Weird relation: its graph is the formula

1. Aug 3, 2007

phoenixthoth

I thought this was really odd. I wonder if they found it by accident or were trying to do this...

Apparently, the graph of this formula is the formula:

graph of formula:

the formula:

Last edited by a moderator: May 3, 2017
2. Aug 3, 2007

ice109

that's ridiculous

3. Aug 3, 2007

4. Aug 3, 2007

HallsofIvy

Staff Emeritus
A "graph" is a set of points. A "formula" is a set or symbols. What, exactly, do you mean by "the graph of the formula is the formula"?

5. Aug 3, 2007

phoenixthoth

I mean that the picture of the graph

http://mathworld.wolfram.com/images/gifs/TuppersFormula.gif

is the formula that generated it

Last edited by a moderator: May 3, 2017
6. Aug 3, 2007

HallsofIvy

Staff Emeritus
And I will assert again that a picture is not a formula!
What exactly do you mean?

7. Aug 3, 2007

ice109

aww comon halls you're being to picky. the graph of the formula/equation/algorithm draws out the formula/equation/algorithm

whatever amorphous concept we are witness to by looking at that picture and reading the wolfram blurb, it's cool.

8. Aug 3, 2007

phoenixthoth

I still mean that the picture of the graph

http://mathworld.wolfram.com/images/gifs/TuppersFormula.gif

is the formula that generated it

So when you say that a picture isn't a formula, that means you don't consider the following picture to be that of a formula?
After all, if formulas are just collections of symbols, do you think that pictures can't contain symbols?

Last edited by a moderator: May 3, 2017
9. Aug 3, 2007

phoenixthoth

Seems like HOI would say that the graph of y=3x+2 is NOT a line because its graph is actually a set of points. In fact, y=3x+2 isn't even a formula because it's not a set of symbols: {y=3x+2} is a formula while y=3x+2 is not.

It's called abuse of language and it happens all the time.

Last edited: Aug 3, 2007
10. Aug 3, 2007

christianjb

Anyone for Goedel?

11. Aug 3, 2007

HallsofIvy

Staff Emeritus
No, I would definitely say that the graph of y= 3x+ 2 is a line. I would object to saying that "3x+ 2 is a line"!

Saying that {y= 3x+2} is a formula because it is a set of symbols while "y= 3x+ 2" is not- not that's picky!

12. Aug 3, 2007

phoenixthoth

Ok, so the graph of y=3x+2 is a line because if you draw a picture of the graph, the result looks like a line. Similarly, if you draw a picture of the graph of the formula http://mathworld.wolfram.com/images/equations/TuppersSelf-ReferentialFormula/equation1.gif [Broken]
the result looks like the formula that the graph is generated by.

Last edited by a moderator: May 3, 2017
13. Aug 3, 2007

christianjb

Since all formulae on these web-pages are composed of pixels- I would definitely have to allow that a picture of a formula is a formula itself.

14. Aug 3, 2007

phoenixthoth

do yall think they found this by accident or were looking for this for some strange reason?

15. Aug 4, 2007

Gib Z

It definitely wasn't found by accident. How they did it though, I can't tell you, or else I'd have to kill you =]

16. Aug 4, 2007

christianjb

I followed the link on the mathworld page to the entry on self recursion. It's got a good puzzle:

What is the volume of a pizza of thickness a and radius z?

17. Aug 4, 2007

HallsofIvy

Staff Emeritus
It took me a moment- I'm kinda slow- but yes, that is good.

18. Aug 4, 2007

HallsofIvy

Staff Emeritus
Okay, I finally broke down and looked at all the references- Yes, its weird and I must agree that in this case the graph is the formula!

19. Aug 4, 2007

uart

No phoenixthoth, the formula is interesting but not nearly as weird or mysterious as you may first think. I have to admit when I first saw it today I was pretty "blown away", until I figured out how it works. It's actually nothing more than a recipe for decoding an arbitrary binary bitmap (monochrome, height=17, width=arbitrary). The bitmap is stored in the huge number n. Imagine each vertical column of dots in the bitmap being represented by a 17 digit binary number. Now imagine all of those 17 bit numbers being concatanated into one large binary number. Well that's what "n" is.

Look closly at the formula. First notice that the whole LHS expression is modulo 2. That just decodes each binary bit to either a "0" ot a "1" (0 if the inequality is false, 1 if it is true). Now look at the 2^(-17 floor(x) ...) part of the expression. Can you see that this is merely shifting the bits in the binary number. The mod(floor(y),17) part is shifting out the bits in each column (that is, as y is varied for a given x) while the 17 floor(x) part is shifting the bits 17 at a time (that is, selecting a different 17 bit column bitmap for each different value of x).

You see it's actually fairly straight forward once you see how it works. The formula can "draw" any bit map that you can make with a height of 17 pixels and arbitrary width, all by constructing the appropriate number "n".

Last edited: Aug 4, 2007
20. Aug 4, 2007

phoenixthoth

It would have been easier on me if PF allowed img tags to produce images... Then you wouldn't have had to go through the trouble of clicking a couple links.

uart: interesting analysis. thanks.

pizza! um, yeah...