Fractals and Brains (Part 1)

This is the first part of two popular science overview articles I have written about fractals. I have split the article in two parts because of the length and the time it would take to read the whole article at once. In this first part here I explain the basic concepts and give plenty of examples of fractals as they can be found and are studied in mathematics, physics, complex systems, chaos theory, quantum theory, nature in general, in the arts, in evolution theory and (molecular and quantum) biology and especially in our bodies and brains.

I will published the second part of this article early next week. The second part will focus more on fractal processes (rather than spatial fractal structures), fractal time- series, fractals and computational complexity and a further fractal analysis of our brains, language, music, oscillations and resonance, our perception and our thinking processes.

Please note: this is an introductory and overview article. Even though I cover a lot of ground and topics, the content must remain a bit superficial as some of the underlying scientific background and theories would be very complex and too difficult to explain here.

I nevertheless hope you enjoy the read - maybe on a rainy day when you have nothing better to do ;-)

Fractals

Fractals can be found pretty much everywhere in nature ! However, fractals as we know and study them today have only become noticed and popular with the general public after the publication of two books by the Polish mathematical genius Benoit Mandelbrot (1924 - 2010; see picture below) who first used and introduced the term "fractals".

His first popular science book about fractals was published in English in 1977 and called "Fractals: Form, Chance and Dimension" and his second and most popular book - and still worth and a recommended reading today - was called "The Fractal Geometry of Nature" published in 1982. With these books Mandelbrot laid the foundations for the new mathematical science of Fractal Geometry.

Mandelbrot at the time and throughout his scientific career studied with new mathematical methods the "roughness" and real, non "smooth" and non idealised shapes and contours of real life objects of nature such as everyday rough surfaces of objects (like the barks of trees), boarders and coastlines of countries and islands, the shape of mountain ridges, the formation of river beds and river deltas, all sorts of plants and flowers and even the shape of graphs representing the fluctuations of stock prices.

Contrary to most mathematicians of his time he did not see these shapes and forms as mishaps of nature or bad quality versions of "ideal" perfect shapes like the Platonists since Plato do. We know and study them in standard mathematics and Euclidean geometry in the idealised forms of straight lines, triangles, circles, 3D polygons etc that we all know from elementary math and school. Mandelbrot saw structure and recurring patterns in real life objects with rough edges and surfaces that few researchers had noticed before or even studied in detail. Instead of neglecting these "rough" patterns of many objects in nature he believed that they are representing some essential features and mechanisms worth studying in themselves.

What are "Fractals" ?

A fractal is any object (not necessarily a concrete physical object found in nature; fractals can also be abstract objects or just mathematical sets or constructs) that can be seen or understood as consisting of parts that resemble the shape and form of the overall object itself just on different, smaller scales, or in other words: a fractal object consists of parts that look like or at least look similar to the object itself just on a smaller scale. This defining property of fractals is called "scale invariance".

The "look like" or "resemble" of its parts does not necessarily mean "identical", it is enough if the parts of the fractal are or look just "similar" to the overall object; the parts can therefore also have (random) variations as long as the resemblance with the overall object is still clearly recognisable (it is actually very difficult to exactly define the concept of a fractal without using complex maths, so we leave it here at that and just use the above intuitive verbal description of what a fractal is; even Mandelbrot himself had difficulties to exactly define fractals and has changed his definition several times over his lifetime).

When one looks around in nature searching for fractals there is no scarcity of them. As a matter of fact, one can easily be surprised how frequent they actually are. Once one is aware of fractals and the concept, it's like in the popular saying: "if you have a hammer, everything looks like a nail". Many natural objects - either alive or non living - are fractals or have at least some fractal parts (or features).

Below are some pictures of classical examples of fractals occurring naturally in the un-organic, non living nature (from the top: a mountain ridge in a canyon, a snow flake, the delta of a river and a lightening strike):

Fractals can also be found in the shape of clouds or the turbulences of flowing rivers, in the atmosphere of Jupiter and the surface of the sun. As a matter of fact, the whole universe can be seen as one gigantic fractal web of similar looking spiral shaped galaxies. We can zoom-in many times on any detailed, high resolution picture of our universe and we will find levels upon levels of similar looking areas each of which again consist of smaller (or further away) galaxies with a similar structure.

