## Classifying Symmetry (in iterated systems)

For the discussion of math. Duh.

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Treatid
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### Classifying Symmetry (in iterated systems)

Introduction

There is a symmetry set in physics called CPT Symmetry.

I have found a pair of systems that appears to have a symmetry set with similar properties. This correlation might be significant. Or it might not be.

As such I would like to get some sort of handle on symmetry with particular reference to iterated systems.

It seems to me that there are three general approaches:

1. Find a way to construct a symmetry set for any given set of rules such that the symmetry sets can be compared for different rule sets.
2. Enumerate the symmetries for different rule sets to discover the frequency of particular symmetry types.
3. (implicitly) Construct all the rule sets that are consistent with a given symmetry set.
4. Something else I haven't thought of.

CPT Symmetry

CPT stands for Charge, Parity and Time. Within (quantum) physics the behaviour of particles may be symmetric if one reverses the charge of the particles (matter becomes anti-matter), reverses the time flow of the interaction or reflects the interaction in the imaginary plane. If all three properties are inverted together - then all particle interactions are entirely symmetric. If any one or two of the properties is inverted then some interactions remain fully symmetric but some break symmetry.

So each of the individual symmetries C, P and T are partial symmetries - but the sum of all three symmetries is a total symmetry.

Note that each symmetry only has two states. Also no account is taken of geometric symmetries such as translation and rotation in space.

Issues

This has introduced the idea of partial symmetries which would seem to make the problem of clearly classifying particular symmetries intractable. With this in mind classifying symmetry sets for given rules seems to be challenging at best.

Some thoughts & illustrations

Z=Z2 + C (the equation for Mandelbrot and Julia sets) can be taken as the rule set of an iterated system. Obviously there are constraints on the geometric symmetry of this rule set (proximity to the Origin is significant). I don't know how reasonable it is to exclude geometric symmetries from consideration. It seems plausible that two distinct systems could have matching rule symmetry sets but differing geometric symmetry sets.

Ignoring adding and subtracting elements of a rule (set) - we could regard any change to a set of rules as creating some form of symmetry. Thus Z=Z3 + C might be one rule symmetry - which in this case implies an infinite number of symmetries in this aspect of the equation. Similarly Z=Z2 * C might be another symmetry class. Here it isn't clear to what extent the operator might be constrained - it seems plausible this might also be an infinite set of symmetries if one allows unlimited complexity of operators. There is also the notional symmetry of Time reversal. Fortunately this can be expressed precisely as Z=(Z-C)1/2. In this case the unit of operation (equation - iteration) is definite and finite and can only occupy two states - forward and backward. This, at least, comes close to the tractable symmetry states seen with CPT symmetry.

On the face of it - any symmetry set that might be constructed for Z=Z2 + C seems unlikely to closely correspond to CPT symmetry. There doesn't seem to be any well defined limit on the number of symmetries or the way they may be constructed.

Cellular Automata (CA) might be more tractable. A one dimensional cellular automata in which the next state of a given cell depends only on itself and its two immediate neighbours is a simple class of CA. Assume initially that a single cell may be in one of two states. The subsequent state of a cell may be defined by a state table. There are 32 possible distinct entries in such a state table (cell + two neighbours = 3, each cell can have 2 states). A rule might be considered a single entry in this state table. Perhaps then the other states in the table are symmetries of that rule? In this case there are 8 symmetry states. The notional time reverse symmetry is not quite the same as one of those 8 table states. Many-to-one mappings are possible in this system.

The symmetry system for this CA seems less inclined to run away than the symmetry systems for Z=Z2+C. If one allowed each cell to have more than two states that might change. Is a tri-state system a symmetry set of the basic rules? or a completely new set of rules? In the latter case this seems much closer to CPT symmetry than Z=Z2+C - although still clearly different.

In a thread no a great distance from here I was raving about simplicity. Does the symmetry set of a system bear any relation to how simple a system may be considered? Or can the rules of a system be arbitrarily expressed in any language such that the symmetry sets depend on the expression of the rules?

