Is Fractual Geometry Theory Real?

[jimharvard]

As an armchair theoretical physics fan, I recently ran across a tv program on the theory of fractual geometry as an explanation for the ordered selection of large and small trees in a forest and the basis of the design for organic systems such as blood vessells, lung aveoli, etc.

1. Is the applacation of fractual mathematics to theoretical explanations for various observable phenomina widely supported within the mathmatical community?

2. Is fractual geometry a real science?

thanks..

jim coster, esq.
pittsburgh, pa

 

[cubzar]

Fractals can have an infinite surface area and a finite volume. Blood vessels and aveoli etc. have shapes approximating those of fractals to maximise their surface areas.

 

[jimharvard]

cubzar,

thank you for your reply.

what I'm trying to survey is whether there is any support within the mathematics community for the theory that numerous natural systems represent the applacation of fractal theory to matter that exists in reality.

i'm intrigued by the explanation of complex natural systems having a design and possible origin in fractal mathmatics. i don't believe that mathematics is the result of random processes within the universe. if fractal geometry is also not the result of random processes within the universe then natural systems that appear to follow the rules of fractal mathematics can not also be the result of random selection.

so i guess, i have a fundamental question: if mathematics is the ultimate example of "order", how could the ordered rules of all mathematics result from random selection and chaos?

thanks again..
jim coster

 

[daveyinaz]

2. Is fractual geometry a real science?

I'm confused by your term here of "real science"...what exactly do you mean?

 

[jimharvard]

hi daveyinaz,

thank you for your reply.

by "real science" i mean theories and opinions that are accepted by recognized scientists in other fields. for example, there are individuals who believe that astrology is a "science." however, no area of orthodox science has ever posited that astrology is a science. the same can be said for the study of ufo's, parapsychology, "ghost hunting", kirean photography, water divination, alien abduction, etc.

most articles that I've read on "fractal geometry" talk about the basic math that produces fractal results that can be graphed or in some cases, photographed. what do legitimate mathmaticians say fractal patterns are? do they believe that there is any underlying meaning in observable fractal patterns? do they believe that the results these mathmatical theorems produce have direct relationships causually to the physically world?

simply put, if one finds that the proportion of large and small trees occurring randomly in a forest is exactly what a fractal mathmatical formula would predict that the forest should look like based upon a given fractal formula, does that suggest that a forest is not the result of random selection but the result of a highly ordered mathmatical process that governs the growth of all natural forests world-wide?

i hope I've explained my point better.

thanks again.
jim coster

 

[zetafunction]

FRACTALS (not fractuals ) are just APPROXIMATIONS in nature (due to the atoms fractals can not be exact) but they work very well as this approximations, they are used in theoretical physics to explain 'scale invariance' and other phenomena of statistical physics.

 

[Raphie]

simply put, if one finds that the proportion of large and small trees occurring randomly in a forest is exactly what a fractal mathematical formula would predict that the forest should look like based upon a given fractal formula

I saw that show too. Nova? Bottom line is that whatever "orthodox" science may state, I dare say (IMHO) that no thinking (open-minded and reasonable...) scientist would suggest that fractals play no role in physical environments, at least as a "tendency." (The coastline of Britain is just how long again? Crystals grows exactly... how?)

The problem as I see it, and I suggest it as a "problem" if only because you felt the need to ask the question you asked (and I am rather grotesquely overgeneralizing here, likely consequence of some bad experiences with "scientists") is when you get "them" (i.e the scientific establishment as an entirety) thinking about human environments; and then all logic goes out the window.

For instance: Can the study of fractals be applied to the study of social and psychological environments? To me it's a no-brainer. Which, in part, is why I view Ken Ono et al's recent mathematical discovery regarding the fractal behavior of partition numbers as truly ground-breaking. How do "we," individually or collectively, partition "thought" in the cognitive and/or social "forest" of mind/society?

Answer: Nobody knows.

But, in tandem with the common sense understanding that a) human cognition is recursive in nature (i.e one thought leads to another), and b) that while history may not repeat, it does "rhyme" (Mark Twain), there is now a powerful new mathematical tool out there, from a modeling standpoint, to (maybe) begin to try to find out.

Yet, from personal anecdotal experience, I can only suggest that notions such as I herein state or imply (i.e that the "laws" of both Mathematics and Physics can at times be applicable within the human domain, and vice-versa...) are regarded as tantamount to "heresy" by at least some working scientists. Go figure. "The Dark Ages" is very much a relative term.

- RF

 

[jimharvard]

hello raphie,

Thank you for your thoughtfull response. I had never considered the possibility that human behavior itself may be the result of highly ordered mathmatical relationships. Although being a Christian, I am a believer in the concept of "free-will" which would argue for the preeminence of random ordering in any application of theoretical mathematics to human behavior and decision making. I suppose that one could argue that the concept of "predestination" in Christian doctrine could in some sense support that idea that all human behavior is actually pre-ordained and that the "free-will" choices that we all make are actually the only choices that are possible for us to make. In that sense, I guess, some omnibus fractal force could actually be selecting the "free-will" choices that we believe we are making voluntarily. But all of that is beyond what I was originally tying to understand.

