The 'Everything' Formula - Numberphile

MATT PARKER: Today I'm gonna show you a... amazing number, and some of you may have come across this before. It's an absolute classic, but I still think there's a lot to be said about the number 9609393... ...four, nine --this is where it gets quite interesting actually, 'cause it goes nine, eight, two... ...three, eight, zero, nine, six, --you'll notice, actually, this is one of the more boring parts of the number-- seven, six, zero... Every time I mess this bit up, hang on, that... that bit shouldn't be there; I put that in by accident. Sorry. Where was I? ...forty-seven, one, nine! And there it is; that's the number I'm going to talk about today. I still find this number absolutely fascinating, and the reason this number is so great is because it relates to a fantastic inequality. A formula if you will, which I will put underneath here. So, the formula we're gonna look at is: ½ less than the floor function of the mod function of the floor function of the y-coordinate on 17, —floor function— times 2 to the power of negative 17 times the floor function of x subtract the mod of the floor function of y again. Oop, that's floor, sorry. And that is mod 17, so I close that mod there and then this whole thing, once you've got that, you then -- that open mod is 2; close that, and then close the floor function. OK, so now we have ridiculous number and we have ridiculous equation. To show you how the two are linked, I thought I would show you the plot of this inequality, because what they're saying is: if you have an x,y axis and you put—for each point on that plane—you put in the x- and y-coordinates into this formula, it would tell you whether or not you should colour in that point. So, in fact, this tells you which part of an x,y plot should have a dot, and which shouldn't. And I've got a printout I have done of that formula, and, so, if you plot that formula, you get this plot here. So, this is the plot of that formula, and, in case you're wondering what is particularly astounding about this, the formula, when you plot it, is itself. It is its own form-- The formula is its own plot. The plot is the formula. This is called Tupper's self-referential formula, and when you plot it, you get itself, and that, for me, is one of the freakiest, most bizarre parts of mathematics. A formula which is its own plot. Okay, there is, however, one slight cheat it does-- it actually does that; if you plot this formula, you get itself as its plot, but the cheat is this 'k' down here, because I haven't actually specified where we are on the vertical y-axis. You can see on the x-axis we're zero to 106; fairly normal territory. But here we're like "'k', what on earth is that?" That k value is this number here. Basically, we are a long way up the y-axis right now. If you go a huge distance up the y-axis of this plot, and you ignore everything else below it, and there's more after it; ignore everything above it. At this one particular zone between this number and this number plus 17, is where you get the equation that we were using to plot them. What's actually amazing about Tupper's self-referential formula is not that it plots itself — it's that it plots everything. As you go up the y-axis, it plots every single 106x17 grid of white and black pixels. So, anything that fits in that grid is somewhere on this plot. So, not only is Tupper's formula in the plot, but also every other formula you can fit into that size. In fact, I found a different value of k which is where it's the same formula, but smiley faces instead of the word 'mod'. And so, that's a slightly different value of k, gives you the smiley-face version of this formula. Also found the point where Pac-Man is eating the formula, as chased by a ghost. And so, that one is... I think that one's gonna be higher up as well It's... yes. It's not far off, though. And so, anything you can possibly fit into one of these appears in there somewhere. And I can actually show you how you can get from any picture you want to that number. The way it actually works is you take the pixelated form of your equation, so let's get our original one. Let's do... Where's the proper one? None of these ridiculous ones. Okay, here were go, right. So there's...there's the original one. And what you do is you start in the bottom left-hand corner, and if you want that to be a black pixel, you put a 1. And then you go to the one above it. And that's black as well, so you put another 1. And the one above that is white, so you put a 0. The one above that is white; 0. 1. 0. 1. 0. 1. 0. 0. Is that two or three? Looks like three. 0. 1. 0. And then you carry on. And you scroll up the first column, and then up the second column, and up the next one. And you write it off as a binary number. And you end up, at the very end, with a very long binary number. You then put that into base 10. You multiply it by 17, and that gives you the k value for where that plot appears on the graph. In fact, this is a bitmap function. And what Jeff Tupper was doing was looking at using equations and being able to plot them automatically using software, he's a computer scientist at the University of Toronto. And this is a kind of a side-effect, it's an interesting outcome of the legitimate proper research he was doing. In fact, Brady, I got you a present. I took the Numberphile logo, pixelated it, turned it into a binary number. Multiplied it by 17. Put it into base 10. And there it is down there. So that is the k coordinate, of the Numberphile logo, on Tupper's plot. Okay, so, when I wrote my book, Things To Make and Do in the Fourth Dimension Brady: Woah, woah you happen to have it on you? Matt: I happen to have it with me at all times as you know Brady. I wanted to put in Tupper's self-referential formula, and so in the book, ah here it is. In the chapter on functions I put in a picture of it, and I'm all like "oh wow that's amazing" and then later on in the book when I get to digital images, because it is just a bitmap (where is it, there we are) here we go I've got it again and I explain the system. So there's the actual plot, there's if you turn it into 0s and 1s And there's when you multiply it by 17. Except, that's not actually the correct number. So the plot's correct, the binary number's correct but this is not, because I thought it would be hilarious to put in a different number. And so if anyone ever does take that number -- and I've never told anyone, Penguin don't know this, FSG who published it in the US don't know this. I've not revealed it to anyone -- this is the first public release of this information. If you reverse that number you get a different plot, I have defaced the plot for that number. This is the world's slowest, longest running troll you'll ever come across. So hopefully, you know, years from now someone may finally get around to doing it. And they will not be faced with that plot.

Channel: Numberphile

Views: 1,725,083

Rating: 4.9731207 out of 5

Keywords: numberphile, tupper, Formula (Literature Subject), Mathematics (Field Of Study), matt parker

Id: _s5RFgd59ao

Length: 7min 10sec (430 seconds)

Published: Wed Apr 15 2015

Reddit Comments

[u/JshWright]

I love Matt Parker. His 'Calculator Unboxing' videos are hysterical.

https://www.youtube.com/watch?v=eaJtjJNrWf0

[u/paypaypayme]

At first I thought he was trolling, until he explained that it's a bitmap function. Pretty cool stuff.

[u/TokenMenses]

Now that's Numberwang!

[u/way2lazy2care]

For people who aren't aware, Numberphile has a sister channel called Computerphile, and it's probably my favorite youtube subscription at the moment. The topics they cover are so varied and increase my base knowledge of computer so much. It's pretty much a must subscribe channel if you're a programmer imo.

[u/pixelrebel]

Does anyone have a python implementation of this?

[u/cleroth]

Output from the number in the book (Python code by /u/smikims):

+/u/CompileBot python 3

num = int(input("Enter your k: ")) // 17

out = []
print()
for i in range(106):
    row = ''
    for j in range(17):
        if num & 1:
            row += '#'
        else:
            row += ' '

        num >>= 1
    out.append(row)

for i in range(17):
    for j in range(106):
        print(out[-1-j][i], end='')
    print()

Input:

960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221619561473865746304976129424093469921873058694492149444647882055806603996307772920108275439090931487231139508469267169521581872227293630931364751681875244141639118844172571080588839278417813855101724217755801034516516847318278139146496085068307449373183066378525002863703739215155304174822734164455483814441481301873381703922338834284527

[u/kwirky88]

I'm losing hair like him but don't sound as smart. I am depressed.

[u/rcxdude]

This one goes one better because it has the coordinates to plot inside the resulting plot itself.

[u/YolocostSurvivor]

Wait a minute...DickButt?