Finite versus infinite universe

[snas]

I've been reading about various possible universes. One question that seems to come up frequently is, is the Universe finite or infinite? As I am not an astrophysicist, I may be missing something here. But, it seems to me that if the BB occurred when a "singularity" containing all of what now makes up the Universe suddenly underwent a period of massive, rapid expansion, then surely the original singularity had an edge,a boundary. If so, it was finite. So how would a growing universe get from a finite singularity on day one of the new Universe, to an infinite state. Surely a finite thing cannot become infinite.

Stuart

[CraigS]

Oh boy .. tough question .. one for a mathematician to answer ..

The way out of the dilemma is to envisage the 'singularity' as having no boundaries and is infinitely small to start with. 'Infinitely small' is the same as 'infinite' … ie: infinite ...

A 'point' in geometry, has zero dimensions and thus has no boundaries because it has no dimensions to establish any boundaries - same thing.

Did that confuse you ?

Cheers
PS: I'm happy to side-step this one .. (I think what I've said is valid, though .. this is how I think of it) . Cheers (I tried, at least).
:)

[bartman]

One thing I have learned whilst pondering the same question you have Stuart, is that we are not yet able to comprehend the concept of "infinite".
The scientists ( all types) and We are trying to understand and make sense of how the BB eventuated.
Experiments, VLT's ,Satellites of all sorts, LHC's etc etc are all being used to make 'sense' of the why, where, when, what and how of the universe as we know it.
I'm sure once 'we' figure that out, your ( and mine!!!!!!)question can be answered!!!!!!
Edge or no Edge ....it's HUGEhttp://www.iceinspace.com.au/forum/../vbiis/images/smilies/eyepopping.gif
Huge as in the recent VLT's discovery of the furtherest ( newest) galaxy as per post: http://www.iceinspace.com.au/forum/showthread.php?t=67249 .
"It has taken 13.1 billion years, travelling at 300,000 kilometres (186,000 miles) per second, for this smudge of infant light to arrive.

[Max Vondel]

The Universe is observably "finite" and I believe "finite" beond too
Also this string theory 10^500 Universes is absolute Cr#p!
A poor way to explain very fine tuning..................
There are many other interesting ways to get fine tuning without creating all these extra universes! What ever happened to "the simplest solution is best"?
;)

[bartman]

Is that a quote from the movie Contact?:

Ellie Arroway: Occam's razor. You ever heard of it?
Palmer Joss: Hack-em's Razor. Sounds like some slasher movie.
Ellie Arroway: No, Occam's Razor is a basic scientific principle and it says: All things being equal, the simplest explanation tends to be the right one.
Ellie Arroway: So what's more likely? That an all-powerful, mysterious God created the Universe, and decided not to give any proof of his existence? Or, that He simply doesn't exist at all, and that we created Him, so that we wouldn't have to feel so small and alone?
[Ellie challenges Palmer to prove the existence of God]
Palmer Joss: Did you love your father?
Ellie Arroway: What?
Palmer Joss: Your dad. Did you love him?
Ellie Arroway: Yes, very much.
Palmer Joss: Prove it.

[avandonk]

Are we talking about infinite bounded Universes or infinite unbounded Universes.

Infinity or it's concept is a really difficult conundrum.

It has troubled better thinkers than you or me.

I will dig up a very good video on just this.

Bert

[Barrykgerdes]

Its time we accepted the fact that infinity means infinity. Just the same as the answer to life, the universe and everything is 42.

Barry:thumbsup:

[bartman]

All I'm saying is we - us humans- cannot fathom the term infinite/unbounded universe(s).
Even if we say " yes I understand that the universe has no end" we still - in the back of our mind- wonder what is beyond the """""end""""".
Tis a coniption at best!
bartman

[sjastro]

Stuart,

Mathematically the visible Universe is finite, the entire Universe is infinite. This is based on the visible Universe being flat or having a zero curvature. We know nothing beyond the boundaries of the visible Universe so an infinite Universe is a speculation.

An analogy which puts a physical perspective on the concept of infinity and how it relates to a flat visible Universe, is to think of the entire Universe as being a sphere. Our visible Universe is an area on the surface of the sphere.

If the radius of the sphere is small we can discern that the surface has a definite curvature. Increase the radius and the surface appears flatter.
Mathematically there is a relationship between radius and curvature, increasing the radius reduces the curvature of the surface.