I was interested to learn that fractal mathematics is, in fact, used in the analysis of certain quantum mechanical theories. I would be interested in learning whether fractal analyses have advanced the understanding of particle physics. For example, have fractal formulas revealed unobserved but probable particle behavior?

Since it appears that the discussion of fractal mathematics borders on the edge of human understanding of "reality", matter and ephemeral numerical relationships, please allow me to posit another question that has often troubled me:

Does 2+2=4 everywhere in the universe and at all times?

If it does, then there are troubling questions regarding the application of Newtonian Mechanics at the Cosmological scale. If it does not, troubling questions of the "unalterable nature of mathmatics" seem to result.

Thanks again to all who have taken the time to read and respond to my pedestrian musings.
jim coster

 

[AlephZero]

Therre are questions about the "unalterable" nature of mathematics, and Newtonian mechanics, which are far simpler than modern quantum physics, though it took a long time for anybody to notice them.

Newtonian mechanics is clearly deterministic, in the sense that if you know the motion of everything in a system exactly at some time T, then in principle you can predict the future behaviour of the system exactly for ever into the future.

Until about 1900, it was generally assumed (without any proof) that there was an even stronger assumption than the above: if you have two systems which are in "almost exactly" the same state at time T, then they will remain in "almost exactly" the same state for ever into the future. With that strong assumption, you don't need to worry too much about the making "small" errors in measurements of the real world when doing science, because (you assume!) the consequences of small errors will always remain "small".

However round about 1900 it was realized that this assumption is simply not true mathematically (quite independent of whether it is true about "reality" or not). Probably the most well-known mathematician who did early work on this was Poincare.

Note, this has nothing to do with uncertainty in the quantum mechanics sense. It is possible to create very simple mathematical systems that behave in this way (usually described as chaotic behaviour). For example the math system may produce an endless series of numbers from some starting number. But to predict accurately say the 100th number in the series, you need to know the starting number to 100 decimial places, and to predict the millionth number you need to know the starting number to a million decimal places.

These are not just a mathematical curiosities, because there are systems in classical physics (again note, classical, not quantum mechanics) that behave in exactly this way. One example is the Earth's atmosphere, and this is why it is impossible to make accurate weather forecasts far into the future.

There are close mathematical links between this type of chaotic behaviour and fractals. In fact some well known "pretty pictures" of fractals come directly from ways of describing the behaviour of chaotic systems using geometry.

In one sense, all this neatly sidesteps the question of free-will and determinism, because it shows there are systems which are completely deterministic, but you can ask simple questions about their behaviour which can not be answered without an "infinite" amount of knowledge about the past history of the system, and therefore can never be answered in practice.

You can make up your own mind how your ideas about God might apply to this type of situation - but that is outside the scope of a science forum I think.

 

[jimharvard]

hello alephzero,

thank you for your excellent response. as i suspected, the discussion of mathmatical theorems is as daunting a task as the understanding of the science of math itself.

i have come to believe that many areas of present science challenge the limits of human language. obvioulsy, there are now serious theories in quantum mechanics that are very difficult to relate in any reasonably understandable manner. so it seems is the case for fractal geometry.

perhaps the more esoteric sciences are not as bewildering and mysterious to those immersed daily in the intracacies of those endeavors. however, for those of us viewing the complex sciences from afar, attempting to build one theory upon another to reach the "inspired" moment of understanding is much like jogging thru quicksand. every subsequent step in the intellectual analysis requires exponentially greater brain power. alas, this humble fan of science quickly reaches his limit of neuronal firing.

thanks again to all who have offered their thoughts.
jim coster

 

[Raphie]

Does 2+2=4 everywhere in the universe and at all times

In a philosophical vein, JimHarvard, here's a question to ask yourself. Let's suppose that 2 + 2 does, indeed equal 4 "everywhere in the universe and at all times." Would the truth of this statement be evident to some arbitrary given living organism, the sentience of which was inseparable from its neighbors? Or, perhaps, would the very question itself seem preposterous?

Metaphor/s: Bose Einstein Condensates / Plasma

Paradoxically, however, without even the need to resort to physical metaphors, if this truth were not self-evident to all observers of "reality", then it would arguably disprove the premise: "that 2 + 2 does, indeed equal 4 'everywhere in the universe and at all times.'"

And what I mean by that statement is the following: How might the skeptic set about "proving" that the awareness of any given deluded individual does not constitute some point in the "everywhere" that you mention?

Meaning? From at least one reference frame -- the reference frame of the deluded observer -- 2 + 2 does not equal 4. No matter how "crazy" and "misguided" that view might seem to be, it nevertheless suggests that there was one point in Space/Time for which that statement was utterly false.