Gird your loins. We're going in deep.

So I saw this image and it reminded me of something i had learned a while ago. I want to share that thing.

This image shows the fact that we can display objects with a higher dimension the the surface we are displaying it on by taking the objects lower dimensional counterpart, translating it, and connecting the vertices

So what do cubes and vertices have to do with it?

First I need to BRIEFLY explain Ramsey Theory. One of the questions posed by this theory is "How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property?" So the 'original structure' for us will be hypercubes (higher dimensional cubes) with colored lines connecting the vertices. the colors in this case will be red or blue

So what's the interesting thing we want to happen?

I think the above picture will explain it better than I can in words. Basically what we want is a flat plane (2D) of lines connecting vertices that all have the same color. Please do not mind my shitty Paint skillz... because there will be more

How do we make the interesting thing happen?

This is the first part of Ramsey Theory that i'll discuss more in depth. It states "How big must the original structure be..." or in Layman's terms: how can we force it happen. For this problem its fairly simple. Make a hyper cube in the Nth dimension. "What dimension exactly though?" I hear you asking. Don't worry, we'll get there

Enter Ron Graham

This dude came up with the upper limit to that question. Keep in mind UPPER limit, the actual answer is either equal to or lower than this but that configuration of colored lines and vertices has to happen at that number of dimensions. "What number of dimensions is that?" I hear you ask, well keep reading.

Yeah that's gonna be a bit difficult...

I'll show you instead of telling you. Lets make sure you have the basics before we get crazy though.

you can combine multiple additions into multiplication:

3+3+3+3+3+3+3 = 3*7

you can combine multiple multiplications into exponents:

3*3*3*3*3*3*3 = 3^7

good. lets go

Let learn new math!

This is Knuth's up arrow notation. It compresses exponents into a 'simpler' form.

Lets apply this to the number three (3)

we'll need this to get to the big number. It's literally a bunch of threes.

This is gonna look like child's play one were through here.

Yeah, were learning a different notation to write these things down (even though we can only really write down one simply).

(To all the haters: I will keep using paint. I like it.

Getting bigger (hur hur hur its an innuendo)

By the way, we ain't even scratched the surface yet.

This is three arrows, god help us all...

three arrows is a compressed operation of two arrows. It's 'A' double arrow 'A' a 'B' number of times

I think you get the picture now

this is a REALLY big number. for perspective, 3 up arrow 4 is a 3.6 trillion DIGIT number and 3 up arrow 5 is 3^(that 3.6 trillion digit number) and DWARFS googolplex.

So now we get too big for the universe we live in.

if we were to write out the entire stack in 2cm font it would stretch 150 million KILOMETERS. That's right. TO THE SUN.

So now four arrows... Why...

We'll call this number G1.

By the way we ain't even fucking CLOSE to Grahams number. We need to go deeper.

BUCKLE UP BUCKAROO CUZ WE AIN"T EVEN CLOSE TO DONE!!!

so let's say G2 is 3(G1 amount of up arrows)3 now what do you suppose G3 is?

That's right 3(G2 amount of up arrows)3

Now how many times do you suppose we repeat this process? Five? Six? Seven?

Oh, honey...

We repeat that process 64 times.

SIXTY. FUCKING. FOUR

That's Graham's Number.

Ron Graham's proof also contained a lower limit to the dimensions the cube can exist in to force that certain outcome.

It's fucking six. Or is Ron himself's words "A small gap in our knowledge"

Mo' Fuggin cat tax

Thanks for reading. Hope your head stops hurting soon.

sauses:

http://waitbutwhy.com/2014/11/1000000-grahams-number.html
https://www.youtube.com/watch?v=XTeJ64KD5cg
https://www.youtube.com/watch?v=GuigptwlVHo
https://www.youtube.com/watch?v=HX8bihEe3nA