When the radius becomes infinitely large, the curvature is zero and the surface of the sphere is flat.

With regards to the singularity it's one of those unfortunate timings that the science of Cosmology predated Quantum Mechanics. The singularity is an artefact of assuming that the laws of physics (in this case General Relativity) extends all the way down to the smallest scales.

In the very early history of the Universe, the Universe was small enough to obey the laws of Quantum Mechanics instead of General Relativity.
Singularities don't exist in Quantum Mechanics as space time is "smeared".

Regards

Steven

[Steffen]

Isn't "infinite bounded" a contradiction in terms?



Yet, if you think about it, a finite universe is actually harder to swallow. As soon as you postulate it's finite (implying a boundary) you can't help but ask yourself "what's on the other side of that boundary?", which whatever it is must be part of the universe as well (the universe being "everything"). It is quite possible that all the "stuff" is huddled together in some corner of the infinite void but there is simply no way one could imagine a bounded void.

Cheers
Steffen.

[snas]

All very interesting points of view, and since I'm better at desexing dogs and cats than I am at getting my head around infinite bounded and infinite unbounded universes, I'll have to be happy with the thoughts you've all put forward and not try and decide who is right or wrong, if anyone is even right or wrong.
Craig, I see your point about the unbounded singularity. I guess it's a bit like saying, how thick is a line, given that a line is one dimensional it cannot have thickness, only length. I see two different types of boundary though, one being a fence or wall, a physical obstruction to going any further; the other being the boundary of "there is no fence, but this is the last star, keep going past here and there is nothing more to see." Perhaps a better description of what I am trying to say there is to go back to Alroy Downs, cattle station on the Barkly Tableland where I jackarooed in 1986. We had about 30 000 cattle on 9 000km^2. Most of these cattle were in fenced in paddocks. They lived in a finite universe. The southern end of the station was unfenced. There were cattle there but they could have walked all the way to Adelaide if the chose to (and could find water and grass on the way). But the lack of water past a certain point is what kept them "in". So these cattle lived in a universe with no physical fence boundary; there boundary was described by that point where you could keep going, but there were no more cattle. There was further "space" out there but this space was effectively a void. I doubt that our infinite universe is like that. If it is infinite then it surely must be similar everywhere to what we see around us.
Dang, my brain hurts.
Max, I agree. In my field the same applies, simplest solution is often the best and most likely.
Bert, I like a lot of the ideas you put forward. Bring on the video.
Steven, I fully understand that the curvature of a golf ball is greater than that of a beach ball. But I'm not even going to try to understand how we measure the curvature of space. None the less I do now get what they are talking about when they talk about flat or curved universes.
Thanks all
Stuart
maybe life was simpler when we were mustering cattle on Alroy!

[CraigS]

Hi Stuart;

Infinity is a tough concept, (probably one of the toughest .. as everyone says), but I think that the concept of 'nothing' has to be considered alongside of it.

According to the thinking, (and that’s what this is all about.. the theory follows on from the initial thought concepts and it isn't necessarily the 'truth'), the Big Bang didn't expand into anything. It expanded into nothing. The ‘thing’ it started from, (at the small end), was also infinite and dimensionless and therefore infinitely dense with energy. Infinite and dimensionless at the small end … expanding into infinite and dimensionless (and nothing), at the big end.

Eventually, (fairly quickly), it then brought into existence, the 'something' - all that we see.

In your example, I notice you're still seeking to describe the boundaries. Let ‘em go and the problem disappears. That’s the trick to all of this. Focussing on the boundaries just brings the problem back. Its only a thought exercise. Following ‘the rules’ and abandoning the boundaries, leads to an appreciation of the profound logic of what follows, and is very cool (IMHO).

It really doesn’t matter whether its true or not ..

Hope this helps.

:)

Cheers & Regards.

[Steffen]

I reckon a lot of people who got into astronomy are very familiar with the concept of "nothing" from watching their bank balances ;)

Cheers
Steffen.

[CraigS]

Even that's not 'nothing'.

.. and that's coming from personal experience!

