Cloudscape

A bit hypothetical: in a system of points, the average distance from one point to its nearest neighbour is equal to the third root of 6 by the point density:

r = ∛(6/n)


Average Minimal Distance.rtf

A simple formula in combinatorics: for a combination (n,2) (the number of possible pairs formed by n):

(n,2) = Tn - n

where Tn is the triangle number (eg T5 = 5 + 4 + 3 + 2 + 1 = 15). Proof in file.

(n,2).rtf

For people interested in combinatorics…

An excerpt from Tempest - The Transition (http://www.lulu.com/content/1929564)

And Function.rtf

If either the age, size, energy or complexity of the universe are infinite, so are the others: if the universe is infinite in age, it must be infinite in size because a finite universe would keep expanding, ergo in energy, because it would otherwise be of infinitely low density due to this expansion, ergo in complexity because this energy would be distributed amongst the universe rather than remaining infinitely accumulated.
If the universe is infinite in size, it must be infinite in age because if it had a starting point it would originate from a singularity rather than popping into existence in infinite dimensions, and infinite in energy and complexity because it would otherwise be of infinitely low density as formerly said.
If the universe is infinite in energy, it must be infinite in age because in order to become of finite density it'd otherwise have to expand with infinite speed (this is the most plausible other possibility, as the absolute speed of light might not be absolute), infinite in size for the same reason, and infinite in complexity because of its infinite size.
Infinite complexity basically equals infinite size because size is relative. If the universe is infinite up or down, it makes little difference. Suppose the universe was contained inside a giant atom (which would have an infinitely complex substructure, so that this would be possible!), we'd still say the universe is tens of billions of light years in diameter instead of one and a half femtometer. Relative to an infinitely small world (at an infinitely complex level), everything is infinitely large. As we've said, infinite size and finite density means infinite energy. However, if the universe is infinite in complexity this does not necessarily mean that it is infinite in age.

Thus, we conclude that either:
1) the age, size, energy and complexity of the universe are all infinite,
2) neither the age, size, energy or complexity of the universe are infinite
3) the size, energy and complexity, but not the age, of the universe are infinite, and the Theory of Relativity is incomplete.

You can scratch the latter two possibilities, however, if you can accept the following argument. Existence cannot have had an actual cause, because that cause would itself have to be part of existence, otherwise it could not have existed because it would then not be part of existence. Why would there be a beginning? Where did it come from? What caused it? Now, I'm talking about the whole of existence, not just our own reality, but any reality at all, including hypothetical realities in which our own universe was created. Unless there already existed "something" which had caused the universe to arise, ie there was already something in *existence* (in other words, in the universe), such cause would necessarily have been acausal, and for something to be science it must obey causality. Causality, then, is its own only exception. Everything has a cause, but the chain of causes and consequences itself hasn't.

Following this line of reasoning we conclude that the universe is infinite in age, size, energy and complexity. This also implies a universe of infinite complexity. In such a universe, there would be an infinite number of phenomena; each of these would in some way have to influence each other, as they would otherwise not form one whole. Whatever phenomenon would not influence the rest of the Universe would not really be part of it, and therefore not really exist as far as we are concerned. In this way, all phenomena would be infinitely connected to one another, that is to say, each phenomenon would be influenced, directly or indirectly, by infinitely many others. One of such phenomena is our own will, or consciousness, which would likewise be causally interlinked to all other phenomena, meaning that it as well could influence and therefore control them. Thus, in principle, if one would find one's way through this web of causal connections, one could, in principle, control any phenomenon in the universe, meaning that one could be omnipotent. This appears to follow logically if the universe is of infinite complexity.

Something you should surely have read if you've visited my blog at all - probably my most shocking idea.

In an infinite universe, everything is real. Probabilities depend on time and space, thus is both time and space are infinite, all probabilities in the universe are equal to one. This is the basic idea of ergodicity.
This infinity has highly bizarre consequences. Think of the most absurd things you can think of - monkeys raining from a clear sky, trees sprouting from nuclear waste in seconds, atoms arranging themselves to a microscopic epic, molecules across a solar system falling into place to form a hi-tech civilization, a live alien popping into existence in space -- these are all configurations of energy which are theoretically possible, and as Feynman said, anything which can happen will happen. It's all happening, right now and every moment, an infinite number of times. How surreal!
The punchline: the very fact that you can think of something happening means that it IS happening. This principle is known as modal realism: all imaginary worlds are as real as our own, "real" world. Remember that when you're reading a book, or slip into a reverie -- you're having a vision of something which is actually happening.
But it's not over yet -- there's something even more amazing about modal realism…
If everything is real, that means everything is true. After all, it would mean that there is a causal association between all things, and in fact all things are connected. This would actually make all delusions true, as anything that would happen would not only have several causes, but all causes one could think of. For instance, if a paper one looks at suddenly floats away, this would be both because the wind blew it away and because one has telekinetic abilities. There obviously appears to be something wrong here.
In principle, according to chaos theory it may actually be possible to cause a paper to float away by looking at it. But this is not necessarily so, and moreover, it can not be caused both by one's thoughts and the wind.
But because of the ergodic nature of an infinite universe, there would simultaneously be a world where it is caused by the wind and a world where it is caused by one's thoughts, quite simply because if the universe is infinite, everything is. These worlds could otherwise be identical, except in the cause of the paper floating away. Because the universes would be perceived in the same way, they would per definition have the same essence, so that there would be no telling in which of these universes one is. In fact, one could well say one is in both universes at once.
As long as you don't know if something is true or false, it is both true and false, because your own perception matches both the universe where it is true and the universe where it is false. Of course, the only condition for something to be true is that you perceive it to be, that is, that your qualia makes it a possibility (for instance, that you discern something is either true or false, not both). Then, there is always a chance that you'll be in the universe where it is true.
Of course, this chance may be very low, as only in a small percentage of the universes which the many identical "yous" would perceive as identical, it would really be true. Still, however, as long as you have no definitive "proof" whether it is or isn't, it is true, because you have all the perceptions of the you who lives in the universe where it is true, and therefore you are essentially that you. To clarify, nothing really exists if it can't be perceived, thus if the difference between the universe where the idea in question is true and the universe where it is false cannot be perceived, it does not really exist as such, either.
And of course, things are even further complicated because the so-called "proof" that it is or isn't true, which would necessarily be part of one's sensorial input, could be hallucinatory, it'd be essentially worthless: you could still be in a universe where the sensorial input of the proof is an illusion and the idea in question is still either true or false. There is no separation between what we know and what we imagine.
This means that everything is in a dual state, from quantum to universe. This explains the principle of Heisenberg, because we can't (yet) know the state of an individual quantum. This version of this principle is known as the many worlds principle, though the latter was perhaps not founded upon modal realism.
Any possibility one can conceive has a 100% probability of happening somewhere in the multiverse. Thus, any story you can think of is true. Somewhere in the infinite numbers of universes in the multiverse, it happens just like you imagined. If the universe is infinite, space and time are all that separate us from that which we imagine. Imagination, therefore, is a discovery.