The fractal fern structure can also be found in some of the earliest life in the ocean. Apparently, this early attempt of iteration of a single cell was not sufficiently sophisticated yet. The species was at a genetic dead end and was replaced with the chromosomal method (itself also fractal in nature).

... and on pond ice and window panes under the proper conditions. ;-)

The fractal fern structure can also be found in some of the earliest life in the ocean. Apparently, this early attempt of iteration of a single cell was not sufficiently sophisticated yet. The species was at a genetic dead end and was replaced with the chromosomal method (itself also fractal in nature).

... and on pond ice and window panes under the proper conditions. ;-)

Fractals can be used in a forest to count trees. It is a whole new dimension added to mathematics (Fractal Geometry, CDT).

I have started a new thread with the question if fractal solar panels might prove efficient.
Cell-phones now use fractal antennas, which gives them more sensitivity and wider bandwidth and extremely small size.

They rose from biology, to explain the growth of plans.

Okay, let’s first construct a simple fractal to use as an example. Starting with a line segment, take the middle third and replace it with a triangular bump consisting of two more line segments as depicted in the top two figures here:

This single transformation is the fractal’s function. Now re-apply this function to each of the four resulting line segments and you get the second iteration depicted in the third figure. The last two figures depict iterations three and four. This is known as a Koch curve. With an infinite number of iterations the original total length becomes infinite. With an infinite number of iterations, we can continuously zoom in on this fractal and always find self-similarity. However, we don’t need to resort to infinite iterations to see my criticism of superficiality.

So let’s say we’ve constructed a Koch curve fractal with some finite number of iterations. As we zoom in on this fractal we keep seeing self-similarity until some point where we zoom in on either just a simple line segment or just a 60° or 120° angle. If we could do this with a romanesco or Lindenmayer weed I would withdraw my complaint. We can, of course, achieve this if we stop zooming at some arbitrary (aside from convenience to your arguments) level.

Imagine if we kept zooming in on a candidate that initially looks like a Koch curve and eventually found something very different, say, something resembling Arabic script, full of disconnected curves and dots rather than straight line segments and certain angles. And if we kept zooming in those features gave way to overlapping ellipses and asterisks. Then I would say that our candidate was only superficially described by fractals. Would you agree with me here?

I say this is the case with our romanesco and Lindenmayer weed candidates. The self-similarity being presented is only demonstrated to an arbitrary degree of zooming in. If we kept zooming we’d encounter scales of proteins, atoms, quarks, (and whatever physics you wish to support beyond that) that have not been demonstrated to have the same self-similarity evident at greater scales.

citizenschallenge.pm - 19 December 2010 08:15 PM

I think I see what you’re saying… but does it need to be mathematically perfect… to be counted as a fractal?

Perfect? No, but then we get into a battle of degrees of self-similarity. Given what I wrote above, I would like to see a convincing degree of self-similarity at all scales before I withdraw my superficiallity complaint.

Write4U - 19 December 2010 05:30 PM

As I understand Loll, fractals work at any level, thus any counter examples on the molecular or atomic scale would also be subject to fractal geometry. Loll seems to think that CDT works at quantum (Planck) level. She proposes that the very curvature of space/time at Planck level is fractal in nature. If we can get there Loll believes it might lead to TOE.

So let’s take Loll’s hypothesis and build upward like we built upward in the construction of the Koch curve. We go from the structure of spacetime to various subatomic particles and from these to various atoms and on to various molecules, cells, tissues, and our candidate plants. Show me that the self-similarity being described by Loll is the same self-similarity being extolled in the romanesco and Lindenmayer weed candidates. The self-similarity isn’t even convincingly similar between these two candidates. So somewhere along line it seems maybe a different fractal function is being employed. Which isn’t to say that some sort of fractal function isn’t being employed, just that the self-similar examples shown here only demonstrate superficial fractal depth. Which was my complaint.

