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**Georgina Parry**: *on* 11/3/12 at 21:46pm UTC, wrote Hi M.C. I looked up Markov chain on wikipedia:Quote: "A Markov chain,...

**M.C.**: *on* 11/3/12 at 19:02pm UTC, wrote I am the editor of the "Markov Chain Universe" website and would like to...

**Andreas Martin Lisewski**: *on* 11/13/09 at 15:39pm UTC, wrote Thanks Tobias. One more response. The embedding of the tree metric into the...

**Tobias Fritz**: *on* 11/13/09 at 11:23am UTC, wrote Thank you for your extensive explanations, this has cleared up a lot! Now I...

**Andreas Martin Lisewski**: *on* 11/12/09 at 23:15pm UTC, wrote One more explanation as you said " I don't understand what you mean by the...

**Andreas Martin Lisewski**: *on* 11/12/09 at 22:39pm UTC, wrote A minor correction: The formula in my previous post should read D*ij = c...

**Andreas Martin Lisewski**: *on* 11/12/09 at 22:36pm UTC, wrote Tobias, Yes, you are right and I was inaccurate in my language when I...

**Tobias Fritz**: *on* 11/12/09 at 18:18pm UTC, wrote danke! I hope you don't mind that I keep questioning your work, being a...

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June 4, 2020

CATEGORY:
What's Ultimately Possible in Physics? Essay Contest (2009)
[back]

TOPIC: The Ultimate Physics of Number Counting by Andreas Martin Lisewski [refresh]

TOPIC: The Ultimate Physics of Number Counting by Andreas Martin Lisewski [refresh]

This essay puts forward the idea that the elementary physical process in the universe is the counting procedure of natural numbers. If true, it would imply that the ultimate possibility in physics is the discovery of this archetypal and fundamental numerical order in nature. In pursuing this astounding idea with methods from modal logic and set theory, it is argued that the number counting process may indeed be sufficient for a complete quantum description of the evolving universe.

The author studied physics and mathematics at the University of Hamburg. In 2003 he earned a Ph.D. in theoretical physics at the Technical University Munich and at the Max-Planck-Institute for Astrophysics. After an intermission as an entrepreneur in artificial intelligence software design he returned to academia in 2004 to pursue research in computational biology.

Infinity is only exciting toward the end - but we have renormalization to ruin even that.

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Mr. Lisewski,

Thank you for an extremely interesting essay. If I understand your ideas correctly, they offer possibilities for gaining a deeper understanding of the processes by which the physical universe evolves, which, in turn, has a profound bearing on our understanding of many aspects of Nature, not the least of those being the nature of time. I believe it can be successfully argued that what traditionally has been thought of and referred to as "the flow of time" is, in reality, nothing more and nothing less than the evolution of the physical universe. The great challenge is to find a way to describe the evolution of the universe in purely Machian, relational terms without introducing a separate external, classical, notion of "time" in the process of doing so. You ideas appear to hold out interesting possibilities along these lines.

Although it may not be immediately apparent, I believe that there is a symbiotic connection between the ideas expressed in your essay and those which may be found in my own essay, 'On the Impossibility of Time Travel,' which appears elsewhere among the current FQXi collection of essays, as well as with related ideas on the nature of time which may be found here.

Cheers

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Thank you for an extremely interesting essay. If I understand your ideas correctly, they offer possibilities for gaining a deeper understanding of the processes by which the physical universe evolves, which, in turn, has a profound bearing on our understanding of many aspects of Nature, not the least of those being the nature of time. I believe it can be successfully argued that what traditionally has been thought of and referred to as "the flow of time" is, in reality, nothing more and nothing less than the evolution of the physical universe. The great challenge is to find a way to describe the evolution of the universe in purely Machian, relational terms without introducing a separate external, classical, notion of "time" in the process of doing so. You ideas appear to hold out interesting possibilities along these lines.

Although it may not be immediately apparent, I believe that there is a symbiotic connection between the ideas expressed in your essay and those which may be found in my own essay, 'On the Impossibility of Time Travel,' which appears elsewhere among the current FQXi collection of essays, as well as with related ideas on the nature of time which may be found here.

Cheers

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J.C.N. Smith,

yes, my idea in this essay is that the fundamental structure in the universe becomes a process rather than an (elementary) particle. As I argued, this process is likely to be the number counting process.

Thank you for pointing to your work.

yes, my idea in this essay is that the fundamental structure in the universe becomes a process rather than an (elementary) particle. As I argued, this process is likely to be the number counting process.

Thank you for pointing to your work.

A copy of this message is being left in each thread of the essay contest forum.

Submitters: Tired of constantly checking the FQXI site to see how your paper's doing in the rankings? I've written a simple program that periodically checks the FQXI site and sends you an email if the status of your paper has changed in any way. Here is a sample email:

The following changes were detected in the status of your paper:

* Number of posts in discussion forum went from 0 to 20.

* Community rating went from 0 to 3.4.

* Community vote count went from 0 to 10.

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* Public vote count went from 0 to 6.

* Community ranking went from being in 0th place to being tied for 16th out of 112.

* Public ranking went from being in 0th place to being tied for 19th out of 112.

You just run the program once, in a command prompt window, and then minimize that window and let it do its thing in the background.

If you're interested in a copy (with source for those who care) drop me a note at ramblinplan@yahoo.com.

Thanks,

Owen Cunningham

P.S. This program requires Windows and the .NET Framework. It has been tested only on Windows XP Service Pack 3 running .NET Framework 3.5, but has a good chance of working with earlier versions.

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Submitters: Tired of constantly checking the FQXI site to see how your paper's doing in the rankings? I've written a simple program that periodically checks the FQXI site and sends you an email if the status of your paper has changed in any way. Here is a sample email:

The following changes were detected in the status of your paper:

* Number of posts in discussion forum went from 0 to 20.

* Community rating went from 0 to 3.4.

* Community vote count went from 0 to 10.

* Public rating went from 0 to 4.3.

* Public vote count went from 0 to 6.

* Community ranking went from being in 0th place to being tied for 16th out of 112.

* Public ranking went from being in 0th place to being tied for 19th out of 112.

You just run the program once, in a command prompt window, and then minimize that window and let it do its thing in the background.

If you're interested in a copy (with source for those who care) drop me a note at ramblinplan@yahoo.com.

Thanks,

Owen Cunningham

P.S. This program requires Windows and the .NET Framework. It has been tested only on Windows XP Service Pack 3 running .NET Framework 3.5, but has a good chance of working with earlier versions.

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Dr. Andreas Martin Lisewski,

I think your essay is excellent. You take a theoretical physics vision of the universe and bring awareness into the fold. I have wondered in the past about the kind of mathematics that will be necessary to go beyond the analysis of mechanical type effects and begin to represent the development of intelligence as a natural process developing toward greater complexity as the universe evolved. Your work is an important contribution to finding a new kind of path for analyzing the complete nature of the universe.

James

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I think your essay is excellent. You take a theoretical physics vision of the universe and bring awareness into the fold. I have wondered in the past about the kind of mathematics that will be necessary to go beyond the analysis of mechanical type effects and begin to represent the development of intelligence as a natural process developing toward greater complexity as the universe evolved. Your work is an important contribution to finding a new kind of path for analyzing the complete nature of the universe.

James

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Dear Andreas,

My congratulations on the challenge you have tried to pick up in this essay.

Unfortunately, I was not able to discern any physical content in the paper. Could you please give some more details on that?

Good luck with the contest!

Steven Oostdijk

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My congratulations on the challenge you have tried to pick up in this essay.

Unfortunately, I was not able to discern any physical content in the paper. Could you please give some more details on that?

Good luck with the contest!

Steven Oostdijk

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Steven,

The essay explores how far can we get if we assume that the fundamental physical process in the universe is the counting process of natural numbers. I argue that with assumption we can reasonably approach fundamental physics problems such as (1) the pointer state problem in quantum mechanics; (2) the wave-function collapse in quantum mechanics; (3) the apparent continuity of the space manifold and its three-dimensionality; (4) the origin of geometry, locality and causality; (5) the quantum-classical transition without an external, classical environment; (6) the self-referential nature of physical observations.

Thank you.

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The essay explores how far can we get if we assume that the fundamental physical process in the universe is the counting process of natural numbers. I argue that with assumption we can reasonably approach fundamental physics problems such as (1) the pointer state problem in quantum mechanics; (2) the wave-function collapse in quantum mechanics; (3) the apparent continuity of the space manifold and its three-dimensionality; (4) the origin of geometry, locality and causality; (5) the quantum-classical transition without an external, classical environment; (6) the self-referential nature of physical observations.

Thank you.

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Dear Andreas,

Natural numbers do not include negative ones. While your approach seems to start from an idea, I dealt with related questions from a quite different perspective. May I ask you for a comparison?

Regards,

Eckard

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Natural numbers do not include negative ones. While your approach seems to start from an idea, I dealt with related questions from a quite different perspective. May I ask you for a comparison?

Regards,

Eckard

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James, thanks for your comment.

Eckard, I briefly read over your essay but could not find any obvious parallels to my text. Can you be more specific?

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Eckard, I briefly read over your essay but could not find any obvious parallels to my text. Can you be more specific?

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Dear Andreas Martin,

very interesting point of view with intriguing observations. Here's one thing I don't understand: the "structural unfolding" of the natural numbers generates a rooted tree. As a relation, this is antisymmetric. On the other hand, the proximity relation is symmetric. So how can these two Kripe frames be bisimilar?

In structural set theory terms, the question is the following: the natural numbers are a deeply nested set, but do not contain self-references. On the other hand, thinking of a proximity relation as a membership relation, it is inherently self-referential.

[I just learned a lot of this stuff for the first time--your essay was a great opportunity for this--so I'm still somewhat shaky, but I hope the question makes sense.]

Finally, the contest rules state that "the entry should differ substantially from any previously published piece by the author". Your essay however has substantial overlap with your paper arXiv:quant-ph/0412047, not only in content, but even in language. So where does your essay provide new insights into any aspect of this?

all the best, Tobias

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very interesting point of view with intriguing observations. Here's one thing I don't understand: the "structural unfolding" of the natural numbers generates a rooted tree. As a relation, this is antisymmetric. On the other hand, the proximity relation is symmetric. So how can these two Kripe frames be bisimilar?

In structural set theory terms, the question is the following: the natural numbers are a deeply nested set, but do not contain self-references. On the other hand, thinking of a proximity relation as a membership relation, it is inherently self-referential.

[I just learned a lot of this stuff for the first time--your essay was a great opportunity for this--so I'm still somewhat shaky, but I hope the question makes sense.]

Finally, the contest rules state that "the entry should differ substantially from any previously published piece by the author". Your essay however has substantial overlap with your paper arXiv:quant-ph/0412047, not only in content, but even in language. So where does your essay provide new insights into any aspect of this?

all the best, Tobias

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Tobias,

thanks for your questions.

First of all, in hyperset theory, the membership relation can well be symmetric (A is member of B, and B is a member of A), and this fact is being used for the bisimulation with the proximity relation. The proposed bisimulation between both Kripke structures is therefore well defined.

Secondly, you are right that some parts of this essay are available as an arXiv preprint but they have not been published. I have taken the opportunity of the essay contest to rewrite, shorten, carve out and further develop those original ideas. In fact, the main idea about the fundamental physical nature of natural numbers was not presented explicitly in the old preprint text.

I appreciate your interest, thank you.

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thanks for your questions.

First of all, in hyperset theory, the membership relation can well be symmetric (A is member of B, and B is a member of A), and this fact is being used for the bisimulation with the proximity relation. The proposed bisimulation between both Kripke structures is therefore well defined.

Secondly, you are right that some parts of this essay are available as an arXiv preprint but they have not been published. I have taken the opportunity of the essay contest to rewrite, shorten, carve out and further develop those original ideas. In fact, the main idea about the fundamental physical nature of natural numbers was not presented explicitly in the old preprint text.

I appreciate your interest, thank you.

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wow, that was an instant reply, great!

So do you mean that the hyperset U discussed in your essay does have a symmetric membership relation? How is U defined it all? Is it defined via its membership graph, which in turn is taken to coincide with the proximity relation? Then the bisimulation principle trivially holds by definition of U.

I have started to study your preprint arXiv:quant-ph/0412047 and find myself having trouble parsing some of the statements and extracting their precise meaning. For example on the bottom of page 21, you state that phi, phi' and psi, psi' are elements of Kripke structures, i.e. possible worlds. But then you also use them as arguments of value assignment functions, just like in section 2 where phi stands for a formula of modal logic. So, what is the intended meaning of these symbols, and what does part (a) of the definition of bisimulation (p. 21) actually state? Just trying to understand...

If I get this right, the evolution of the system is governed by transition probabilities to specified basis vectors. In other words, the system is described by a Markov chain? This sounds a lot like a non-contextual non-local hidden variable model.

Finally, what is the physical motivation behind the choice of self-test operator on page 35? IMHO the Euclidean distance matrix, and therefore also the lambda_i and the self-test operator, depend on the chosen embedding of the tree metric into l_1.

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So do you mean that the hyperset U discussed in your essay does have a symmetric membership relation? How is U defined it all? Is it defined via its membership graph, which in turn is taken to coincide with the proximity relation? Then the bisimulation principle trivially holds by definition of U.

I have started to study your preprint arXiv:quant-ph/0412047 and find myself having trouble parsing some of the statements and extracting their precise meaning. For example on the bottom of page 21, you state that phi, phi' and psi, psi' are elements of Kripke structures, i.e. possible worlds. But then you also use them as arguments of value assignment functions, just like in section 2 where phi stands for a formula of modal logic. So, what is the intended meaning of these symbols, and what does part (a) of the definition of bisimulation (p. 21) actually state? Just trying to understand...

If I get this right, the evolution of the system is governed by transition probabilities to specified basis vectors. In other words, the system is described by a Markov chain? This sounds a lot like a non-contextual non-local hidden variable model.

Finally, what is the physical motivation behind the choice of self-test operator on page 35? IMHO the Euclidean distance matrix, and therefore also the lambda_i and the self-test operator, depend on the chosen embedding of the tree metric into l_1.

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Tobias,

I am glad we can discuss this.

First, what you might call "trivial" is a central result in hyperset theory, i.e. that every graph depicts, up to bisimulation equivalence, a set. Thus, in my work, a central point is to identify the tree graph of structural unfolding with the proximity relation of experimental outcomes in the universal quantum system.

Second, every possible world in a Kripke structure (a graph node) represents a modal sentence. In my preprint you are referring to, as a general rule, \varphi stand for modal sentences in set theory and \psi stand for their bisimilar counterparts in the preferred basis.

Third, with regard to the embedding, every tree metric can be uniquely transformed into an ultrametric, and any ultrametric can be isometrically embedded into l_2. The resulting distance metric (self-test) in l_2 then becomes independent of the emdedding vectors (see, e.g. the work of Deza and Laurent, ref. [10] in preprint). The old preprint you refer to does not explicitly mention this.

I am not sure if I understand your Markov chain remark.

Thanks und viele Gruesse nach Bonn,

Andreas

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I am glad we can discuss this.

First, what you might call "trivial" is a central result in hyperset theory, i.e. that every graph depicts, up to bisimulation equivalence, a set. Thus, in my work, a central point is to identify the tree graph of structural unfolding with the proximity relation of experimental outcomes in the universal quantum system.

Second, every possible world in a Kripke structure (a graph node) represents a modal sentence. In my preprint you are referring to, as a general rule, \varphi stand for modal sentences in set theory and \psi stand for their bisimilar counterparts in the preferred basis.

Third, with regard to the embedding, every tree metric can be uniquely transformed into an ultrametric, and any ultrametric can be isometrically embedded into l_2. The resulting distance metric (self-test) in l_2 then becomes independent of the emdedding vectors (see, e.g. the work of Deza and Laurent, ref. [10] in preprint). The old preprint you refer to does not explicitly mention this.

I am not sure if I understand your Markov chain remark.

Thanks und viele Gruesse nach Bonn,

Andreas

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danke! I hope you don't mind that I keep questioning your work, being a generally extremely skeptical person towards my own and other people's ideas. Overall, I find the paper pretty hard to read since the distinction between postulates, derived results and their proofs is hard to make out. So far I have not been able to spot anything which resembles a non-trivial proof.

