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The Infinite Lattice

11/22/2006 1:43 PM

Most of us know M.C. Escher's artful and 'impossible' drawing constructions. One that I found very applicable to cosmology is Escher's Infinite Lattice. My son, Coburt, produced this replica of Esher's lattice in color. First 'enjoy the view' and then we will discuss it briefly.

Credit: Coburt

This looks like a very static arrangement. Now, just suppose that every blue bar connecting two red cubes is stretching (increasing in length) at a steady rate, say at 1 meter every second. What will happen to the distance between the red cubes?

I think it is clear that the more bars there are between two red cubes, the faster they will be moving apart. Two bars between them and they move apart at 2 meters per second. Ten bars between then and they move apart at 10 meters per second.

This is effectively Hubble's law: the apparent recession speed of distant galaxies (from us) is directly proportional to the distance of those galaxies from us. Now this is not a hypothesis - Hubble measured it with good precision and then formulated his (empirical) law.

If you look at that bottom-left 'shaft' that goes down to (almost) invisibility, it represents the limit of our observational ability. There is actually an 'horizon' down there, where the red blocks are moving away from us (up here) so fast that light did not have time to reach us yet over all those stretching blue bars!

This is about as close as one can get to the real universe, as perceived today. It is a much better analogy than the historically favored 'balloon' analogy. Forget the balloon, consider Escher's infinite lattice!

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#1

Re: The Infinite Lattice

11/22/2006 2:50 PM

Well done!

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#2

Re: The Infinite Lattice

11/22/2006 3:31 PM

Unlike most of Esher's drawings this lattice appears to be a physicaly realisable structure at least for the section that is shown.

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#3
In reply to #2

Re: The Infinite Lattice

11/23/2006 2:50 AM

syhprum, you wrote: "Unlike most of Esher's drawings this lattice appears to be a physically realisable structure at least for the section that is shown."

You are right - the only "unrealisable" part is that it is an 'infinite' lattice. I used this particular view to show that there may be a horizon to what one can see, even in an infinite universe.

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#4
In reply to #3

Re: The Infinite Lattice

11/23/2006 4:30 AM

Jorrie, if you can say "infinite universe" I can take an "infinitely strong" telescope and look "infinitely far". How do you reconcile your "horizon" with this?

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#5
In reply to #4

Re: The Infinite Lattice

11/23/2006 6:43 AM

Guest wrote: "... I can take an "infinitely strong" telescope and look "infinitely far" ...."

This would be possible, in principle, if the universe was infinitely old. To our best knowledge, it is ~13.7 billion years old and light could only have traveled a finite distance in that time.

In any case, at great distances, we look 'back in time' and our best telescopes look back to when the universe was completely 'fogged over'. It is unlikely that we will ever observe anything more distant than that.

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#7
In reply to #5

Re: The Infinite Lattice

11/23/2006 5:50 PM

A game of find the fallacy: the universe started from a singularity, yet we are so far from the other objects in the universe that no light from them has yet reached us. So the universe expanded faster than light?

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#8
In reply to #7

Re: The Infinite Lattice

11/23/2006 11:24 PM

Guest wrote: "the universe started from a singularity, yet we are so far from the other objects in the universe that no light from them has yet reached us. So the universe expanded faster than light?".

1. The singularity means infinite or near-infinite density, not a point in space.

2. Space is expanding 'faster than light' - nothing moves through space faster than light. Think about the lattice. No cube moves, yet even a very small expansion of each blue bar will cause an infinite lattice's ends to apparently move away from each other at superluminal speed!

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#9
In reply to #8

Re: The Infinite Lattice

11/24/2006 2:44 AM

Further to my "1. The singularity means infinite or near-infinite density, not a point in space." to clear up possible misunderstanding.

Our current models break down at time zero and there is consensus that a reconciliation of quantum physics and general relativity is required to investigate the epoch near the 'singularity'. It is reasonable to think that infinite density probable never happened. However, from observations it is clear that the Big Bang must have happened from something very near to a singularity.

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#10
In reply to #9

Re: The Infinite Lattice

11/24/2006 12:02 PM

Have I got the following right?
The initial singularity is the situation where the density becomes so great that you do not trust the existing formulation. I.e. it is not a mathematical singularity.
You need dark matter to get the correct values for gravity.
You need dark energy for the model to match the expansion rate.

Is it time for Fr. Occam to hang out his razor to dry?

Apropos of almost nothing, in Fred Hoyle's science fiction oeuvre "the Black Cloud", from time to time a cloud would announce that it had stumbled on the secret of the universe; shortly after that it would explode. Was this just an illustration of Hoyle's view that absolute understanding was impossible, or was he also saying something about cosmological theories?

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#11
In reply to #10

Re: The Infinite Lattice

11/24/2006 1:47 PM

Hi Guest, you asked: "Have I got the following right?
1: The initial singularity is the situation where the density becomes so great that you do not trust the existing formulation. I.e. it is not a mathematical singularity."

Right! Nobody ever said it was a mathematical singularity, AFAIK! This singularity must simply have been infinitely dense, which is hard to accept - hence the quest to find a better formulation.

2: "You need dark matter to get the correct values for gravity."

Slightly wrong! Cosmologists need dark matter to get the correct value for the mass that holds galaxies together and to generally agree with other observations. Do you know that they have recently made measurements that can only be explained by dark matter? Tell me if you don't, and I will give you the link (posted on CR4 quite recently).

3: "You need dark energy for the model to match the expansion rate"

Almost right! Not quite the expansion rate, but rather the increase in the observed expansion rate. Many simpler models also matched the expansion rate, but not increasing expansion rate.

My questions to you, Guest:

1. Do you have a better explanation for these observational facts?

2. Do you understand the cosmological theory that you are "attacking"?

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#12
In reply to #11

Re: The Infinite Lattice

11/24/2006 4:57 PM

1. No, and nor at the moment do I suppose has anyone else
2. Obviously not, or I wouldn't be asking such basic questions (it's as far away from my specialism as you can get). But I would comment that the history of science is littered with situations that require more and more additions to fit the data until some youngsters come along with new insights that make them all unnecessary - and were cosmology my field, that would be what I wanted to do (old as I am).

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#13
In reply to #12

Re: The Infinite Lattice

11/25/2006 3:50 AM

Hi Guest,, you wrote: "But I would comment that the history of science is littered with situations that require more and more additions to fit the data until some youngsters come along with new insights that make them all unnecessary -"

I agree, and the 'young guns' are attacking the singularity problem by means of quantum gravity via strings, branes, etc.., and who knows, this may even solve the dark energy mystery! For dark matter, the hope is on the Large Hadron Collider project.

Regards

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#14
In reply to #11

Re: The Infinite Lattice

11/26/2006 6:21 AM

Returning to this now I have a few more minutes:
I think non-uniform infinite density would imply mathematical singularities. In an universe of non-zero dimensions, presumably an infinite number of them. Probably locally dense in the spatial domain, and possibly generally dense and uncountable. So perhaps the term should be "singularities". Also, this sounds like an analytic nightmare?

"Measurements that can only be explained by dark matter". Given the number of people involved who (presumably) should be trying to find other explanations, I expect this statement is as near correct* as any such statement can be.
Given that there are realistic potential candidates, I don't see any reason to doubt the existence of dark matter. And, if the 'dark energy' could be accounted by kinetic energy of dark matter, it would presumably not be a problem. What I find worrying in the present state of the theory is the need to create/destroy matter/energy.

This brings me to a fundamental concern: observations naturally tend to be interpreted in the light of current theories. In addition, funding tends (as it should) to go to recognised centres of excellence. Which leads us to the law of unintended consequences: positive feedback. Young Turks get funding - but what about young mavericks?

*N.B. When working in my own (much more straightforward) field, I treat any statement that can be interpreted as "no other possible explanation" as extremely suspicious. Part of the reason is (naturally) that the statement is usually made when suspicion is either expected or has been implied (as in this case). Whatever the reason, my experience is that (in >90% of cases in my environment) sufficient probing has found alternative explanations that were more consistent with subsequent observations.

TSPG (the same pesky guest)

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#15
In reply to #14

Re: The Infinite Lattice

11/26/2006 6:53 AM

Infinite density would imply infinite temperature and pressure so that raises the question of how matter could exist? Since matter couldn't exist under these conditions it then raises the question of how long it took and how much did the universe need to expand so that matter could condense into existence?

If matter condensed to early then what would stop the universe from collapsing back in on itself like a giant black hole?

Jorrie please help this is starting to give me a headache!

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#16
In reply to #15

Re: The Infinite Lattice

11/26/2006 7:17 AM

I think we would also have infinite gravitational fields, so infinite temperature does seem unavoidable. I suppose that matter could "exist" at infinite temperature - but it would need to be moving at the speed of light. If so, the question reverts to a headache-inducing problem with time.
However, it is just possible that Jorries comment about 'young Turks solving some of the problems' means that this could simply be a case of undue extrapolation.

Help is clearly needed...

TSPG

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#17
In reply to #15

Re: The Infinite Lattice

11/26/2006 8:33 AM

Infinite temperature?I thought molecules had to be "excited" and moving to create heat.Can there be heat if there is no movement?Is there movement within the "singularity", under infinite pressure? No movement,= no temperature= absolute zero.

I know, pressure and temperature ordinarily go hand in hand, but perhaps they disassociate under these thoretical conditions.

Under infinite gravity, nothing can escape.If nothing can escape, no big bang, unless the "containment" of the singularity had a flaw, a weak spot, a non-uniformity.

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#18
In reply to #17

Re: The Infinite Lattice

11/26/2006 8:53 AM

I'm probably missing something here - but where did the idea of "no movement" come from. BTW, when I refer to "matter" under these conditions, I don't think of molecules - or even nuclei. I shouldn't even think we should assume hadrons. I think the working definition for "matter" would be (otherwise undefined?) material with non-zero rest mass. All this is presumably beyond the reach of meaningful theories at the moment? But perhaps Jorrie can elucidate present thinking at the nearest time to this for which we have some sort of model?

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#19
In reply to #18

Re: The Infinite Lattice

11/26/2006 9:28 AM

So you are saying there IS movement in a singularity?

Perhaps at the exact center, where the gravity from all directions is equal, there may be a small spec of zero-gravity?

Where does the "infinite heat" mentioned come from?I do not think heat is possible till the singularity explodes, nor is any matter as we know it possible. IMHO

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#20
In reply to #19

Re: The Infinite Lattice

11/26/2006 10:22 AM

Before replying to your contribution, please note that, for me, this is pure hypothesis based on Jorrie's semi-throw-away remark implying non-zero extent and infinite density and the interpretation of reply 14.

I wasn't writing about what happened at/in a singularity - though if the singularity is pole-like, I imagine movement would include speed-of-light velocity (whatever that might mean at the singularity) with infinite curvature, resulting in zero displacement.
What I was presuming was a space of non-zero extent in which there were disposed an infinite number of singularities in the fields/gradients. Movement is clearly possible everywhere that is not identical with a singularity - even if the distances from such a singularity is vanishingly small (it all depends on the relative measures of the zero distances and the infinite gradients).
IMHO, this is all speculation beyond the bounds of any theoretical basis - but I could be a long way out of date. Nevertheless, I believe that matter (as I defined it in #18) and infinite heat are mathematically possible under these constraints. (It's probably also essential that the extension of matter becomes zero when it is travelling at the speed of light). [But I'd be equally happy to hear that work had been done that proved this to be incorrect]

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#21
In reply to #15

Re: The Infinite Lattice

11/26/2006 2:54 PM

Hi Masu, since we had such good discussions in the past, I feel I must rush in to try and assist with your: "Jorrie please help this is starting to give me a headache!"

Firstly, I must say that as an engineer, I'm pretty uncomfortable with cosmic inflation theory, simply because I do not understand the physics well enough!

However, I 'cured my headache' about it by accepting the following basic interpretation: during the first 10-34 seconds or so, there was no matter and, in fact, not a great deal of energy (some photons?) and a very slow expansion rate, if at all.

Then something like a 'false vacuum' popped up as a quantum fluctuation, causing a run-away inflationary expansion of space. During this period, virtually all of the energy of the quantum fluctuation was tied up in the expansion rate.

Then, at about 10-32 seconds, a phase transition happened and the 'false vacuum' went over into a 'normal vacuum' (whatever that may mean) and, as phase transitions do, the excess energy went into something - in this case into the formation of matter and anti-matter. It happened to be a lot more matter than anti-matter - why, no-one knows (apparently).

After the inevitable annihilation of matter and (some lesser amount of) anti-matter, creating a huge quantity of photons, there was a fair amount of matter left. These particles (quarks, or what?) were stationary in a space that was expanding at one tremendous rate, taking them apart faster than what they could ever hope to move together. So, from then on, the forming of a universal black hole was no longer possible.

Ouch!! Now I'm getting a Sunday evening headache as well! So let's just say that, simplistic as it sounds, this scenario is not completely out of line. It lacks some detail, but it eases the headache considerably!

I tried to convey this picture in my chapter on cosmic inflation on my website and eBook.

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#22
In reply to #21

Re: The Infinite Lattice

11/26/2006 3:24 PM

Apologies for being a continuing pain (head or neck?). I can't align this with the statement "This singularity must simply have been infinitely dense" in your (Jorrie) earlier contribution. I had obviously msunderstood it completely, as I took it to mean that the universe at that time was very full.

TSPG

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#24
In reply to #22

Re: The Infinite Lattice

11/27/2006 12:17 AM

Hi TSPG, fortunately, I left myself a back door out when I wrote in past 8 (I think):

"1. The singularity means infinite or near-infinite density, not a point in space."

The stage normally referred to as 'the singularity', could be just after inflation, when normal Frieddman expansion 'laws' began, because the Friedmann equation runs into difficulties before that.

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#23
In reply to #21

Re: The Infinite Lattice

11/26/2006 11:29 PM

I am starting to get the picture, I think. Now where did I put that aspirin?

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#6

Re: The Infinite Lattice

11/23/2006 3:21 PM

Jorrie,

Thanks! That is the very best analogy I have ever seen, and allows one to easily imagine an otherwise very elusive (for some) concept.

If you are not a teacher, you missed your calling.(imho)

exit 0 HTRN

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#25

Re: The Infinite Lattice

03/16/2012 4:30 AM

Greetings from Benoni :-) L-o-n-g time passed since this discussion being held and this short note to add that I highly recommend attending a presentation given by Jorrie (as I had the privilege a doing a few weeks ago [March 2012] at a Pretoria Astronomical Society meeting) if the opportunity is presented. EXCELLENT discussion with potential to induce MUCH more than a headache (more like a migraine <grin>!). That's all for now. Remember to KEEP SMILING :-)

VBR Nigel R

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