6. Petey says:

I will never feel guilty about writing stories, poetry or lame philosophical ramblings again.