I will discuss the fractal structure of our universe and a modern mathematical theory explaining the possible fractal roots of nature in the second part of this article in more detail.

Fractals as Art

Obviously, due to various physical limitations of the real world, natural objects are never fully fractal structures on all levels of their parts and scales. The fractal properties of physical objects usually only go up or down a few levels of their sub-parts (usually not more than 3 to 5 levels of scale).

To understand the underlying and inherent structure of fractals, Mandelbrot therefore studied not only naturally occurring fractals but also purely abstract mathematical fractals that can have an infinite level of detail and scales (which means one can zoom-in or out on these objects to any depth and still finds self-similar sub-structures). In such fractals one can pick any area and zoom in pretty much to any level of degree a computer simulation allows to represent and is calculable in a reasonable time.

As a result Mandelbrot discovered the most famous fractal of all which also has been named after him and in his honour, the so called Mandelbrot Set:

It has later been graphically enhanced with colouring effects and animated in the highest resolutions possible for computer graphics and to levels of extreme depth and detail admired by scientists and artists alike. Various fractals are now regularly being used in many movies for animations of more realistic looking landscapes, scenes, textures and complex backgrounds.

The original Mandelbrot set as shown above is modelled and represented usually as a 2-D structure in the 2-D plane of complex numbers. However, around 2007 some very impressive renderings of 3-D versions and variants have been created, called Mandelbrot Bulbs or Mandelbulbs (also Mandelmorphs), as shown in the pictures below.

For further more beautiful and impressive samples and 3-D fractal artwork see: www.mandelbulb.com or www.skytopia.com

Fractals as mathematical Objects

When Mandelbrot first discovered the Mandelbrot set, he was shocked and surprised by its unlimited detail and variety - and its complexity. Since his discovery and Mandelbrot's publication of the first images of the Mandelbrot Set it has instantaneously become a famous and important mathematical object to study.

This is due to the fact that the Mandelbrot Set and many other fractals discovered afterwards are often just the result of very simple iterative mathematical operations. The Mandelbrot Set for example with all its detail is generated by a simple function which maps a (complex) number x in the 2-dimensional plane of complex numbers onto x^2 + c (the square of x plus a constant c). This extremely simple recursive operation:

f(x) -> x^2 + c,

that can be applied any number of times to its initial values of c, can be implemented and programmed with an algorithm of just a few lines of code that any high school student could implement. Still, it will generate an unending variety and complexity of never before seen shapes and forms which are self similar on all levels and scales of detail.

Mandelbrot (and most mathematicians after him) was very surprised to see that the initial constant "c" in the iteration can have dramatic effects on the outcome of the repeated iteration even when "c" is only very slightly changed in its initial value. This effect is since studied and used for the definition of chaotic systems, which are dynamic (feedback) systems that are highly sensitive to their initial states when iterated many times (this happens in most dynamic systems with feedback loops; the effect is also known as the "butterfly effect").

Up and until Mandelbrot's discovery mathematicians had mostly all tacitly assumed that any mathematical or real life system or set that is very detailed and complex must also be generated by a function or algorithm that is itself very complex and difficult to understand and hard to program. The Mandelbrot set provided a famous counter-example of a highly complicated and infinitely detailed structure that was nevertheless generated by an extremely simple reiterative function (and algorithm).

