>>BRADY HARAN: I think most Numberphiles are fascinated by the idea of Graham’s number, which is supposedly unimaginably big – but what actually is it? What does it count? Well, I think the best person to explain that is none other than the world famous mathematician himself, Ron Graham.>>RON GRAHAM: Suppose you took four vertices of a square – and, this is two dimensons, so it has how many vertices? Two squared (four vertices). Now, what I’m going to do is take these four vertices, and put – two more edges connecting all possible pairs of these vertices. Now how many possible edges would you be able to draw? Well, there’s something called a binomial coefficient, and there are four vertices, and you choose two of them, and this turns out to be six. So you see there are six edges here – all the ways you could join these four dots. And now, what we’re going to do – is – take the four dots again and look at each one of these possible six lines and color them red or blue in any way that you like, so for example – I might color this one red, and maybe this one red, and then I’d say, well ok, maybe I’ll color this one blue, and this one blue, and this one blue, and this one blue. Ok, so that’s one way, and you might say, well, okay, so what? I’m looking for certain special configurations – and, I’m not going to find it here in a square because this is too small; we need more dimensions. What about three dimensions? Well, what does a three-dimensional cube look like? Well, I can’t really draw it on a piece of paper, but you can kind of draw a version of it, namely, if you take your four vertices that form your square and then you kind of move it a little bit, well, that’s a picture you might recognize, how many vertices on these corner points here? Well, you can see there are eight, which is two cubed. So the three-dimensonal cube has – two cubed – eight vertices. 28 possible line segments. And if you color them, say, this one’s red, and this one’s red, and this one’s red, and that one’s red, but – you know – a lot of the others are blue, and what you’re trying to do is do it in a way that avoids something. Now, what is this thing you’re trying to avoid? That’s the whole point. You’re trying to avoid the following configuration: namely, four points in here someplace and all the six line segments have the same color (in this case it’s red) and, furthermore, these four points lie in a flat plane. They don’t stick up in three dimensions. Usually, if you took four points, you’d have three dimensions. It kind of spans a three-dimensional volume. Here, I want four points to be flat, and they form a square, and all six line segments are the same color. In three dimensions, you definitely can avoid such a configuration, because this only seems to happen on, just kind of the faces. And what you have to do – your challenge – is to go in a much larger dimension. And what do you mean by a cube – in say – four dimensions? Well, in four dimensions, you could take this picture, and translate it, and join the lines, but it gets kind of messy, but yeah. I’m not afraid; I think I can do that. Let’s see. Here’s a three-dimensional cube, and now I have to shift it, okay, well, I’ve gotta be a little careful here – and now I join the corresponding things. Well, you can see it’s going to be, as we can call it, a tesseract. Now how many different ways can you color these 120 different lines with red and blue? Well, with each one of them, there are two choices. So, it’s two times two times two times two 120 times, or 2 to the 120 (2^120). Already beyond what computers can do, and no computer can do 2^120 things right now. But the problem we’re trying to avoid having these four points that form a square lie in a plane and have all six line segments the same color and in four dimensions, you can do it. Not too hard to do. Well, what about the five-dimensional cube? In five you can avoid it, six – it takes more work, seven – yes, it takes a lot more work, the question is, can you always avoid it? The answer: no. If the dimension is large enough, you cannot avoid it. No matter how hard you try to color it, you can’t avoid it. And when is that dimension finally reached? Well, you know it’s reached by the time you get to a very large number, sometimes known as Graham’s number, a pretty big number, and you know for sure, in that dimension, you cannot avoid it. And the proof that you can’t avoid it is pretty complicated, pretty recursive, and that’s what causes this number to be so large. Now, you might say, well, what’s the truth? Sure, if you go as far as Graham’s number, in that dimension, you can’t avoid it, but maybe you can make it unavoidable in a smaller dimension. And it turns out that it appears that already in thirteen dimensions you can’t avoid it. Now, how could you check that by computer? Well, okay, no problem really, in thirteen dimensions, so “n” is thirteen dimensions, so how many vertices in your thirteen-dimensional cube? Well, 2^13 vertices, which is, what, that’s about 8192. Okay, a little over eight thousand vertices. Well, how many line segments? Well, 8192 times 8191 over two – you have to check – no matter how you color them with red or blue – you’ll make one of these things. Well, it’s uncheckable. You can’t check that; it’s impossible. You need to think! You can’t just compute. Right now, we don’t know. It’s between 13 – we know, maybe – uh, 12 is not enough, it turns out, and this other, slightly bigger number, which uh – I can – I can try to write it,>>BRADY HARAN: So there we go. As cubes move up the dimensions, from three dimensions, that I think we all understand, to four dimensions, to five – eventually we reach a number of dimensions in which one of these configurations must exist. And the upper bound on that, the point where it has to happen by, is this “Graham’s number”. Now, we know this number is unimaginably big, it was even in the Guinness Book of Records. But just how big is it? Well that deserves a video all of its own, and again, we’ve got Ron Graham to explain it. The links are on the screen here, I’ve also put them in the description under the video. So, why don’t you check it out? It’s a pretty big number.