:)

Cheers

[mswhin63]

:lol::rofl:

[snas]

So Craig, if I have to get used to the idea of no boundaries in the Universe, does that mean I've got to go back up to Alroy Downs and cut down all the fences so the cattle are then living in a boundary-less universe.
(See my previous post in this thread if that all sounds a bit weird)
Stuart

[CraigS]

I'm not so sure I'm up with your thinking about the cattle but what is the need for the boundary fence, if they truly live in an unbounded universe ? As you have mentioned, there are other things which stop them. Whilst I don't think like cows/bulls, their boundaries seem to be limited by where they think they can roam .. and their will to survive (ie: proximity to water/food).

We can remove the thought boundaries, if we try.
Survival ? … Well ..?.. hmm .. dunno ..



Not the best of formal quotes but you can kind of get where they're coming from. No voids beyond your 'last star' boundary (a thought barrier), everything the same … we see just a sample.

Cheers
PS: Now my brain hurts … give me a beer !! :) Cheers

[avandonk]

Here is a very good BBC doco that tries to explain the unexplanable

Part one of ten

http://www.youtube.com/watch?v=Cw-zNRNcF90

Well worth a careful look!


Bert

[CraigS]

Great documentary, Bert !

I just watched part one and I'm hooked !!

I might launch another thread in the Books & Media forum .. folk there would be really interested in this.

Thanks.
Cheers

[avandonk]

Yes Craig all these blokes were all over the Physics and Maths that I with great difficulty finally partially understood after studying this at RMIT from 1968 on.

Bert

[CraigS]

I've just finished Part 3 and I'm having a chuckle about what I said to Stuart about 'don't take it as the truth'. It seems that Cantor did, and it, (unfortunately), drove him to obsessiveness and breakdown.

There's so much gold in this documentary ! (I'm excited just watching it !)

These guys that have come before us truly were pioneering greats !

Just fantastic.

Cheers

[avandonk]

It gets far worse!

You may now understand my thoughts with the brain as a quantum computer and being connected in some way to the whole Universe by quantum entanglement.

Bert

[sjastro]

Unfortunately with country speed dial up I can't view the documentary but anything involving Georg Cantor would be very interesting.

From what I can recall from my Uni days Cantor came up with the concept of "countable" and "uncountable" infinite sets.

He also defined ideas that were intuitive but still difficult to comprehend due to the definition of infinity. For example the set of natural numbers and integers are both infinite sets but the set of integers contains twice as many elements...

Regards

Steven

[CraigS]

I think Robh also pointed this out recently, in the 'New Habitable Planet' thread.

Boltzmann's done himself in, too (I'm starting to see what Bert means).

They move onto Kurt Godel and Alan Turing.

Great doco.

Cheers

[mswhin63]

So far very interesting doco, and will continue to watch the whole series. Thanks for the link.

[avandonk]

Wait there is more! Chaos theory and fractals!

We live in an indeterminate Universe. Every act we perform changes the future.

Fate is a myth!

It is up to each one of us to make our future.

Bert

[sjastro]

Set theory is intuitive compared to other concepts pure mathematicians have come up with particularly in the field of algebraic and geometric topology.

I find it difficult to visualize how a Klein bottle (a hollow bottle with no inside;)) when cut in half lengthwise will form two Mobius strips (a strip twisted once with the ends glued together.;);))

Mathematically however it is straightforward.:rolleyes:

Regards

Steven

[CraigS]

Don't know about that one Steven but the areas this doco is covering include:
Godel's incompleteness theorem:
"No matter how large you make your axioms (logic bases), there'll always be statements that are true but cannot be proven to be true .. no matter how much data you have."

Consequences of this is that Logic is a 'failure' when it comes to proof.

So, when applied to Cantor's Continuum Hypothesis about infinite infinities, Godel couldn't prove it, then he ended up in a sanitorium .. lost in intellectual problems. (They mustn't have had enough beer in those days).

So .. there is no way of telling which are the unprovable problems and which are the provable ones.

(Sorry if you're already across this Steven .. this is more for anyone who cannot see the doco or to add to add some Science value to Stuart's thread).

Cheers

[avandonk]

You have to see this doco in the sense that these blokes were products of their times. To stand up and say everything is not real takes guts. To admit it to oneself takes more guts.

I have come to the conclusion all great scientists and thinkers are marginally autistic if not they were manic depressive.

We are emotional beings and without it we would just rape pillage and burn. Ok it is allright to do it to the enemy but not our neigbours.

Bert

[CraigS]

Bert .. yep .. I'll come back to that.