[ Edited: 20 December 2010 09:29 AM by the PC apeman ]

A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.

Mathematical fractal:

A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

However, in reality, it is impossible to have infinite iteration:

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns.

One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. It could be said that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the speculation that its dimension might best be described in a sense by a number between one and two.

Hence, there are no mathematical fractals in reality. However, this does not imply fractality does not exist in reality.

In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is fractal. More generally, it relates to the usage or appearance of fractals in the study of the universe and matter. A central issue in this field is the fractal dimension of the Universe or of matter distribution within it, when measured at very large or very small scales.

Fractals in observational cosmology:

A debate still ensues, over whether the universe will become homogeneous and isotropic (or is smoothly distributed) at a large enough scale, as would be expected in a standard Big Bang or FLRW cosmology, and in most interpretations of the Lambda-CDM (expanding Cold Dark Matter) model. Scientific consensus interpretation is that the Sloan Digital Sky Survey suggests that things do indeed seem to smooth out above 100 Megaparsecs.

OTOH:

In May 2008, another paper was published by a team including Pietronero, that concludes the large scale structure in the universe is fractal out to at least 100 Mpc/h.

In theoretical cosmology:

Since 1986, however, quite a large number of different cosmological theories exhibiting fractal properties have been proposed. And while Linde’s theory shows fractality at scales likely larger than the observable universe, theories like Causal dynamical triangulation and Quantum Einstein gravity are fractal at the opposite extreme, in the realm of the ultra-small near the Planck scale. These recent theories of quantum gravity describe a fractal structure for spacetime itself, and suggest that the dimensionality of space evolves with time. Specifically; they suggest that reality is 2-d at the Planck scale, and that spacetime gradually becomes 4-d at larger scales.

The work of Connes with physicist Carlo Rovelli suggests that time is an emergent property or arises naturally, in this formulation, whereas in Causal dynamical triangulation, choosing those configurations where adjacent building blocks share the same direction in time is an essential part of the ‘recipe.’ Both approaches suggest that the fabric of space itself is fractal, however.

However, the concept of the fabric of space as a continuum is intuitive and some would find it preposterous to consider it as fractal.

They rose from biology, to explain the growth of plans.

Okay, let’s first construct a simple fractal to use as an example. Starting with a line segment, take the middle third and replace it with a triangular bump consisting of two more line segments as depicted in the top two figures here:

However, we don’t need to resort to infinite iterations to see my criticism of superficiality.

You think that the transformation from straight lines into a single meandering (curvature) line (occupying a greater plane) is superficial?

Imagine if we kept zooming in on a candidate that initially looks like a Koch curve and eventually found something very different, say, something resembling Arabic script, full of disconnected curves and dots rather than straight line segments and certain angles. And if we kept zooming in those features gave way to overlapping ellipses and asterisks. Then I would say that our candidate was only superficially described by fractals. Would you agree with me here?

I don’t. If we found Arabic script, we probably would find that the script itself is fractal in nature. But at the greatest possible reduction (zero state), individual characteristics are “washed out” (Loll) and apparently no longer of importance..

Write4U - 19 December 2010 05:30 PM

As I understand Loll, fractals work at any level, thus any counter examples on the molecular or atomic scale would also be subject to fractal geometry. Loll seems to think that CDT works at quantum (Planck) level. She proposes that the very curvature of space/time at Planck level is fractal in nature. If we can get there Loll believes it might lead to TOE.

So let’s take Loll’s hypothesis and build upward like we built upward in the construction of the Koch curve. We go from the structure of spacetime to various subatomic particles and from these to various atoms and on to various molecules, cells, tissues, and our candidate plants. Show me that the self-similarity being described by Loll is the same self-similarity being extolled in the romanesco and Lindenmayer weed candidates. The self-similarity isn’t even convincingly similar between these two candidates. So somewhere along line it seems maybe a different fractal function is being employed. Which isn’t to say that some sort of fractal function isn’t being employed, just that the self-similar examples shown here only demonstrate superficial fractal depth. Which was my complaint.

