As above so below: a short journey through fractals
Beautiful, isn’t it? This abstract artistic expression is not, as you might mistakenly think, the invention of some penniless artist. This strange object is called a fractal. In particular, this is the famous Mandelbrot set.
If at this point you were wondering what the heck is a fractal, imagine indefinitely enlarging the edge of this figure. By repeating an experience perhaps already lived under massive doses of acids, you would discover that the shape of the edge is repeated endlessly.
In fact, these are self-similar structures, that is structures that are similar to themselves even if we observe them on different scales.
Let’s try to understand the concept of self-similarity a little better. Let’s take a Y, yes exactly, you got it right, something Y-shaped, just like below.
And now let’s try to repeat the exact same pattern, that is another Y-shaped object but scaled by 70%. For example, if the stem of Y measures 1, the stem of the new scaled Y will measure 0.7.
Now we can take 2 of these scaled Ys and replace them with the branches of the previous larger one.
Let’s try to repeat the process once again.
Now you should have understood: we are getting a tree! Of course, it is quite abstract and vaguely similar to a real tree, but the concept is that: trees in nature resemble fractals.
With a few more steps we can achieve this
While with the animation and changing some parameters, we can obtain a slightly more suggestive effect.
There is a quantity that can roughly indicate how full the fractals are. It is known that to calculate the volume of an object it needs 3 characteristic lengths. In the simple case of a cube with an L side, its volume will be L³. If you then take a solid, a liquid or a gas and want to calculate its mass within a certain volume, you will notice that the mass m will also be proportional to the volume, that is, it will scale with the same exponent: therefore we say that m is proportional to L³. But it won’t always be this way.
In the case of a fractal, its mass or, to be more generic, the number of self-similar substructures grows with different power of its characteristic dimensions. This exponent is called the fractal dimension. For example in the Koch curve, the number n of substructures grows as L ^ D where D, the fractal dimension, is equal to log (4) / log (3), slightly larger than 1. Translated into more understandable terms: the Koch curve is neither a line nor a plane!
But why are we interested in fractals? First of all, you can make beautiful paintings and T-shirts. But fractals often appear in the world and knowing how to use them allows us to construct interpretative models for different phenomena. For example, the cliffs draw beautiful shapes on the maps. But if we enlarge them we would discover that they are fractals, although not in a very regular way.
Now that you know more about fractals, let’s find out other interesting examples, perhaps different from the usual cabbage or the shell of a snail.
A very picturesque example is the existence of fractals in the maps of the attraction basins of phenomena described by nonlinear differential equations… What?!
Maybe it’s better to start from the bottom and reread everything more slowly. Let’s take the famous example of the magnetized pendulum. If you have what you need, you could try doing this very colourful experiment on fractals in this way: take a ferromagnetic pendulum (just a piece of iron, free to swing attached to a rope). Keep it in the rest position by attaching it to a rod and letting it stop in a vertical position. Place a few magnets (3 or more, depending on how colourful you want your final fractal) symmetrically to the pendulum’s resting position. Make sure that the magnets have the same pole facing upwards and then assign each magnet a different colour.
Now let the pendulum swing by starting it from a different position each time. You can neglect the height from the plane and mark the coordinates of the starting point from the pendulum which represent its position with respect to the vertical position. When the pendulum stops near a magnet, you can colour the starting point using the same colour as the arrival magnet. In this way, point by point, a map is created of the areas that lead the pendulum to the three different balances that are approximately placed over the 3 magnets.
The set of starting points that lead to a precise final equilibrium position is called the basin of attraction so the map we obtain is called the map of the basin of attraction.
Ok, but why all this? The final map is extremely chaotic. There will be large areas of the same colour, and more or less large filaments and spots of other colours.
Now, if you want to enlarge the narrow areas, you may find a fractal structure: on the border between two colours, there is always an interposed line of the third colour! Ideally, you could enlarge the image to infinity to always find the same pattern, but, in reality, physics sets us limits, first instrumental, then quantum (we’ll talk about this again!).
Obviously the appreciation of such a scheme depends on how precise one is in selecting and writing down the starting position, which is why you may prefer to use computer simulations (as you can see in the video below).
Okay, okay… you can make many beautiful drawings with physics, but are there other fractal objects? Sure! Some are obvious, others a little less. Polymers are among the less evident but no less fascinating. Polymers are chains of more or less similar pieces called monomers. Of course, we usually talk about protein chains and similar small strands, but to get clear ideas to consider a polymer as a pearl necklace. If you take this necklace and unfold it, the number of monomers (pearls) will be proportional to the length:
[number of pearls]= [a constant] * [length of necklace]
If we then decided to curl the necklaces in the most compact way possible, the number of monomers would grow proportionally with the cube of the radius of the ball. To give an example as similar as possible to the environment in which the proteins are found, we could dip the necklace into boiling soup. Because of the agitation inside the soup, the pearls will be pushed here and there but they will try to stay more or less all united for entropic issues. The random distribution of the pearls in the soup means that there are many in the centre of the approximate ball of pearls and gradually less outside.
At this point, you may want to count (starting from the centre of the ball), how many pearls there are. Surprisingly, the number of monomers grows in the radial direction with the following formula:
[number of monomers] = [a constant] * [distance from the centre]^1.67
And this is nothing but the fractal dimension of the polymer!
Fractals are not only a topic related to physics or computer science: they can also be traced in the biological field. Taking, for example, the human body, this has many fractal structures: the nervous system, the liver, the kidneys and even the bones or the intestine seems to recall the concepts of geometric repetition already described. This phenomenon is fundamental in the development of life as we know it, and it is found at the molecular level as well as at the macroscopic level.
In this video produced during a study, it can easily be seen how the network in which collagen is organized (one of the most present proteins in the human body, which is going to form the cellular support of tissues) follows these same principles.
Recursion in living systems has historically been classified into 3 categories:
- fractal structures isolated but related to others of higher-order
- completely isolated fractal structures
- fractal structures that stop abruptly to continue in another system, always fractal and of the same order of magnitude, but different.
As regards the first class, numerous studies are still ongoing. Indeed, it appears that this typology is traceable in the dispositions of neurons both in the deep brain nuclei and in the cortical regions. Interesting studies seem to highlight, for example, how alterations in the fractal arrangement of white and grey substances within specific brain areas underlie diseases such as schizophrenia.
Other studies instead try to trace an evolutionary history of language development by evaluating the expansion of the brain areas that concern it. Indeed, it seems that these, although located at a distance from each other, have followed the same fractal growth model in terms of the number of neurons and their connections.
The second class is of more marginal biological interest: the structural arrangement of feathers in birds belongs to this, for example.
The third class is the most represented and has a crucial role in all biological systems. This includes most human tissues, such as the arrangement of nephrons in the kidneys and hepatocytes in the liver. It also includes the wider organization of systems such as the digestive, vascular and respiratory systems.
In fact, the digestive system of cattle has a succession of different stomachs which each have different cell structures in succession, still organized in a fractal way.
Also as regards the human respiratory system, it can be seen that the rings that make up the cartilaginous skeleton of the trachea disappear after its branching in the formation of the bronchi but are immediately followed by the ramifications of the bronchial tree. Recursion is traceable here also in the structure of the epithelium, that is the shape of the cells that form the walls of the bronchi, which passes from pseudostratified to simple cylindrical. The bronchi, shrinking and branching, maintain a constant relationship between surface area and volume.
At this point, it is interesting to note how insects, which use oxygen transport systems different from those of larger organisms, find their limit to growth in their respiratory system: once certain dimensions are exceeded this is no longer able to ensure adequate oxygen supply to all districts. It, therefore, seems that this is the basis of their impossibility to reach enormous dimensions. That’s why the giant spiders of Harry Potter if they existed, would collapse to the ground lifeless due to tissue hypoxia (without having to use particular spells).
The same relationship is also preserved in another fractal system belonging to the third class, the vascular one. Here it is calculated taking into account the diastolic and systolic blood pressure (i.e. the pressure present when the heart muscle contracts and when it relaxes), the viscosity of the blood, the gravity and the resistance of the walls of the vessels. Some studies show that a vascular development where this relationship does not fall within certain limits is the basis of multiple pathologies. Furthermore, this constant seems to constitute a limit also for the growth of tumours, which promote to feed a process that leads to the formation of new vessels (neoangiogenesis).
It is fascinating to see, therefore, in all these systems, alongside an established and repeated order, the counterpart of chaos is always present, necessary to counterbalance perfection and allow a certain degree of flexibility and variability between organisms. The fractals generated by computers, not taking into account this random variability, appear to us just unnatural. In conclusion, biology, therefore, seems to prefer the use of constants capable of conserving during growth rather than exactly replicating identical geometric structures on different scales.