Fractal Dimensions

When Mandelbrot studied the "roughness" of real life objects of nature he noticed something strange. When for example looking at the border of countries or the coast line of an island he noticed that they usually have a fractal structure. When one looks at any part of the boarder or coast line of a country or island from a large distance it usually first appears quite smooth. However, once one zooms in closer it turns out that the shape of the coastline is actually rough and when one zooms in even further it appears more and more rough and this process can be repeated down to the individual grains of sand:

By analysing this further Mandelbrot noticed that the total length of the coast line cannot be considered as invariant of the ruler that is used to measure the length ! The length of a coast line varies with and is a function of the length of the ruler used to measure the length. If one wants to have a more accurate measure of the length, one has to reduce the length of the ruler step by step to more appropriately fit to the curvature of the coastline. This leads to a limit process where the shorter and shorter ruler scales have to be added up to get the overall length. However, when doing this repeatedly the overall measured length usually increases the more rough the coastline is (at least for real life natural cost lines like visualised below using a simplified British coast line).

Mandelbrot checked systematically how the length of a line changes when more detail and fractal features are added. He noticed especially that when certain fractal constructs are used lines can be generated that slowly with the number of iterations can cover nearly any point of a given plane. Lines can somehow morph and transform from 1-dimensional objects stepwise into objects that are more like 2-dimensional objects covering a full region of a plane 2-D surface.

A 1-dimensional line can hence slowly be transformed using fractal iterative operations to fully cover a 2-dimensional area of a Euclidian plane. Such a line therefore somehow changes its dimension from 1 to approximate 2 - or potentially any value in between 1 and 2. With this insight he generalised and revolutionised the classical concept of dimension which usually measures physical objects only in natural numbers: 0 (for points), 1 (for lines), 2 (for planes), 3 (for 3-d objects). Mandelbrot however allowed dimensions that were not just full natural numbers like 0, 1, 2 and 3 but could also be fractions of natural numbers like 1,56 or 2,74.

See the pictures below for examples of fractal curves that iteratively fill a 2-D plane or a 3-D volume:

A fractal can have a natural number "n" as its fractal dimension or any fraction of a natural number. If the fractal dimension of a line is close to 1, say 1,15 then it is still considered similar to a normal straight line. If however the fractal dimension is closer to 2, say 1,95, then the line is more like part of a plane than a 1 dimensional line. For example, the Mandelbrot Set itself has a fractal dimension of 2.

In general, if a structure has a non natural number as its dimension, then it is a fractal. However, the reverse is not always true: fractals can also have natural number dimensions (as the example of the Mandelbrot Set shows). As a matter of fact, the typical Euclidian "smooth" objects of lines, planes and cubes can be considered as limit cases of fractals. If one zooms in on a line to any degree the result will still be a straight line. So any segment of a line is again a line. The same is true for any part of a plane. Cut a plane in pieces, then each piece will still be a smaller plane; and the same applies to cubes, every sub-cube of a cube is still a cube.

There are several, well established mathematical definitions of the fractal dimension of a structure. These definitions do, however, not necessarily produce the same value for a given object. Depending on the complexity of the fractal and the known details the most appropriate calculation for its dimension can be selected and used. Some of these are quite simple and straight forward like the classical "box counting" method (see below) whereas others are much more complex.

As a first and simple approximation we will use the following definition of the fractal dimension:

The fractal dimension D of a structure is the logarithm of the number of self-similar pieces of the fractal divided by the logarithm of the scaling factor.

Or more formally:

D = Log (number self-similar pieces) / Log (scale)

Here two concrete examples: the fractal dimension of the famous Sierpinsky Triangle is: 1,58 and the fractal dimension of the Koch curve is: 1,26. Therefore, the Sierpinsky Triangle is considered more like a rough plane whereas the Koch curve is considered still more like a line (see below).

The reader may note that this definition has a few variables and undefined terms. The dimension depends on the scaling factor and the number of self-similar pieces of the structure. Hence when zooming in or out of such a structure the dimension may change slightly. It also means that a full definition of the dimension requires a mathematical limit process and will only be very precise for structures that have an unlimited level of scales (up or down). This is not possible for physical objects, only for mathematical structures. In addition, the term "similar" in "self-similar" is not defined and can be very complex and difficult to define depending on the underlying structure and fractal generation process.