The iteration of self similarity does not necessarily produce a self similar result. Your example of a straight lines morphing into a meander is proof. By your critique all we would get is triangular shapes at any level. That is true individually, but the reduction in size alone modifies the selfsimilarity. The term of self similar iteration maybe misleading or incomplete. There is a reduction of size involved. By your example #4 derived from selfsimilar (but reduced) iteration has a completely different shape as #1. Moreover, when you keep iterating, the result may include spiral formations which are completely absent in the original.

But why should we assume that molecules and atoms are not fractally constructed themselves? One can devise several different secondary fractal models within the superior fractal model, each coming into play when that stage of reduction is reached. Some sets of fractals may be a combination of subsets. There are also “attractors” which apparently are able act complimentary to the basic fractal geometry modifying the fractal set to include subsets.

If the universe is truly fractal in nature, then it becomes a matter of setting proper parameters to include the molecular, atomic, and sub atomic fractal geometries. Are all atoms not identically constructed (the only difference being the number of neutrons and electrons)? Same with subatomic particles, each kind is identical in structure.

Fractals are not things. They are a geometric function in mathematics, which theoretically can be applied until a zero state is reached. If I take an apple and keep halving it (equally dividing into two halves), when I come to the molecular (or atomic) level, I may still be able to halve it, but am I still slicing an apple?

The iteration of self similarity does not necessarily produce a self similar result. Your example of a straight lines morphing into a meander is proof. By your critique all we would get is triangular shapes at any level.

No, I didn’t claim that. First of all, don’t get the construction of a fractal confused with the exploration of it by zooming in, as you seem to be doing in your reply. Now, look at the example of the romanesco photo. Why do you suppose that photo was chosen. It’s because there seems to be a self-similar pattern in it: slightly spiral spikes covered in slightly spiral spikes. Zoom in on it down to the quark level and, if this we’re exploring a single fractal function, there should be some version of that repetition of that pattern or the pattern being held up as evidence is superficial.

The iteration of self similarity does not necessarily produce a self similar result. Your example of a straight lines morphing into a meander is proof. By your critique all we would get is triangular shapes at any level.

No, I didn’t claim that. First of all, don’t get the construction of a fractal confused with the exploration of it by zooming in, as you seem to be doing in your reply. Now, look at the example of the romanesco photo. Why do you suppose that photo was chosen. It’s because there seems to be a self-similar pattern in it: slightly spiral spikes covered in slightly spiral spikes. Zoom in on it down to the quark level and, if this we’re exploring a single fractal function, there should be some version of that repetition of that pattern or the pattern being held up as evidence is superficial.

And obviously there is (with some minor differences caused by external pressures.) The fact that the ultimate expression of similar spiral pyramids of a romanesco indicates that there must be a general similarity in the properties and and growth function throughout the plant.
But the romansco is by no means the only example available. The fern fractal is present in modern land based ferns as well as the earliest ocean organisms. Moreover nothing in the macroworld is identical. But at subatomic levels every particle is identical to another particle of the same kind. Fractals would not work if it was different. The limited number of different (but still relatively similar) building blocks allows for a standardized mathematical function.

The iteration of self similarity does not necessarily produce a self similar result. Your example of a straight lines morphing into a meander is proof. By your critique all we would get is triangular shapes at any level.

No, I didn’t claim that. First of all, don’t get the construction of a fractal confused with the exploration of it by zooming in, as you seem to be doing in your reply. Now, look at the example of the romanesco photo. Why do you suppose that photo was chosen. It’s because there seems to be a self-similar pattern in it: slightly spiral spikes covered in slightly spiral spikes. Zoom in on it down to the quark level and, if this we’re exploring a single fractal function, there should be some version of that repetition of that pattern or the pattern being held up as evidence is superficial.

