# Would one actually find their doppleganger in a “Googolplex Universe”?

I've recently become a fan of Numberphile, and today I happened to watch their video regarding Googol and Googolplex. In the video, I found a rather confusing claim that I'm hoping someone here can help me sort out.

It's first stated in the introduction, and then explained in more detail at around 4:10. The claim can be summed up as this:

In a universe which is "a Googolplex meters across", if you would travel far enough, you would expect to eventually begin finding duplicates of yourself.

Getting deeper into the detail of this, Tony Padilla explains that this is because there is a finite number of possible quantum states which can represent the volume of space in which your body resides. That volume is given as roughly one cubic meter, and the number of possible states for that volume is estimated at $10^{10^{70}}$. This is obviously much less than the possible number of quantum states that could be represented within each cubic meter of a Googolplex Universe, and so the idea does make some sense.

But I believe the idea relies on a false premise. That premise would be that the universe as a whole is entirely comprised of random bits of matter. We can easily see that this is not true. The vast majority of our universe is occupied by the near-vacuum of space, and those volumes which are not empty are occupied by some fairly organized objects which all interact according to certain laws and patterns.

So, I'm curious to know two things:

1. Who originally proposed this idea? Is it something that perhaps Tony just came up with to include in the video, or is there a noted physicist or mathematician who actually wrote about this at some point?

2. Given the possibility that a universe similar to ours could exist and be one Googolplex across, would this actually be probable? Or, would the natural order and organization of the universe prevent this from being as likely as it might be in a more random universe?

A note regarding the size of the universe, in respect to this question. In the linked video, the following constraints are mentioned:

• The number of particles in the universe is $10^{80}$
• Stated at 1:38 in the video.
• The number of the grains of sand that could fit within the universe is $10^{90}$.
• Stated at 1:55 in the video.
• The number of Planck volumes in the universe is $10^{130}$
• Stated at 2:22 in the video.
• The size of the universe is $(10^{26}m.)^{3}$
• Stated at 6:36 in the video.

A few times in the video, Tony does qualify his statements by referring specifically to the observable universe. However, sometimes it's not quite so clear.

So, to simplify the problem for the purposes of this question, let's assume:

• Our universe is finite, and much less than a Googolplex in diameter.
• The "Googolplex Universe" suggested in the video, and in question here, is also finite.
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Why do you feel that the premise is 'random bits of matter'? It doesn't look like he assumed that. He only took a worst-case scenario (no organization). Organizing the universe makes this more probable, if anything. –  Manishearth Mar 15 '12 at 4:52
I'd be interested to learn how an organized universe makes this more probable, especially in a universe such as ours where the organization calls for most of the universe's volume consisting of little or no mass. –  Iszi Mar 15 '12 at 12:22
Oh that's easy. In the worst case scenario, if we have a $l\times l\times l$ 'grid', with $N$ quantum possibilities at every vertex, then the worst case scenario for number of possibilities is simply $N^{l^3}$. Now, if we dictate that there must be some sort of organization in this system, we are applying a constraint (never mind that its not mathematically definable). If there's a constraint, number of cases reduces. –  Manishearth Mar 15 '12 at 12:31
@Manishearth I'm still having a hard time understanding how that applies as a general rule. In our universe's case, it would seem the constraint is one that specifically reduces the possibility of this occurrence. If we're looking to find a certain quantum state which is about a cubic meter in volume, and that quantum state contains matter - and, more specifically, a human - and it is known that a very large percentage of the universe does not contain matter, how does this make the likelihood of finding that state more probable? –  Iszi Mar 15 '12 at 12:37
I used I used $\TeX$ formatting. To use TeX, enclose the the TeXcode in $...$ or $$...$$. The latter makes the code get its own line. To get you started, superscripts with ^, subscripts with _. These can be combined, eg 1^2_3 gives $1^2_3$. If you want more than one character in a super/subscript, use braces. 10^{100} gives $10^{100}$. For fractions, use \frac{abc}{def} giving $\frac{abc}{def}$. Greek letters like this: \delta \Delta gives $\delta \Delta$. That's all you'll need often, for more complex things, ask me and I'll find a link. –  Manishearth Mar 15 '12 at 13:42

Regarding the source of claims such as those, I believe they are due to Max Tegmark, likely from this article. He has curious ideas, though discerning whether they are correct, useful or at least in the realm of possibility is far beyond my capacity. He is a strong proponent of the idea of a mathematical Universe. If I recall correctly, he goes to the point of stating that anything exists at all merely because mathematics/logic exists, and everything we perceive as reality is logical statements playing out in a marvelously complex fashion very far down the line. Why does it happen? Because it can, basically.

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The answer is not at all undue. Feel free to post an answer any question here, so long as you are answering it (and preferably correctly, with citations if needed), regardless of how old it may be. This is not a "traditional" forum, where old topics are considered generally un-useful. –  Iszi Apr 20 '13 at 2:26

I think this assertion would be more correct if it was something like:

There are more 1 cubic-meters of matter units in a googolplex-meter-wide universe than there are possible quantum states of 1 cubic-meters of matter units, so at least one of those quantum states of 1 cubic-meters of matter occurs in more than one 1 cubic-meters of matter somewhere.

In fact there are at least as many duplicates of some kind, to make up the difference between the number of possible states, and the number of total volumes.

But, this does not imply that any particular configuration is duplicated. Heaven forbid there be more of me out there. Maybe a rather simple configuration accounts for 99.99% of all the duplications.

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Don't forget that the quantum state in the cubic meter would be "you" for one planck moment only, after which it might rearrange to become anything. For it to remain "you" it would have to be constrained by whatever properties normally constrain you.

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If you mass 120 kg (264 lbs) and the average atomic weight in your substance is, we'll be generous, 25, then you contain $(120,000/25) \times (6.022 \times 10^{23})$ atoms or $2.9x10^{27}$ atoms. They can be arranged in $(2.9x10^{27})!$ ways.
is a big number, dwarfing string theory's $10^{500}$ acceptable vacua.