It would go too far for the purpose of this article to get deeper into the mathematical details of how to calculate fractal dimensions. The interested reader is referred to and may check the links and literature list as a starting point for further readings at: https://en.wikipedia.org/wiki/Fractal_dimension .

Fractals and Thermodynamics

What is surprising is the fact that fractals actually occur in the non living nature at all ! There is no known classical physical force that should produce fractals in the innate nature. The obvious existing fractal structures in the non organic world are a hard problem to be resolved and explained in classical physics. Except for fractals generated by more or less random effects like erosion fractals should not be so ubiquitous in the non living world without a known force of nature that causes their existence - or at least, if they exist so abundantly they should vanish and morph over time into random, non structured objects with non detectable patterns. But this does not seem to happen even over long periods of time.

According to the Second Law of Thermodynamics all physical objects and structures should eventually fall apart and should not leave any statistically significant patterns behind. According to contemporary physics, the world will end up in a pattern free unstructured state. However, fractals - and especially living organisms in general for that matter - are the opposite of what the second law of thermodynamics requires or predicts. Biological creatures are highly organised structures with all sorts of simple or complex self-similar and organised patterns and their building blocks - organic and un-organic macro molecules - can be found spread out all over the whole universe.

However, the basic of thermodynamics is the seemingly random Brownian motion of atoms. The atoms fly around and randomly bump into each other (depending on the density of the medium) which causes the speed of the atoms to increase or decrease and thereby heating up or cooling down the medium in which they move around. The combined impact of all the small motions of the atoms can actually cause effects on much larger macroscopic scales and objects and push them around and hence cause also chemical reactions of molecules etc which then can lead to organic highly organised forms of matter and - life.

Interestingly, the origin of this Brownian motion was unclear at first. It was already noticed by Lucretius in the ancient history who mentioned the random looking movements of small dust particles in the sun light. Later this seemingly random motion was rediscovered and analysed by Robert Brown (hence the name) in 1827 when he observed the jittery motion of tiny particles under his microscope.

Albert Einstein used this discovery to argue for the existence of atoms. Atoms hadn't been experimentally confirmed or observed by then even as late as 1905. In one of his famous papers in his "annus mirabilis" of 1905 Einstein formulated a diffusion equation that (assuming Boltzmann's kinetic theory of heat) implies that the mean square of the displacement of the observed jittering particles is proportional by a constant factor (the diffusion coefficient) to the observation time. This theory of Einstein was experimentally confirmed in 1908 and indirectly established the existence of atoms as the cause of these seemingly random movements.

Now, surprisingly the Brownian motion as a seemingly random process can itself be seen as a fractal process (random walk). It can even be measured and assigned a fractal dimension. The Brownian motion movement patterns are from our modern point of view a fractal as is illustrated by the charts below (without a reference frame and scale one cannot tell how far apart the direction changes in the movement are):

Depending on the medium, density and pressure the Brownian motion path of an atomic particle can be more or less "jittery". The fractal dimension of the path generated by an atom therefore can vary substantially and range from a point like narrow cloud with a dimension just above 0 when an atom is stuck in a rigid crystal at one location where the atoms just jitter around a fixed position.

The motion path may be approaching fractal dimension 1 when an atom moves more freely or in a straight line in a near empty space. If there are more atoms around potentially frequently crossing each other the fractal path dimension can be any fraction bigger than 1 and may approach a dimension of 2 if the direction changes are frequent and strong as is illustrated in the lower, second picture above. And the fractal dimension of the path can get even close to a value of 3 when the medium is very dense and hot and the atom path approaches a 3-D volume filling curve (see below).

Even though the Brownian motion is or might be considered random for all practical purposes, the paths of the atoms may still show some detectable patterns. The particle may be pushed more frequently to certain areas of space than to others. In this case the movement may not be completely random but rather pseudo-random and may be caused or influenced by some non detected or unknown force(s). Since the randomness of the motion can be measured by the fractal dimension of its path any variance of it up or down may indicate gaps in our understanding of the underlying physical causes for the motion. In the end, fractals can therefore potentially be used to describe and detect basic causal processes ingrained in the structure of the universe.