And obviously there is (with some minor differences caused by external pressures.)

The iteration of self similarity does not necessarily produce a self similar result. Your example of a straight lines morphing into a meander is proof. By your critique all we would get is triangular shapes at any level.

No, I didn’t claim that. First of all, don’t get the construction of a fractal confused with the exploration of it by zooming in, as you seem to be doing in your reply. Now, look at the example of the romanesco photo. Why do you suppose that photo was chosen. It’s because there seems to be a self-similar pattern in it: slightly spiral spikes covered in slightly spiral spikes. Zoom in on it down to the quark level and, if this we’re exploring a single fractal function, there should be some version of that repetition of that pattern or the pattern being held up as evidence is superficial.

And obviously there is (with some minor differences caused by external pressures.)

Really? Down to the quark scale? Show me.

Within the plant itself? Are quarks in a romanesco plant different than quarks in a tree (another fractal expression)? At that level there is more similarity than difference. But if I place a small pebble on a growing plant, of course it affects the growth pattern. But that does not mean it is not trying to follow its fractal geometry.

The concept of self duplication (cell division) is fundamental to the creation of complex forms. And when you have self duplication (in all directions) it will always follow a fractal geometric pattern.

Within the plant itself? Are quarks in a romanesco plant different than quarks in a tree (another fractal expression)? At that level there is more similarity than difference. But if I place a small pebble on a growing plant, of course it affects the growth pattern. But that does not mean it is not trying to follow its fractal geometry.

Within the plant itself? Are quarks in a romanesco plant different than quarks in a tree (another fractal expression)? At that level there is more similarity than difference. But if I place a small pebble on a growing plant, of course it affects the growth pattern. But that does not mean it is not trying to follow its fractal geometry.

Sounds like a no then.

I can’t answer your question in formal terms. Perhaps someone else here can. My knowledge of fractal geometry is superficial, not the math itself as far as I understand it. All I can do is refer you to Loll who, I am sure, can answer your question.

Within the plant itself? Are quarks in a romanesco plant different than quarks in a tree (another fractal expression)? At that level there is more similarity than difference. But if I place a small pebble on a growing plant, of course it affects the growth pattern. But that does not mean it is not trying to follow its fractal geometry.

Sounds like a no then.

I can’t answer your question in formal terms. Perhaps someone else here can. My knowledge of fractal geometry is superficial, not the math itself as far as I understand it. All I can do is refer you to Loll who, I am sure, can answer your question.

Let me make it considerably easier for you. We started with the example frame filled with one romanesco. Now zoom in to where the frame is filled with just one romanesco cell. Is it your contention that we’ll see self-similarity in this frame that at all resembles the self-similarity observed in the original frame? Will the surface of this cell have anything close to resembling slightly spiral spikes covered in slightly spiral spikes? Keep in mind that we have a tremendous amount of additional zooming ahead of us to consider so we shouldn’t be seeing a significant unravelling of the pattern due to finite iterations.

[ Edited: 20 December 2010 10:31 PM by the PC apeman ]

First of all, don’t get the construction of a fractal confused with the exploration of it by zooming in, as you seem to be doing in your reply.

Then literally fractals do not exist at all, except as mathematical objects. I would say a romanesco has a fractal structure, because it is somehow built according a fractal algorithm. If you zoom in on a fractal on a computer, you also get at a limit: the precision of floats of the processor.

the PC apeman - 20 December 2010 07:55 PM

It’s because there seems to be a self-similar pattern in it: slightly spiral spikes covered in slightly spiral spikes.

It not just seems like that, it is that. But only to a certain level. I agree with the facts you give (how couldn’t I?), but obviously draw the opposite conclusion. I call ferns, romanescoes etc fractals because the construction is fractal. It explains why such regular structures can be determined by only a few parameters. If I remember correctly the fern generated with a ‘Iterated Function System’ has only 4 parameters. Change one a bit, and you get a completely different end result: