Visualizing Graham's Number

Visualizing Graham's Number

Anomie

Member #9,403

January 2008

I've been thinking about Graham's number for a little while now, and I've been trying to visualize g₁. (that being the first of 64 layers leading to Graham's number (g₆₄)

I started on paper, but quickly gave up. Then I installed the GMP library, with the hopes of writing a simple program to calculate the value of g₁. I'm sure I don't have to tell you that it's still much, much, much too large. Even just that first tiny layer is incomprehensibly massive.

So I wonder, how do you visualize g₁? Or even Graham's number? What analogy could possibly stand to help someone appreciate the size of this number?

(I apologize to the people who aren't fascinated by large numbers. I'm finding that's it's sort of an on/off switch with people: either it's really interesting or not at all.)

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Jakub Wasilewski

Member #3,653

June 2003

Quote:

So I wonder, how do you visualize g1? Or even Graham's number? What analogy could possibly stand to help someone appreciate the size of this number?

You don't. You simply run out of universe while trying to imagine it visually .

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Anomie

Member #9,403

January 2008

I read somewhere that if all of the material in the universe were turned into a pen and ink, there wouldn't be enough to write the number. How many universes would it take? My bet's on 42.

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I am the Lorax. I speak for the trees. I speak for the trees for the trees have no tongues.

Jakub Wasilewski

Member #3,653

June 2003

My bet is on g₆₃ .

Neil Black

Member #7,867

October 2006

My bet is on 2. Our universe is just really, really tiny compared to an average universe.

Johan Halmén

Member #1,550

September 2001

If all of the material in the universe were turned into a pen and ink, there wouldn't be any paper to write the number on.

Our average universe is very small compared to other sets of universes.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.

Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.

Anomie

Member #9,403

January 2008

I'm not sure I could bet on 2. Unless the second universe is 10^{476^{8497^37894483}} times bigger than ours, or something like that. Is it? That would be way cool.

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I am the Lorax. I speak for the trees. I speak for the trees for the trees have no tongues.

Neil Black

Member #7,867

October 2006

It is.

Roy Underthump

Member #10,398

November 2008

IIRC, even a google (as opposed to googleplex) is much larger than the number of particles in the known universe.

http://en.wikipedia.org/wiki/Googol said:

A googol is greater than the number of atoms in the observable universe, which has been variously estimated from 10^79 up to 10^81

so each atom would need ~= 10^20 particles to overflow 10^100.

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Onewing

Member #6,152

August 2005

I bet Graham was pompous. "My number is larger than yours."

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nonnus29

Member #2,606

August 2002

Bah, there's more than a googleplex in the open interval (0, 1).

Universe? Look to the space between you thumb and index fingers, FOOLS!!!!!

link to my blog

gnolam

Member #2,030

March 2002

Goddamnit.
Googol: 10¹⁰⁰
Googolplex: 10^googol
Google: search giant.

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Anomie

Member #9,403

January 2008

gnolam said:

profanity!

I was waiting on that.

Let's say that g₆₄ would fit in 4.2 x 10^{googolplex^{googolplex^googolplex}} universes the same size as ours. Given that, I challenge you all to be the first to compute g₁ into it's full integer form.

[edit] In ten lines of code or less. And it has to be able to work on at least one computer in existence today, however long it takes.

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X-G

Member #856

December 2000

Graham's number is way worse than that. Not only aren't there enough atoms in the universe to write the number, ther aren't enough to write the number of digits in the number. If you tried to express it in terms of x^y^z^w... - a power tower - there still would not be enough to write that number. You can't even write the number of powers involved! Or, in terms mentioned earlier...

Quote:

I read somewhere that if all of the material in the universe were turned into a pen and ink, there wouldn't be enough to write the number. How many universes would it take? My bet's on 42.

Unfortunately, you can't even write the number of universes you would need with the atoms in this universe, let alone 2 or 42.

Graham's number is huge.

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Johan Halmén

Member #1,550

September 2001

Graham said:

My number is larger than yours.

It's not the size, it's how you use them. Really!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.

Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.

GullRaDriel

Member #3,861

September 2003

Johan said:

It's not the size, it's how you use them. Really!

Johan, do you have a lil' one ?

In the same meaning as yours: The size of the wand does not count, only it's power.

_
Edited

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Allegro Wiki, full of examples and articles !!

Evert

Member #794

November 2000

Never having had any reason to think about, or even be very much aware of, Graham's number, the best I can do in terms of visualising it is $<math>G \approx \infty</math>$ .

But hey, at least we know it's odd and not prime!

Anomie

Member #9,403

January 2008

The notion hit me earlier today that Graham's number is likely to include the full text of several well-known books, encoded in integers.

And thinking of 'G ~= inf' did more to expand my vision of infinity than it did to expand my vision of g₆₄.

And no one has conquered my challenge yet! Mwahaha!

______________
I am the Lorax. I speak for the trees. I speak for the trees for the trees have no tongues.

Johan Halmén

Member #1,550

September 2001

This is not Graham or Googol, this is our own common numbers:
Million = 10⁶
Billion = 10¹²
Trillion = 10¹⁸
Quadrillion = 10²⁴

some_numberillion = 10^{6*some_number}

Well, Americans don't count like this, but that doesn't count.

Centillion would be 10⁶⁰⁰, which long ago was noted in the Guinnes book as the biggest number word. My suggestion would be millillion = 10⁶⁰⁰⁰. So if one million is 10⁶ and one millillion is 10⁶⁰⁰⁰, for each "ill" you add in the word, you can add three zeroes in the exponent. I know it's far from Graham, but at least it's (vaguely) based on our convention of number words.

[edit]
For real big numbers you just stress every second "ill" syllable. That way you can count how many "ills" there are. Try:
One millillillillillillillillillillillillillillion. How many zeroes?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.

Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.

alethiophile

Member #9,349

December 2007

I adapted the layer definition on the Wikipedia page into a series notation that I like better. Maybe it'll help.
$<math>x \uparrow_n y = (x \uparrow_n y-1) \uparrow_{n-1} x</math>$
$<math>x \uparrow_n 1 = x</math>$
$<math>x \uparrow_1 y = x^y</math>$
$<math>g_1 = 3 \uparrow_4 3</math>$
$<math>g_n = 3 \uparrow_{g_{n-1}} 3</math>$
Graham's Number is g₆₄.

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Johan Halmén

Member #1,550

September 2001

Too bad it wasn't Conway, but Graham. We would be talking about C64.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.

Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.

Anomie

Member #9,403

January 2008

I can't get over the size of this damn number. It's tweaking with my head. According to the Wikipedia page on up-arrow notation, 3^{3^3³} (that is, a three-level power-tower of 3's) would require ~1.37TB of storage to write out in full integer form. 1.37 terabytes of space for that number.

g₁ expands into a power-tower that's trillions and trillions of levels tall.

That number came from 3^^^^3(four arrows). g₂ has g₁ arrows between those two threes.

I wonder why that's so relevant. If someone came up to me a said 'How massive is infinity times 2?! It's crazy!' I wouldn't care at all. Graham's number seems more substantial in some way.

[appendectomy] I think it's because g₆₄ represents the point at which some random thing must - necessarily - occur. In some extrapolated way.

______________
I am the Lorax. I speak for the trees. I speak for the trees for the trees have no tongues.

Neil Black

Member #7,867

October 2006

What is the purpose of Graham's Number?

Evert

Member #794

November 2000

It's an upper bound for some number in a problem in mathematics.

wikipedia said:

Graham's number is connected to the following problem in the branch of mathematics known as Ramsey theory:

Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices which lie in a plane?

Graham & Rothschild [1971] proved that this problem has a solution, N*, and gave as a bounding estimate 6 ≤ N* ≤ N, with N a particular, explicitly defined, very large number

ixilom

Member #7,167

April 2006

Its a friggin big number, get over it. You're not going to be able to use it anyways

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