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Let's take symmetry of functions as an example of regularity, others may be their periodicity, etc. If one analyzes it, one can conclude that even functions, which are by definition those for which f(-x)=f(x), examples of these are polynomials consisting of even powers of x, and odd functions, for which f(-x)=-f(x), examples of these are polynomials consisting of odd powers of x, are actually exceptions, rather than a rule, ie that functions are generally speaking asymmetric objects with respect to the x=0 axis (or plane in 3D), that do not necessarily have anything to do with those that are symmetric (even and odd). However, the fact is quite the opposite, every asymmetric function can be represented as a sum of an even and odd part, like this:
f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(even) + f(odd)
So, for even functions, odd part equals to zero, and vice versa. That may be surprising, that such a simple logic shows the truth that may seem counterintuitive. Interesting is however, that symmetry in a microscopic world, for example in the world of elementary particles, is exact, while in a macroscopic world, for example in biology, it is only approximate. Why is it so?

By that I mean that while hydrogen molecule is perfectly symmetrical consisting of two identical atoms, neither our bodies are perfectly symmetrical, nor we can produce any macroscopic object that is perfectly symmetrical. Is there a mathematical explanation for that fact, or does this question belong to a philosophy forum?

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I am afraid that before anyone can answer that, you will have to say where you draw the line between "regular" and "irregular".

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On 4/8/2019 at 12:13 AM, Hrvoje1 said:

By that I mean that while hydrogen molecule is perfectly symmetrical consisting of two identical atoms, neither our bodies are perfectly symmetrical, nor we can produce any macroscopic object that is perfectly symmetrical.

Perfect crystals can be of any size.

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6 minutes ago, Strange said:

Perfect crystals can be of any size.

Can be, but I would expect the difficulty of growing a perfect crystal to scale with size in some way.

As to the OP, there is noise in any natural process, and there are nonlinear effects in nature. These tend to confound the symmetries we see in small objects. Once you have more than one or two particles, you have entropy to worry about.

And I agree with HallsofIvy — you need to define irregular vs regular. Personally, I would not associate it with perfection/symmetry.

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Just now, swansont said:

Can be, but I would expect the difficulty of growing a perfect crystal to scale with size in some way.

It is a stochastic event and the larger it gets, the bigger the likelihood of growth defects. The likelihood for each event is dependent on the type of molecule as well as the physical conditions, of course.

The relative simple answer to that is that there are only a limited (and sometimes only singular) configuration in which a perfect system can develop. Stochastic elements that interfere with it, are a function of the number of elements that are required to form the structure. As such, larger structure are more prone to imperfections overall. Specifically biological entities usually consist of more complex subunits with more potential interactions with each other and the liquor. As such a protein crystal is much more difficult to obtain than e.g. a simple salt crystal of comparable size.

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# What is more common in nature, regularities or irregularities?

This is surely more a philosophy of meaning question.

Now by definition one point cannot be regular (or irregular)

Difficult to say if two points can be regular or irregular.

It takes many points to show regularity.

By definition an irregularity is that which is not regular, unless there are too few points to distinguish.

So in order for there to be irregularity there must be many points of regularity to establish that regularity.

If there are many more non regular points than regular, there is no pattern and therefore neither regularity nor irregularity.

On 4/8/2019 at 12:13 AM, Hrvoje1 said:

or does this question belong to a philosophy forum?

Yes

Edited by studiot

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On ‎4‎/‎22‎/‎2019 at 10:59 PM, studiot said:

This is surely more a philosophy of meaning question.

True, but while this is trivial for mathematicians, it may not be for philosophers. First of all, one should show that this is actually true for any f(x):

f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(x)/2+f(-x)/2 + f(x)/2-f(-x)/2 = 2*f(x)/2 = f(x)

Then, one should prove that this actually represents a decomposition to an even and odd function:

f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = feven(x) + fodd(x)

feven(x)=[f(x)+f(-x)]/2 => feven(-x)=[f(-x)+f(-(-x))]/2=[f(-x)+f(x)]/2= [f(x)+f(-x)]/2=feven(x)
fodd(x) =[f(x)-f(-x)]/2 => fodd(-x) =[f(-x)-f(-(-x))]/2=[f(-x)-f(x)]/2=-[f(x)-f(-x)]/2=-fodd(x)

Finally, one can check that for a symmetric function f(x), one of such parts is zero. So for odd f(x):

feven(x)=[f(x)+f(-x)]/2=[f(x)-f(x)]/2=0

For even f(x):

fodd(x)=[f(x)-f(-x)]/2=[f(x)-f(x)]/2=0

If I wrote all of this in a philosophy forum, they would assume I am pretentious.