If I have a set of axioms that define a system, then one might imagine that symmetries for those rules (axioms) can be generated by taking each rule in turn and looking at the ways in which that rule can be changed (with some vague constraint that the changed axiom must relate to the original rule). Thus non-euclidean space is a symmetry of Euclidean space through a symmetry of the fifth Pythagorean postulate. This might give us an approach to defining the symmetry set for a set of rules based on the number of distinct rules (this is completely ignoring the issue of how to determine the symmetry for a given rule). But... The language used to express a set of axioms might allow the rules to be expressed with fewer or more axioms. Does this mean that the symmetry sets for a set of rules can vary depending on how the rules are expressed? Is there no underlying consistency independent of language?

Conclusion

I'm all kinds of lost. It seems obvious to me that there is some structure to the symmetries possible for a given set of rules. The symmetry set of the rules of a CA is not arbitrary and is not dependent on the language used to express those rules. As such there is some means of classifying and comparing the symmetry sets for different rule systems. Something akin to Korogorov Complexity may show that my intuition is wrong.

1. Is it possible for any set of rules to specify a unique symmetry set that is unambiguous (does not depend on the language used to express those rules)?

[Semi-answer to self - I probably need to define a set of axioms to provide a solid foundation for the meaning of 'symmetry'... from there it should be possible to define a unique symmetry set for any given set of rules... I am then left with an issue of deciding whether CPT symmetry falls within my axioms - could be interesting].

Umm - well - looks like expressing my problem has made it clearer in my mind. Who would've thunk?

I'd still much appreciate any comments - At the moment I've little idea of how to choose a good set of axioms for this...

Talith
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### Re: Classifying Symmetry (in iterated systems)

I'll admit that I was lost pretty quickly in your post, although that is probably because I'm not a physicist. However, The wikipedia page on symmetry in physics seems to suggest that the symmetry of a system is related to the study of Lie groups and finite groups. I think I would need to know more about how physicists model 'systems' in a rigorous mathematical way in order to suggest how the relation between 'symmetry' and the theory of Lie/finite groups arises. I would guess though, that it has something to do with the isometry group of Riemannian manifolds and subgroups thereof (possibly the range of spaces is extended to Pseudo-Rimennian manifolds to account for the study of Lorentzian manifolds in general relativity).

It might be that what I've just written has nothing to do with what you want to study, but I hope it helps.

PM 2Ring
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### Re: Classifying Symmetry (in iterated systems)

I suspect that Category theory may be relevant to the OP's questions.

Vieneoume
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### Re: Classifying Symmetry (in iterated systems)

The same, infinitely

Treatid
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### Re: Classifying Symmetry (in iterated systems)

Talith wrote:I'll admit that I was lost pretty quickly in your post, although that is probably because I'm not a physicist. However, The wikipedia page on symmetry in physics seems to suggest that the symmetry of a system is related to the study of Lie groups and finite groups.

The above link is a useful intro to some symmetries.

However, I'm don't really want to stray into physics. While my specific interest is provoked by an aspect of physics - I really want a mathematical approach to describing symmetry - something more abstract than looking at specific symmetries in specific models. I would like a formal way of constructing symmetry sets for the rules of a given system such that the symmetry sets can be compared even when the underlying rules appear to be quite distinct. For example, there may be a set of rules in set theory and separately a set of rules applied to the complex number plane. These sets of rules might describe very different behaviours - however, the symmetry sets for the rules might be very similar (or not).

PM 2Ring wrote:I suspect that Category theory may be relevant to the OP's questions.

I'm working through this at the moment. I can see how this sort of approach might be relevant - It provides a framework to talk about rules (functors) in an abstract way - and thus looks like it provides a way to think about meta properties of rules from different systems in a similar fashion. To that extent it is in the right ball park. On the other hand, the abstraction seems to be too great in that it rules out examining (at least some of) the symmetry that might be specific to the functors.