Time travel paradoxes

What if you colored the vertices red and blue as well? What's the fewest number of dimensions where it's impossible not to have a 2-D square of vertices with all red or all blue lines

andvertices?I don't understand why 2 to the power 33,550,336 posibilities need to be checked for 13 dimensions. Doesn't one configuration whereby the 6 same-colored vertices does not occur suffice? Or do these valid configurations become very scarce with higher dimensions?

The reason that it does not matter if you are using plank times or millennia is not actually just because the numbers are big. It is because, the way the numbers are written, a third significant figure would alter the result many orders of magnitude more than any unit conversion could. The calculation is not precise enough to make units relevant. To demonstrate, type 10^10^1.5 and compare it to 10^10^1.52. Notice that those small significant figures add two orders of magnitude to the answer! In the chain written in the video, the difference between 1.1 and 1.11 would be huge enough to make units irrelevant.

On a scale of 13 to Graham’s number…

whats the application for this?

Quick mathzzz

Is he alive today?

Am I ignorant because I wonder, if you cut the in diagonal halves, you would have four points with six egdes on one plane, as stated to avoid? I ask because there would be six of these possible planes on any cube, two for each axis. Or are they not counted because these would not be square?

He talked about coloring the lines red and blue, but Numberphile colored them red and cyan. That just ruined it for me.

So what are the applications of knowing this information? How exactly is it that this multidimensional "cube" HAS to be completely one color and it is completely unavoidable?

Its just a little larger than 13

He is trying to make a surface out of edges which is huge somewhat to the order 13 because 5 plus something and that something is Phi. .

G64 + 1 BOOM, I'm better

WHAT MAKES THE LOUDEST SOUND?

GRAHAM('S) BELL

Harrison Wells has taken another body

Base is ten extra dimensions are two so twelve. So twelve is possible Phi × 4 is the max. Four is the constraint.

Instructions not clear, mind permanently disappeared

This guy gives me hope in mankind….

IMPORTANT: The upper bound is not Graham's Number anymore, and, strangely, NEVER WAS. The original bound was proposed in 1970 and was about G(12) in terms of Graham's function. Only in 1977 was the paper about G(64) published. In terms of up-arrows, the new upper-bound is 2^^2^^2^^9, which is still REALLY big, but it's actually less than even G(1).

Aham's number

Three13d—125375757748856786788789707896958967787

a+1=counting

a+b=addition

a×b=multiplication

a^b=exponentiation

a^^b=tetration

a^^^b=penetration

a^^^^b=intercourse

a^^^^^b=sigulation

How many possible lines of chess moves can be played from the first move onward?

I’m confused.. why is the pattern that their trying to avoid not possible to avoid at some point?

DAMN!! my mind is dying

ok

THIS is epic

Graham's number + 1

It is awkward to know the graham number Is Explained by Graham itself who discovered it. Lolz

The biggest number is 9

So what's the proof here? Graham's number is where it's unavoidable says what?

how about if you made all the diagonal edges red and the strait (in x,y,z,…) edges blue??

wouldnt that avoid having any squares of all one color??

Wait doesn’t the newest version of string theory call for 12 spatial dimensions and it said 13 may be unavoidable for crossing lines coincidence probably but SCIENCE

So what?

0:57 tho

who else cringed

Couldn't you clone using this?

3:32 That's from Wikipedia but with the colors reversed, right? A shame you can't even make your own rotating tesseract.

I avoid this

Imagine it'd be 42d. WW3 would start

I love his voice and want more videos narrated by Ron Graham.

Huh, he's still alive… That's unusual for having something named after you

They say that at dimension (Graham's number)^Graham's number it can be avoided, but only if you ate a banana in the last 24 hours

this sounds an awful lot similar to the ramsey theory graphs, is there some correlation?

Grandma's number…

3:28 it s magic

Well what if 13 dimensions is enough to make the 1-colored plane unavoidable? Would Graham’s number be useless then

Why can't you just replace one of the center crossing lines with the other one?

GP means googolplex

GP^^GP = F1

GP^^…F1 number of arrows…^^GP = F2

GP^^…F2 number of arrows…^^GP = F3

F100 = GP(F99^)GP

Surely this beats Graham's number.

THIS VIDEO IS MY TOP 5 ALL TIME FAVORITE YOUTUBE VIDEOS SINCE I FOUND YOUTUBE IN 2006!