Fractals in Biology and living Organisms

So far we have only considered fractals as part of the non living, un-organic nature or as mathematical and computational constructs. However, obviously many fractals exist also in life and living biological systems and organisms like trees, plants, flowers, animals and even within us humans. Some of these bio fractals are among the most beautiful examples of fractals one can think of as can be seen in the selected pictures below (a cauliflower variant called Romanesco, a farn, a plant leaf and the branches of a tree):

Fractals and Evolution

As a matter of fact, all multi cellular living organisms including us humans can be considered fractals because they are all generated by one of the most important processes in biology: the self-division and replication of cells (during myosis and mitosis) !

Our embryo-genesis (and that of all animals) shows that our body evolves from the fractal process of cell division. We all start out as just a single fertilised cell, then this cell self-divides into 2 identical copies, then these 2 copies divide into 4, then these into 8 and so on - up to a certain point when they mature and start to differentiate, but even then, the differentiated cells start their own fractal sub-multiplication processes. We are indeed living fractals !

We all would not be alive if it were not for the existence and action of these powerful fractal branching and self-similar reproduction processes. Starting as a single fertilised egg which then multiplies by self-division billions of times during our life spans thereafter to produce somewhere around 100 trillion (i.e. 100,000 billion !) self-similar cell copies in a mature human body. Every day, every hour, every minute and every second our trillion of cells produce self-similar copies and it is this process that keeps us alive as a biological living system. It's a very sensitive process that has to be kept in balance. We die whenever this process stops - and we also die when the process gets out of control (cancer).

In the initial stages of the cell division process after inception and during the initial gestation period the produced cells are usually identical copies of the original cells. Only in later, more mature stages of our life and our bodies these originally identical copies may vary slightly caused by genetic mutations due to the evolution process or environmental factors.

Fractals can also explain some theoretical problems for evolution theory. As a matter of fact, the process of evolution itself can be seen as a fractal constantly branching process represented in the so called "tree of life" (see below a strongly simplified but still heavily branching tree of life version).

In classical evolution theory adaptation to the environment and the change of organisms is achieved by self-reproduction in combination with small and relatively rare mutations of the genome. Too many mutations in the genetic code is usually fatal - the animal will usually not survive and hence not reproduce. Too few mutations will lead to a stagnant adaptation to the environment and will over time also reduce the chances of reproduction when the environment changes. Therefore, evolution depends on a small band of mutations in the genetic code for each generation to succeed.

But even when a perfect mutation rate is given, evolution theory has great difficulties to explain how new types of animal and organisms can suddenly pop up that have major new biological features or body parts and structures. It is also a well known fact, that even though evolution predicts small step by step changes and adaptations of organisms in the evolution process there are many gaps in the known fossil chains. Often new types of animals or new senses or body parts suddenly appear in nature without the missing intermittent variants and fossils linking them to prior fossils. This is a hard and unresolved problem for classical evolution theory. An interesting proposal to solve this problem has been made by Stuart Kaufman who postulated that evolution drives our genome to the "edge of chaos" (see chart below).

The edge of chaos is the area in the fitness space of biological creatures that allows a rapid or instantaneous transition between complete order and regular structure on one side and chaos on the other side in complex systems (with feedback). At this assumed edge of chaos sudden and major changes in the genome are possible even with small and rare mutations in affecting genes that control the expression of other genes.

Alternatively, fractals can also be used to explain this gap problem in evolution. As we have seen, even very slight variations (like rare mutations in the genetic code) can easily lead to vastly diverging results (new types of animals or completely new body features) in the iterative self-replicating process of the myosis and mitosis. Hence, the rare mutation of a single gene during the iterative self-replication process can produce a completely unexpected outcome and new biological functions, tissues, organs etc. explaining possible "jumps" in evolution.

In addition, we can also be considered fractals not only on the levels of our self-replicating cells, but also on the macro level of our human bodies as we produce similar offsprings (children). These offsprings usually are self-similar copies of ourselves and our mating partners with undeniable and easily detectable similarities in body structure, personality traits, capabilities, abilities and features.