Alan Turing: Uncalculability Theorem: incompleteness in computing .. are we or aren't we logic based computers ? Are our minds like computers ? If so, incompleteness will be our burden .. our problem because logic is flawed with uncalculability. (Turing's gone).

Back to Godel: reaching truth outside of logic was his new quest. He thought he was 'divine'. He and Einstein were colleagues and talked about deep philosophical issues. Godel latched onto human intuition which extends our ability to think beyong the provable and logic (of Turing's computers). However, he attempted to use "maths to show the limits of maths" .. very contradictory. He's gone also, with that thought.

End conclusion of the doco is that we have to learn to live with the problem. There are no absolutes .. we are incomplete and finite.

Stuart: You've gotta see this doco. It hits the nail right on the head.

Bert/Steven: Thank you, both.
Fantastic ! Time for a beer ! Before I go completely nuts !!

Cheers & Rgds.

[CraigS]

These guys were passionate about their work. Their passion led to obsession (perhaps with the exception of Turing).

They extended human thinking beyond most boundaries. Its interesting that Einstein didn't although, he became fairly obsessed with unification. In the end, I think he ignored the evolving Bohr/quantum mechanics revolution, which may ultimately, one day, lead to unification.

Its all about balance eh, Bert ?

Cheers

[avandonk]

It is even scarier than that Craig. Your brain has already computed all the necessary steps to move your arm before you have consciously thought of moving it!

NMR scans have proved this.

Are you in really charge of your own body?

It seems we have an auto system that does all the movements automatically and absolutely blindly without reason. We just go along for the ride thinking we did it!

Bert

[avandonk]

Craig there is a very good book called Godel Escher and Bach I read many years ago that covers these problems of reality and consistency in our meagre efforts to to confront reality.

I lent my copy out and never got it back.

Bert

[CraigS]

I'll have to snoop that one out !
Add it to a list that could end up putting me into an asylum !

:)

Cheers

[avandonk]

Escher used to make drawings of continuous water falls that defied gravity and perspective. Bach used to cheat our senses that we could not detect his cunning ever rising pitch as an illusion. Godel knew why we were fooled.

Bert

[CraigS]

Sounds dangerous !
:)

[CraigS]

So Bert, your point has always been that the human mind can take us way beyond anything which may be real in nature, right ?

And we are only just at the beginnings of understanding the universe because we've only just begun to develop the tools to do so.

Did you see in the General Chat section the thread about the headset that is capable of reading one's brainwaves and taking actions based on that.

Does this mean we are beginning to tap into how the brain really works ? Or is this just some from of trick ?

The universe may be a hologram, and us the projection of that hologram. (I haven't even begun to work this one out yet .. so don't ask me about it !!)
I think you've mentioned that a good smack will sort this one out !!

Cheers

[avandonk]

There are a few assumptions that cannot be avoided. Can a finite brain understand an infinite Universe? Alright a very big one at best!

I do not have any answers. Godel says we cannot! We cannot even define the simple number system without paradoxes.

The Universe is built from the bottom up ie quantum and particle theory.

The foundations at best look very dodgy! Indeterminate and suffering from a reality crisis.

Yet the stone in my shoe really hurts.

Bert

[CraigS]

I keep coming back to a perspective that none of this universe theory stuff has to be real (or the truth). Attempting to understand the concepts simply leads to an expanded view of it all, and can be fun.

Most of the theories predict the behaviours of things. People take comfort in predictability. They get to be right .. which is a buzz for them. The theories also result in discoveries, and technologies, which improve all our lives. They also satisfy our own insatiable curiosity.

So many take it all far too seriously, though.

I get a lot of comfort from these perspectives.

Each to his own, I guess.

Cheers

[avandonk]

Here is a doco that builds on the first.

http://www.youtube.com/watch?v=HACkykFlIus


Bert

[CraigS]

Thanks for that Bert. Summary of the doco follows:

So what we have here follows on from Alan Turing’s, (see post 30), previous theories leading to his publication of his work called ‘The Chemical Basis of Morphogenesis’. Morphogenesis is the biological process that causes an organism to develop its shape.

This leads to the observation that shapes in nature are irregular and unpredicatable. But how can this be if Newtonian Physics is all about predicting nature’s behaviour with certainty?