I'm having a think about whether I can cast the rules of a given system as objects (X or Y) and define possible morphisms of X to {Yi, ... , Yj}... e.g. Z2 is a rule (technically Z=Z2) for which a set of morphisms exist to the rules {Z2, Z3, Z4, ... , Zinfinity}...

This might be an approach to constructing a set of sets for a given rules set (function) - and then the set of sets for given rule sets can be compared.

I still don't know how to ensure that I create just one unique set of sets for a given function... taken to the extreme, every function can be mapped to every other function which leaves no distinction between functions - I need some way to constrain the mappings (or to order the mappings)...

Hmm... getting closer - thank you.

Talith
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### Re: Classifying Symmetry (in iterated systems)

The link I gave to symmetries in physics was really just to try and give some context for the rest of my post which is more conceptually important. Classically in mathematics, the set of symmetries of an object can be described by a group, for instance, the capital letter 'B' has symmetry group ℤ2 generated by the map from B to itself which reflects in a horizontal line going through the obvious 'centre' of the letter (I've used a very loose definition of symmetry group here for simplicity. Ordinarily one would have to specify the category that we're considering the set 'B' to be an object in).

This is obviously a trivial example but the generality of the theory of groups is really quite amazing. It seems to me that any theory of symmetry that you are interested in studying would find a good starting point as first a study of groups. Once again, I have to admit that it's hard to understand what exactly you're trying to formulate as a theory. It might be that category theory is a good language to formulate your theory around. Category theory is a powerful and robust language.

You also make regular reference to iterated systems - the obvious mode of study for investigating iterated systems is the theory of dynamical systems. Specifically, discrete dynamical systems. You'll find that after studying discrete dynamical systems for a bit, you quickly meet your favourite example, the Mandelbrot set.

On rereading your first post, I simply have to ask a question. What is a 'symmetry set'? Wikipedia gives a definition of a symmetry set to be something pretty unrelated to what you're talking about. I feel like if we can give a good definition for what a symmetry set is, it might make communication a bit easier. For that matter, what do you mean when you use the term 'system'? Are you just using it to describe an element of some set of mathematical objects that you're not quite sure of? It would be nice to know what objects we're looking at here. Feel free to make your own definitions if they don't already exist to your knowledge - just make sure they're formulated with reference to well known definitions and not more hand-wavey terms with ambiguous meaning.

Treatid
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### Re: Classifying Symmetry (in iterated systems)

TL;DR: After some thought I have discovered that what I was trying to do can't be done. This is progress. I still want to be able to compare the symmetry(s) of two systems - but I need to take a different approach than the one I was imagining.

Groups and Categories still look like they are going to be useful tools.

Only read further if you are morbidly curious...

Talith wrote:The link I gave to symmetries in physics was really just to try and give some context for the rest of my post which is more conceptually important. Classically in mathematics, the set of symmetries of an object can be described by a group, for instance, the capital letter 'B' has symmetry group ℤ2 generated by the map from B to itself which reflects in a horizontal line going through the obvious 'centre' of the letter (I've used a very loose definition of symmetry group here for simplicity. Ordinarily one would have to specify the category that we're considering the set 'B' to be an object in).

It looks like I skipped over the concept of Group too quickly. A quick look at groups on wiki does look interesting. I shall take a closer look...

Talith wrote:You also make regular reference to iterated systems - the obvious mode of study for investigating iterated systems is the theory of dynamical systems. Specifically, discrete dynamical systems. You'll find that after studying discrete dynamical systems for a bit, you quickly meet your favourite example, the Mandelbrot set.

Yes indeed. I use Mandelbrot merely as a reasonably compact illustration. It is quicker than trying to explain Graph Dynamical Systems which is my main area of play.

Talith wrote:On rereading your first post, I simply have to ask a question. What is a 'symmetry set'? Wikipedia gives a definition of a symmetry set to be something pretty unrelated to what you're talking about.