In biology fractals can be understood as powerful data compression algorithms created as a side effect and consequence of evolution (it is much easier and way more likely for nature to use and encode certain basic fractal features and repetitive processes in our genetic code than for example a full mechanism of how to build a working and functioning body).

As is illustrated by the Mandelbrot Set and the 3-D Mandelbulbs, very complicated and detailed 3-D "bodies" and shapes can be produced by a very short and simple repetitive process which could be implemented using some chemical, autocatalytic biological feedback mechanism. Fractals can produce complex bodies like our human body (see below) using a very compact, short genetic code. This is probably one of the key reasons why fractals are so abundant in all areas of biology.

The development of the embryo during the gestation in the womb of the mother is a continuous fractal iterative process as well. First comes the self-divison of the original fertilised cell over and over. Then the embryonic tissue grows and morphs into a series of distinguishable shapes that are also self-similar in varying degrees combining the growth process and maturation process of the cells with the fractal replication process thereby re-generating the phylogenetic development sequence of our evolution every time an embryo is created.

This combination of the maturation of the cells with the mitotic self-replication of the cells that results in a constant morphing of the shape of the generated biological structure is characteristic for bio fractals in the living world. Fractals in the non living world can also change their shape and structure over time but the causes for these changes come usually from outside the fractal (wind, erosion, electrical currents, magnetic fields etc) and are not an integral part of the fractal process itself.

Another advantage of using such recursive, fractal genetic procedures and encoding self-similar structures in living organisms is that a recursive production of cells is much less prone to mutations in the genetic code. It is in general more compact and much more efficient than a lengthy linear non recursive code would be. The gene sequence needed for creating molecules and proteins that cause fractals and autocatalytic processes to emerge can be very short and dense and hence will usually be more efficient and stable. Therefore, evolution will probably give a higher chance of survival to proteins, enzymes and autocatalytic molecules that themselves produce further concise fractal protein generators and autocatalytic effects.

Fractals in our Bodies

Fractals occur abundantly in plants and animals and as we have seen above also everywhere in our bodies. Our human body relies actually heavily on the existence of internal fractals and is indeed packed with fractal organs and fractal tissues. Here are some obvious examples (lungs, kidneys, our artery network and capillaries):

Fractals in our Brains

The main and most complex fractal organ in our body is our brain. The brain is actually a fractal structure on may different layers and levels. I will argue below in the second part of this article, that even the overall function of our brains, our thinking processes, our language skills and even our consciousness are all based on complex, nested fractal processes used and generated by the brain.

A first simple visual inspection of our brain immediately reveals its fractal structure. The human brain does not have a smooth surface underneath the skull but rather shows an outer layer and a "rind" which is the evolutionary youngest part of our brain, the so called neocortex, which is responsible for most of our intelligence and conscious thinking.

This tissue is strangely folded and highly convoluted and covered with a pattern of self-similar structures that have developed during evolution to increase the overall surface area of the brain and the neocortex while keeping the volume small enough to still fit inside the skull. When unfolded and straightened out to a flat sheet the surface of the brain would cover about the same area and radius as a 30 cm diameter pizza and hence would be much too large for our skulls.

In the past (and even often still today), when studying the brain scientists often needed to physically dissect and cut the brain tissue (of diseased people) in very thin slices horizontally or vertically (or any other orientation) and then scan the so obtained many essentially 2-D slices with a computer scanner and re-model and re-construct the brain virtually as a 3-D image in the computer for further more detailed investigation (see below).