Regarding the first...

view entire post

Regarding the first...

view entire post

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Tobias,

Yes, you are right and I was inaccurate in my language when I called the AFA a result (in the sense of a theorem). What I meant, however, was the observation that every rooted graph depicts a set is a generalization of conventional set theory and thus a "conceptual result" with non-trivial implications. To call it "trivial" is misleading; in the same sense it would be misleading to...

view entire post

Yes, you are right and I was inaccurate in my language when I called the AFA a result (in the sense of a theorem). What I meant, however, was the observation that every rooted graph depicts a set is a generalization of conventional set theory and thus a "conceptual result" with non-trivial implications. To call it "trivial" is misleading; in the same sense it would be misleading to...

view entire post

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A minor correction: The formula in my previous post should read

D*ij = c + 1/2 (Dij - Dir - Drj)

with c = max(Dij}.

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D*ij = c + 1/2 (Dij - Dir - Drj)

with c = max(Dij}.

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One more explanation as you said " I don't understand what you mean by the statement that "every possible world [...] represents a modal sentence"."

In structural set theory, every node (possible world) in the graph representing a Kripke structure represents a set and every set is satisfied by a modal sentence (formula) in modal logic , i.e for all sets A there exists a modal sentence \varphi such that A |= \varphi.

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In structural set theory, every node (possible world) in the graph representing a Kripke structure represents a set and every set is satisfied by a modal sentence (formula) in modal logic , i.e for all sets A there exists a modal sentence \varphi such that A |= \varphi.

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Thank you for your extensive explanations, this has cleared up a lot! Now I find myself not being able to resist another reply ;)

First, good, I agree about AFA and that it shouldn't be called "trivial". From your paper I just had the impression that you claimed to have a derivation of this, therefore the misunderstanding, sorry.

Second, I know what a Kripke structure is. And certainly yes, at any node there is a sentence turning true at that node; for example, any tautology will do. This doesn't answer my question, but I can see the direction and will have to do some more reading and think it through.

Third, thank you for the explanation of "transforming". I don't doubt that one can then isometrically embed this into Hilbert space; I just don't see the point of doing it.

Fourth, unitary evolution is deterministic in the sense that if you know the exact state at the present time, you can know with certainty the state at any future time. So if we apply quantum theory to the whole universe as a closed system, it becomes deterministic. If my understanding is correct, this is not the case in your model. Then, in particular, your model should have somewhat different physics than ordinary quantum theory. Are they different observationally? (There is good reason to be suspicious about any theory which assigns a state to the whole universe, but that is a different story.)

Fifth, fine, this is a nice simplification.

It was very interesting to read the referee reports!

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First, good, I agree about AFA and that it shouldn't be called "trivial". From your paper I just had the impression that you claimed to have a derivation of this, therefore the misunderstanding, sorry.

Second, I know what a Kripke structure is. And certainly yes, at any node there is a sentence turning true at that node; for example, any tautology will do. This doesn't answer my question, but I can see the direction and will have to do some more reading and think it through.

Third, thank you for the explanation of "transforming". I don't doubt that one can then isometrically embed this into Hilbert space; I just don't see the point of doing it.

Fourth, unitary evolution is deterministic in the sense that if you know the exact state at the present time, you can know with certainty the state at any future time. So if we apply quantum theory to the whole universe as a closed system, it becomes deterministic. If my understanding is correct, this is not the case in your model. Then, in particular, your model should have somewhat different physics than ordinary quantum theory. Are they different observationally? (There is good reason to be suspicious about any theory which assigns a state to the whole universe, but that is a different story.)

Fifth, fine, this is a nice simplification.

It was very interesting to read the referee reports!

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Thanks Tobias. One more response. The embedding of the tree metric into the Hilbert space has the purpose of obtaining pointer states, i.e. the elements of the preferred basis, as the eigenvectors of the euclidean distance matrix in the Hilbert space (the self-test).

One point regarding time. The ordinal alpha is the exotime, or stage time, in the sense described in the essay and introduced also by Jaroszkiewicz, Bucchieri, among others. This is not the local (internal) time in quantum theory which appears in the Schroedinger equation. Physical time is therefore two-fold: it has a discrete stage character and it is a local, continuous parameter in unitary dynamics. Both characters of time are radically different, of course. I recommend further reading about "endophysics" which opens this new dimension of physical time.

I think, a the very least, the essay write-up and the discussion here have already motivated me again to write up these ideas in a shorter, concise scientific manuscript. I'll keep you updated on the results, if you wish.

Good luck also with your ideas!

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One point regarding time. The ordinal alpha is the exotime, or stage time, in the sense described in the essay and introduced also by Jaroszkiewicz, Bucchieri, among others. This is not the local (internal) time in quantum theory which appears in the Schroedinger equation. Physical time is therefore two-fold: it has a discrete stage character and it is a local, continuous parameter in unitary dynamics. Both characters of time are radically different, of course. I recommend further reading about "endophysics" which opens this new dimension of physical time.

I think, a the very least, the essay write-up and the discussion here have already motivated me again to write up these ideas in a shorter, concise scientific manuscript. I'll keep you updated on the results, if you wish.

Good luck also with your ideas!

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I am the editor of the "Markov Chain Universe" website and would like to comment on the above discussion from the context of Markov Chain Universe theory. (If you are curious what it is, please google the term).

The flow of time represents the transitions of the universe from one state to another state. The future of potential states is determined completely by the current existing state of the universe (and there is no secret hidden markings somewhere outside of the current state dictating its future trajectory). Hence, the universe is a Markov Chain.

It unfolds through a time history where the past is distinguishable from the future by a set of distinctly available states. The states of the future are not the same available states as the past. This is the primary distinguishing feature of the passage of time- movement into a new state space.

Quantum theory does indeed prescribe a specific deterministic movement through time to the future- excluding the "decoherence" which is physically observed that places the outcome of the future into only one of the possible sets of that deterministic evolution of the probability function of quantum theory.

The exciting aspect of Markov Chain Universe theory is that some states are lower probability than others. It appears that living organisms themselves are lower probability structures propagating themselves through time.

In any event, whether or not you accept that you do have to accept the universe behaves as a markov chain, providing you describe the set of states of the system coherently with regards to what is observed by physical law.

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The flow of time represents the transitions of the universe from one state to another state. The future of potential states is determined completely by the current existing state of the universe (and there is no secret hidden markings somewhere outside of the current state dictating its future trajectory). Hence, the universe is a Markov Chain.

It unfolds through a time history where the past is distinguishable from the future by a set of distinctly available states. The states of the future are not the same available states as the past. This is the primary distinguishing feature of the passage of time- movement into a new state space.

Quantum theory does indeed prescribe a specific deterministic movement through time to the future- excluding the "decoherence" which is physically observed that places the outcome of the future into only one of the possible sets of that deterministic evolution of the probability function of quantum theory.

The exciting aspect of Markov Chain Universe theory is that some states are lower probability than others. It appears that living organisms themselves are lower probability structures propagating themselves through time.

In any event, whether or not you accept that you do have to accept the universe behaves as a markov chain, providing you describe the set of states of the system coherently with regards to what is observed by physical law.

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Hi M.C.

I looked up Markov chain on wikipedia:Quote: "A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes."

I have a problem envisioning the material universe in this kind of way because it does not consist just of simple linear processes and structures. Structure and function go hand in hand. Current structure imposes constraints and guides development rather than it just being -random- events. This is an extremely important concept in biology.

Could get into a long debate over the wording of your first Axiom but will resist doing so: )

You say in your Axioms page under number 7. "More improbable structures are more complex." Isn't that because you have selected the Markov chain as foundational and it assumes random construction. But complex forms are more probable not more improbable if potential energy is minimised by their formation. I do not expect a lightening bolt or trickle of water or tree to be a straight line. ??? The shapes are complex but not more improbable. I would be interested in your explanation if you care to enlighten me.

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I looked up Markov chain on wikipedia:Quote: "A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes."

I have a problem envisioning the material universe in this kind of way because it does not consist just of simple linear processes and structures. Structure and function go hand in hand. Current structure imposes constraints and guides development rather than it just being -random- events. This is an extremely important concept in biology.

Could get into a long debate over the wording of your first Axiom but will resist doing so: )

You say in your Axioms page under number 7. "More improbable structures are more complex." Isn't that because you have selected the Markov chain as foundational and it assumes random construction. But complex forms are more probable not more improbable if potential energy is minimised by their formation. I do not expect a lightening bolt or trickle of water or tree to be a straight line. ??? The shapes are complex but not more improbable. I would be interested in your explanation if you care to enlighten me.

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