Agreed - that is not what I mean by symmetry set.

Talith wrote: I feel like if we can give a good definition for what a symmetry set is, it might make communication a bit easier.

This is almost certainly a key issue. Usually a symmetry is some transformation that leaves the initial object unchanged. However, I'm playing with Partial Symmetries - this confuses matters somewhat. I'm also looking at symmetry of the rules of a system rather than the symmetry of the space - maybe.

[Turns out I'm thinking aloud for the following bit - probably not worth trying to understand it...]

I have two data points... CPT Symmetry in physics and a similar structure of symmetry in a Graph Dynamical System (GDS). In the GDS the symmetry arises through symmetries of the rule system. Time symmetry arises through logical reversal of the rules (and is common to all discrete dynamic systems); the other two symmetries arise from inverting the direction of uni-directional relationships, and rotating a quality in the inverse direction respectively. Changing any one or two of these parts of the rules slightly changes the behaviour of the system - changing all three together is a full identity of the rules (back at the starting set of rules).

Obviously it is unclear how the CPT symmetry arises in physics. However, understanding the GDS symmetry may provide insight into the CPT symmetry of physics.

The rules of the GDS are such that there are only a limited number of ways of changing those rules without adding additional rules to the system. Each of the individual changes to the rules can be considered symmetrical in a conventional sense in that they are directly inverting/reflecting an existing rule (inverting the arrow of time, inverting the direction of a uni-directional edge, inverting the direction of rotation of a rotation operation).

There are other changes to the rules that can be made without adding anything to the rules - these changes are not as symmetrical as the described changes.

[/thinking]

I was working on the idea that the symmetry set of a given set of rules is achieved by substituting each symbol of the rule with a corresponding symbol (A number would be a single symbol no matter how many digits are used to represent it - and could be replaced by any other number - possibly with distinction between integers and reals (for example)). Unfortunately that gives rise to Z=Z2+C having a different symmetry set to Z=Z*Z+C.

I'm stumped (and confused). Umm...

For any given description of a Turing Machine there are an infinite number of equivalent systems (have the same output). This means that it is impossible for me to breakdown a system into components (e.g. rule set) and construct a symmetry around that subset that is in anyway meaningful - Conversely, for any given system I can find an equivalent system with any symmetry (for the rules) I choose.

Good - What I thought I was trying to construct can't be constructed.

Talith wrote: For that matter, what do you mean when you use the term 'system'?

Since my interest is primarily with dynamic systems; a system tends to be some equivalent of a Turing Machine.

Talith wrote: Are you just using it to describe an element of some set of mathematical objects that you're not quite sure of? It would be nice to know what objects we're looking at here. Feel free to make your own definitions if they don't already exist to your knowledge - just make sure they're formulated with reference to well known definitions and not more hand-wavey terms with ambiguous meaning.

I agree that my use of (mathematical) language tends to the informal and frequently strays from the mainstream use. Mathematics isn't my first language and I tend to pick it up in a rather informal manner. This doesn't help my communication. I am grateful for your encouragement to express myself so that I can be understood [seriously].

PM 2Ring
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### Re: Classifying Symmetry (in iterated systems)

The Wikipedia page on elementary group theory is quite good, but it is rather dense. I'm rather fond of Introduction to Group Theory.

Seeing that you're interested in the Mandelbrot set, did you know that you can calculate pi via the Mandelbrot iteration? See Pi and the Mandelbrot set. It's not a very efficient way to calculate pi, but it's nice to see pi turning up in the M set. Another interesting site is Mandelbrot Bud Maths, which discusses the location & size of buds on the Mandelbrot structure.

EDIT
Here's a small illustration of the Mandelbrot - pi connection. Just paste this JavaScript bookmarklet into a browser address bar & hit <enter>.
javascript:e=1/33102;z=c=.25+e*e;for(i=3;z<2;i++)z=z*z+c;i*e