The tiniest difference in initial conditions can dramatically alter the outcomes of a system defined by simple, deterministic, mathematical equations and result in indeterminate behaviours. (The Butterfly Effect).

Its all dependent on feedback. In nature, the environment provides the feedback mechanism.

Pattern formation: enter Mandlebrot and fractal geometry .. self-similarity .. complex systems following simple rules can spontaneously generate complex patterns or organisms (like humans).

The culmination of this documentary is summed up as:



The combination of Evolution + Complexity + Timescale results in extremely complex systems .. like human beings and all the way up to the complexity we see looking through our telescopes.

It all has mathematical foundations (and proofs) all originating from Georg Cantor's original deliberations on the concept of Infinity.

I am left with a deep concern for those who pass judgement on theoretical mathematics and physics on the basis of superficial queries such as:
"Whatever happened to the simplest solution is best ?"

Cheers

[avandonk]

Yes no organizing principle or being is needed. If you invoke an organising being then where did he/she come from? As a solution it is therefore a complete cop out!

Is this the final bit of the puzzle? Or is reality even stranger?

http://www.youtube.com/watch?v=E7FV9aaiwKQ&feature=related


Only time will tell with some definitive experiments. The LHC may give a glimpse. The gravity wave detectors being planned now may give more insight. There is a long baseline interferometer that will be built in WA Australia or South Africa.

Astronomy with the analysis of rotating binary black holes or black holes/neutron stars may give us more data.

Closer to home if Eta Carina goes SN the gravity wave signal should be detectable.

I started studying Applied Physics in 1968 at RMIT. All through my working life in science I have kept up with these developments as they happened in journals such as Nature, Science etc. as a sort of hobby apart from my 'real' work that paid for the bills. It makes you realize how smart all these people are when their discoveries are difficult to understand let alone originate them.

Bert

[CraigS]

Yep. String Theory leading onto M-Theory … unproven and with even more data, it may still be unprovable .. but this may not mean that it is 'untrue'.

The point to be made here is that (Godel's incompleteness theorem about Logic):

"No matter how large you make your axioms (logic bases), there'll always be statements that are true but cannot be proven to be true .. no matter how much data you have."

This may or may not be the case for String and M-Theory

Astronomy provides the data. Without the theoretical context however, it is simply data. Humans need a context to make sense of the universe. 'Tis a symbiotic relationship .. one can never be real without the other. Intuition, or dare I say it .. 'faith' often bridges the gap.

Judicious forethought and caution is paramount in this approach, however.

:)

Cheers

[avandonk]

A leap of faith may lead to insight but by itself it is meaningless. It leads to such deep truths as the tooth fairy, easter bunny and witches!

All Godel did was to shake the mathematicians out of their smugness and complacency. It was also another nail in the coffin of the clockwork or totally predictable Universe already put there by Quantum Mechanics and Relativity.

Chaos theory and fractals have forever in our understanding ruled out fate or predeterminism.

Ever heard of the three or more body problem? Differential equations can describe a mass and spring interaction but falls in a heap with more masses and springs without strict boundary conditions.

Physics can exactly predict the spectral lines of a single Hydrogen atom (Proton) and Electron. It fails for a Hydrogen molecule!

Bert

[CraigS]

I agree with all your statements there, Bert. I 'get' it. (I'm not sure Quantum Mechanics describes predictability, though .. perhaps 'probabilities' which isn't exactly predicatability ..)

So we live in the midst of unbounded complexity .. so that is what we see around us.

We're trying to work it out from the midst of that perspective.

And yet at the largest and smallest scales, things may ultimately be described quite simply. Who knows ?

Andrei Linde's ideas are to do with bubble universes. Guth has wrangled with him about flatness so he now seems to speak of 'expanding infinite sheets' rather than 'finite spherical surfaces' the difference is purely because our brains aren't fully capable of dealing with the dimensions separating the two analogies.

I'd prefer to keep with the directions outlined in this thread:
Finite/Infinite (Cantor, Godel) => BBT/Inflation => Chaos/Fractal/Complexity => String/M-Theory.

It seems that skipping the steps of thinking through all of these concepts can lead to a premature perspective on it all. Does this ever stop ? I don't think so.

I'm intrigued with the concept of our brains being entangled with something else out there … sounds like fun to ponder that one !! (Especially considering the difference between the sexes !!)