On ‎4‎/‎22‎/‎2019 at 9:38 PM, HallsofIvy said:

I am afraid that before anyone can answer that, you will have to say where you draw the line between "regular" and "irregular".

I draw that line by mentioning something that appears to me as examples or regularity, such as symmetry and periodicity. How about you?

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1 hour ago, Hrvoje1 said:

True, but while this is trivial for mathematicians, it may not be for philosophers.

If I wrote all of this in a philosophy forum, they would assume I am pretentious.

I think the philosophers are better able to handle your proposals since these proposals do not match any of the many meanings mathematicians put the term 'regular' to. As you can see these are not 'trivial'.

Since you want to stay mathematical which of these  are you referring to?

When we have established this we can further examine your use of the term "In Nature".

Note that a crystal (or any array) can be imperfect yet have symmetry.
Symmetry is not regularity.

Edited by studiot

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1 hour ago, studiot said:

Symmetry is not regularity.

True dat, but, for example, regular polygons are more symmetrical than those which are not, so I figured it would not be that wrong to talk about symmetry considering it as an indicator of regularity. And all other cases in which mathematicians use the term "regular" sound interesting, about some of them even I have a clue, but I don't want to talk about it. I would like to discuss is the ability to pose a philosophical question, as a result of reading a mathematical text, a positive or a negative trait?

Edited by Hrvoje1

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14 minutes ago, Hrvoje1 said:

True dat, but, for example, regular polygons are more symmetrical than those which are not, so I figured out it would not be that wrong to talk about symmetry considering it as an example of regularity.

But symmetry does not imply regularity, as I have already noted and you have agreed.

In fact the symmmetries of regular polygons are unbounded and approach infinity as the number of sides increases.

This is not true of polytopes in general.

Symmetry is an interesting subject in its own right.

20 minutes ago, Hrvoje1 said:

I would like to discuss is the ability to pose a philosophical question, as a result of reading a mathematical text, a positive or a negative trait?

I offered you a philosophical analysis in my first post and you did not comment on it.

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7 hours ago, studiot said:

But symmetry does not imply regularity, as I have already noted and you have agreed.

It meant that I understand the definition of regularity with respect to polygons, which is binary. They are either regular (equiangular and equilateral), or they are not. However, if we change that definition, and admit some degree of regularity to cases that are somewhere in between two extremes, strictly regular on one side and totally irregular (in a sense that they have no equal sides) on the other, then symmetry would be an indicator and measure of such weakened regularity, maybe even the same thing, so that it might seem redundant to introduce the notion of "loose regularity" when there is already a notion of symmetry. Anyway, in case of triangles, by that definition, isosceles triangle (defined as the one that has exactly two sides of equal length) would be such case, more regular than totally irregular, and less regular than strictly regular. Since it has one axis of symmetry, while strictly regular have three, and totally irregular have none. Right?

8 hours ago, studiot said:

I think that if the question is pretentious, ignorant, vacuous, such that a mathematician would gladly leave it to philosophers to answer it, then it is a negative trait, otherwise, if it has some merit, for science or even for mathematics itself, then it is a positive trait.

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On 4/22/2019 at 9:40 PM, Strange said:

Perfect crystals can be of any size.

Not really. Because you didn't take into account mass of crystal. i.e.  large enough will deform and collapse due to gravity and eventually in extremity will become black hole..

Another thing is bombardment of macroscopic object by external particles. In extremity, cosmic ray particle can decay inside and destruct or damage internal of your "perfect" crystal..

Edited by Sensei

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I came up with the question reading this article: https://mysite.du.edu/~jcalvert/phys/groups.htm , but such and similar questions were raised and reflected upon thousands of times, books were written on similar topics https://www.cambridge.org/hr/academic/subjects/physics/general-and-classical-physics/symmetries-physics-philosophical-reflections?format=HB&isbn=9780521821377 , so I am afraid I didn't do a good job.

Edited by Hrvoje1

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2 hours ago, Hrvoje1 said:

I think that if the question is pretentious, ignorant, vacuous, such that a mathematician would gladly leave it to philosophers to answer it, then it is a negative trait, otherwise, if it has some merit, for science or even for mathematics itself, then it is a positive trait.

If you don't want to discuss, then why come to a discussion site and post in the first place?

You, sir, are avoiding the issue.

3 hours ago, Hrvoje1 said:

It meant that I understand the definition of regularity with respect to polygons, which is binary.

No it meant that you either did not understand or avoided what I said about symmetry, in order that you can claim symmetry to be synonymous with regularity, which it is not.

I gave you one example of something irregular, yet possessing symmetry.

I will repeat it more generally.

Take any irregular shape and reflect it in an axis.
You now have a symmetrical irregular shape.

Point proven. They are different.

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I thought everything would have some characters of the patterns of the atoms .

Edited by MandanMaru39

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In terms of nature, there are no regularities at all. There are no perfect circles, no straight lines, no triangles or spheres, no perfect symmetry.

And in terms of abstract math, there are far more irregular and random objects than regular ones, if by regular we mean at the very least computable; that is, capable of being generated by an algorithm. Most functions are highly discontinuous, most real numbers are not ocmputable, and so forth. One shouldn't confuse the nice functions they meet in calculus class with all the wild functions that are out there.

One should also not allow oneself to be confused by the apparent regularity of physical law. What we mean by physical law in this context is NOT the true nature of the universe; if there even is such a thing.

Rather, by physical law we mean the historically contingent human-created scientific theories of the universe. All such theories are at best clever approximations. Our best physical theories are good to 12 decimal places or so. That's terrific as physical theories go. But they're not exactly what nature does.

The apparent regularity of nature may arguably be telling us more about our own minds than it does about nature.

Edited by wtf

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8 hours ago, wtf said:

In terms of nature, there are no regularities at all. There are no perfect circles, no straight lines, no triangles or spheres, no perfect symmetry.

No, not so much.

There is symmetry, but it's at a very small scale.

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1 hour ago, swansont said:
9 hours ago, wtf said:

In terms of nature, there are no regularities at all. There are no perfect circles, no straight lines, no triangles or spheres, no perfect symmetry.

No, not so much.

There is symmetry, but it's at a very small scale.

Well I suppose it come back down to what is meant by regularity.

If you note that every electron is the same as every other electron (spin apart) in the universe, I'd call that pretty regular.

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1 hour ago, studiot said:

Well I suppose it come back down to what is meant by regularity.

If you note that every electron is the same as every other electron (spin apart) in the universe, I'd call that pretty regular.

That was one of the things I was thinking about. Not just electrons. Protons, neutrons, and atoms of the same isotope, too.

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Matter on Matter interaction has created a lot of patterns

I think its OK to call that patterns regularity

The regularity in Twins for example is one such example .

The orbit , The symmetry of human beings , the rainbow etc are some regularity .

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29 minutes ago, MandanMaru39 said:

Matter on Matter interaction has created a lot of patterns

I think its OK to call that patterns regularity

The regularity in Twins for example is one such example .

The orbit , The symmetry of human beings , the rainbow etc are some regularity .

I was decorating the bathroom ceiling today.

So if I spill a pot of paint on the floor, creating a matter on matter pattern, you would call that 'regular'?

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OK , i was wrong

I think i made a mistake here thinking regularity and patterns were the somewhat the same thing .

Does electromagnetic waves have regularity then ?

I think the force acting on it has to be somewhat uniform too for regularity to happen

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22 minutes ago, MandanMaru39 said:

OK , i was wrong

I think i made a mistake here thinking regularity and patterns were the somewhat the same thing .

Does electromagnetic waves have regularity then ?

I think the force acting on it has to be somewhat uniform too for regularity to happen

Patterns can be regular, or not.

But since there are an uncountable number of both regular and irregular patterns, who is to say which there are more of in Nature?

Why does some kind of force or generator have to be involved?

Well for my example of electrons there is no small or large, just regular.

So it is important to explain what you mean by regular when you use the term.

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10 hours ago, studiot said:

So it is important to explain what you mean by regular when you use the term.

I thought to leave it open to discussion, and gave examples of what my intuition tells me the regularity is. That may or may not be identical to established mathematical terminology or to your idea of regularity. I buy medium size, and I guess that could be another interpretation of regularity. I also think that everyone intuitively understands that symmetry of polygons requires that lengths of at least some of its sides are identical, and as such is related to their regularity, that requires all of their sides to be equal.

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11 hours ago, studiot said:

Patterns can be regular, or not.

But since there are an uncountable number of both regular and irregular patterns, who is to say which there are more of in Nature?

Why does some kind of force or generator have to be involved?

Well for my example of electrons there is no small or large, just regular.

So it is important to explain what you mean by regular when you use the term.

Irregular pattern is an oxymoron, a bit, if we assume regularity is needed to establish patterns.

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