Can someone please explain me how did Mr. Graham able to conclude this "If the dimension is large enough you cannot avoid same color vertices configuration". Is this merely a hypothetical assumption or do we have a concrete evidence of this? Thx in advance. 🙂

I just don't know why the they don't teach us this kn School.

School makes us hate everything….

With what respect(according to what) is he colouring the lines(edges) plz reply??????

{4,64,1,2} is bigger than g_64

Well if Graham found it then how come he doesn't know it??

What if I colour ALL the lines blue except ONE which is red? Haven't I avoided it? There was no rule of using one colour multiple time

His hands moves via sine-wave function.

So is the 14th dimension just completely made up of these uni colored squares?

What's interesting to me is not how big the number is, that's just a

number, it can always be bigger. What's interesting is the application.

1) how would a 14 dimension cube even look like??

2) how is he so sure it's unavoidable????

3) what exactly triggers AVOIDABILITY on the 13th and 14th ????

at least at Grahams number one can't avoid that kind of configuration. But how does he know?

Love videos on Graham’s number. Graham is a genius!

Wait, so it's not avoidable in the 13 dimensions. Is that why string theorists say, there's only 10-11 dimensions, because everything would collapse in the 13th dimension? Or is that just a fishy

coincidence?my man is shakiiiiiiiiiiiiiiiiing

This guys voice reminds me of Bob Ross for some reason

I have trust issues with people who have an unsolved Rubik's cube.

Would love to know why it is so big, and what is the reason for the recursion. The only thing I noticed is (4 choose 2) is 6, and 2^6 is 64, and G64 is Grahams Number. What is the connection?

Honestly I think mathematicians just dream up this BS because they have nothing constructive while all the engineers are making space engines, biologists are curing cancer and chemists are making hangover free alcohol or some such things…

( 2^(x-1) ) * ( (2^x) -1 )

where x is the number of dimensions, represents how many lines will be created in the given dimension.

I started with this formula:

( 2^( 2x-1) ) – ( 2^x )

Using algebraic manipulation I ended up with the former. On paper, at least, this first formula is much simpler than what I started with, but the second formula is more tangible to perceive.

Why didn't numberphile explain this in their first video about grahams number?

Why is his hand so unstable

Whats this thing that they have about writing on brown paper??

Why select such a conservative Progressor as 64 ???????? G65 is so much larger

so graham's number is an upper bound to how many dimensions must a complete 2-colored graph of a cube have to be forced to contain a complete single-colored coplanar 2-dimensional graph of a cube. what if the graph could be 3-colored? 4-colored? graham's number-colored? or what if it could have as many colors as dimensions? what if the single-colored graph we're trying to avoid was 3-dimensional, not 2-dimensional? 4D? graham's number-dimensional? we could make a function from this: f(c, d), where c is the number of colors of the main graph and d is the number of dimensions of the subgraph. we can also put countable ordinals as c, where if n is the number of dimensions in the main graph, then c becomes the nth member of the fundamental sequence of the ordinal, so f(omega, 2) means that the graph could have as many colors as it has dimensions

2^33550336 easy

5D, 496 possible edges.

I got Graham's Number tattooed on my ankle today. Original instead of an infinity tattoo 😂😂

What we avoiding again ?

I couldn't understand it

3:48 " far beyond what computers can calculate" I did the calculator, and it did mate

Wtf I'm been watching this everyday for the past 5 days.

2＾120 equals 100000000….(120 zeros) on base 2

I confess – I prefer Numberfile's videos about numbers I can get a handle on!

Very cogent explanation by Ron Graham.

3:28 Hey, I'm on the video !

What happens if you call the Ackerman function with Graham's number as the arguments?

What did i miss? Wich config you have to avoid? Same color in a side or differnt colors in.a.side?

The best person to explain Graham's number is day[9]

So is grahams number the dimension or that ways to color the lines????

It like watching Bob Ross teach Math.

what is the purpose of finding those specific configurations?

World famous mathematician but can't draw a straight line….

''Already beyond what computers can do.'' And then they show the number. Hahaha!

This other, slightly bigger number.

Basically it is known for being the worst upper bound in history.

So the upper bound on the number of dimensional cube where a plane with a square that is all one color is unavoidable is somewhere between 13 and graham's number. Physicists: "the cosmological constant problem is a theoretical prediction that differs from observation by a factor of 10^120. It is the worst prediction of the worst precision in physics." Mathematicians: "Hold my beer."

Sometimes maths just seems so pointless

2019 from Still Untitled on Tested

It just got over my head 😴

3:06–3:10 aaahaha, Yupp😂😂😂😂

wow he has the most wobbly straight lines ever

5:59 famous last words