:lol:

Cheers

[Karls48]

Yet the stone in my shoe really hurts.
Bert

This statement, in the context with rest of this discussion summarise our ability to comprehend concepts such as infinity at this time and age, for most of people. Yes there are exceptions, but again as Bert pointed out, those exceptional individuals seem to be either Autistic or Manic depressive. Great observation, as the deficiency in some human abilities is compensated by increase in other abilities in those individuals It seems that our understanding of “reality” is tied up to where we are in the time line. From historic point of view, we as the humans come long way from the stone hand axe, lets say 100 000 years ago to the hydrogen bomb in our present. The scale of our development seems to be logarithmic. Most significant development occurred in last five hundred years. The speed of development seems to accelerate during the times of great stress (such as war). That seems to be demonstration of ours survival instinct. After all, all of our pondering about the Universe, Infinity and Reality is the result of our survival instinct at work. Thousand years in the future it is most likely that our we are going to have quite different view of the infinity (and other things) that we have now.
Bert’s comments about intuition deserve separate tread.

[avandonk]

I have said this before but I will say it again in the interests of clarity.

Quantum entanglement is real. Look it up. If at the instant of the BB all we know today was in the same place and time then we are forever all linked in some way we do not yet understand to all of the Universe(s) and obviously to each other.

What if the feed back is even more subtle than we already think. Mere existence reinforces further time dependant existence. Sheldrakes hypothesis comes to mind.

The devil dodgers could be correct but they have so many beliefs without proof, some of them must be correct if only by betting on every horse in the race.

Bert

[Steffen]

Definitely get Gödel-Escher-Bach. Hofstadter does an excellent job at demystifying the incompleteness theorem while at the same time illustrating the brilliance of Gödel's proof. The theorem is surrounded by an aura of hocus-pokus and frequently gets sensationalised by pop-science media (btw, it's not claiming that formal systems are always incomplete, but that they're either incomplete or self-contradictory, the latter actually being the more interesting case). Hofstadter shows how it can be found in every-day life and that the power of self-contradiction is in fact what makes the human mind possible.

Get it, read it and witness your mind becoming cleansed. No other book has ever given me a rush like this one :thumbsup:

Cheers
Steffen.

[avandonk]

Does this mean we are beginning to tap into how the brain really works ? Or is this just some from of trick ?

I will try to answer this question seperately.

It is a bit like looking at the heat distribution or production in a computer and then glibly saying 'we know how it works!'. The hot bits is where all the action is!

Neural connections and their firings measured as electrical waves have as much meaning as measuring the amplitude of waves on a beach to describe a storm hundreds of miles away! All you can say is when the waves are bigger there is a storm or a prevailing wind.

I have seen programs that can 'evolve' electronic circuits to perform better. The resulting circuits work superbly but no known theory can explain how.

We have been 'designed' the same way by blind random evolution and hence we are an embodyment of the Universe we find ourselves in.

Bert

[CraigS]

Thanks Steffen .. I'll have a snoop around for it.
You've got me interested .. (again) ..

Cheers

[Archy]

I agree, a finite thing cannot become infinite.

[avandonk]

Get it, read it and witness your mind becoming cleansed. No other book has ever given me a rush like this one :thumbsup:

Cheers
Steffen.



Yes Steffen it made me get out of a very simplistic 'groove' where my education had firmly put me.

Bert

[CraigS]

Breaking news … I actually sat in a bookshop and had a skim thru of it today.

There's a lot of text.

And a lot of philosophy …

Cheers

[avandonk]

Did you notice the whole chapter written in palindromes?

Philosophy is fine as long as it is based on testable ideas.

Arguing about the number of angels that can dance on a pin comes to mind!

Bert

[CraigS]

Nope .. I didn't have much time .. but a WHOLE chapter ??

Going to have to get me a copy ..

[avandonk]

Craig I do not have all the answers. I just thought that I would make your quest for them shorter by pointing you at the many people that were trying way before us.

Bert

[CraigS]

And I thank you for the 'heads-up' .. great stuff.

Was thinking today about the fractal nature of the Universe.

Self-similarity may cause me to mellow my stance about exo-planet life possibilities .. so far, I've been pretty sceptical .. also considering entanglement .. well, there's another spanner in my scepticism ..!….

Hmmm ...


Cheers

[avandonk]

That is the point of fractals even with self similarity there is an infinite number of variations!

Bert