As a result, neuroscientists can use the various slices of brain tissue to also study the contours of the brain and study each slice like a Mandelbrot island and analyse the folding and convolution patterns and the form of the brain's outer layer given by the individual slices. This allows an analysis and estimate of the fractal dimension of individual brains to compare this to other brains or to averages of certain populations (for example compare the shape of the brain of younger and older populations). Such fractal dimension analysis is nowadays already frequently used to medically diagnose and analyse brain damages that cause physical changes of the brain structure as they occur for example in dementia or Alzheimer's disease (see below):

The fractal layers of the brain go far beyond just this outer layer of the brain. There are many more layers of fractals inside the brain. For example, to secure the blood flow to and from the brain the brain tissue needs a highly interwoven network of dense arteries and veins that supply blood and nutrition to the billions of nerve and glial cells in the brain. This network of arteries and veins has a highly complex nested fractal structure which fills the whole 3-D space of the brain (see below).

The vascular system inside the brain is connected to the vascular system of the face and the outside of the scull in an intricate mesh as shown in the picture below:

And the vascular system obviously does not stop with the head and the brain. It covers the whole body for the obvious reason that the blood supply is needed everywhere in the body to allow the distribution of blood, nutrition, hormones, oxygen etc to all tissues and organs in the body. Hence the whole body is covered with this highly complex fractal network underneath our skin which is shown in the schematic picture below:

A similarly but independent further complex network is our nervous system which is connecting our brain and the central nervous system and glial cells in our brains to the rest of our body. This network is needed to control our actions and movements and works on transmitting electro chemical signals between the nerve cells and our muscles and other tissue.

The nervous system also has an obvious fractal structure as well and looks like the branching roots of a plant or a tree with the brain as the organ on top to which the nervous system connects via its spinal cord to deliver and receive and send information from and to the body and senses (see the simplified schematic below).

Coming back to the brain itself, another fractal structure inside the brain is the so called white matter that makes up the massive internal "wiring" of the brain. The white matter consists essentially of all the myelinated axons of the nerves that connect the brain with the spine, connects the inner deeper sub-cortical regions of the brain with the outer layer of the cortex and also connects the many distributed functional regions of the neocortex itself with each other (see below):

Fractal Sub-Structures of the Brain

The brain and our bodies have developed over millions of years during our evolution. The brain therefore is not a clearly and optimally designed coherent single structure but rather a step-wise grown sub-optimal layered structure with the phylogenetic older parts of the brain at the deeper levels inside the brain originating on top of the spine and brain stem and the newer structures of the brain added on top as depicted below.

One evolutionary older part of our brain and often neglected and underestimated is the so called cerebellum. It is located at the lower back of the head and part of the hindbrain. It is much smaller and lighter than the cerebrum on top of it but in some animals the cerebellum can be as large or even larger than the cerebrum. Its main function is the motor and movement control of the body as well as coordination of movements and precision movements (a feature needed especially by athletes). Modern research however has found that the cerebellum is also involved in several important higher level cognitive functions like language processing, attention and control of emotions such as fear and pleasure.

Even though the cerebellum is quite small and only makes up around 10% of the brain's volume and weighs only around 130 grams it still contains a massive and staggering amount of nerve cells. The number of nerve cells in the whole brain is currently estimated to be around 86 billion cells. Of these, the cerebellum alone contains around 60 billion nerve cells, i.e. around 70% of the total number of nerve cells of the brain !

The surface of the cerebellum is covered with finally aligned parallel grooves in stark contrast to the roughly convoluted structure of the cortex. The surface is still tightly folded but not with large scale convolutions like the cortex but rather with a thin layer folded more like the flexible air pockets of an accordion. Most of the nerve cells in this layer are the highly fractal and computational powerful Purkinje cells (already studied by Cajal) and a massive amount of tiny nerve cells called granule cells.

Surprisingly, despite the massive amounts of neurons in the cerebellum, its role for us is quite limited. Actually, we would not even need the cerebellum to live and survive ! There are people born completely without a cerebellum and they can live quite normal lives and one would barely notice any difference to healthy people if we met such a person (this is owed to the fact that our brains have an important property called "plasticity" where some regions of the brain can take over functions of other regions if a brain region is missing or damaged).

The cerebellum has a clear fractal structure as can be seen from the outside and especially when looking at a slice of the cerebellum disclosing the clear branching internal fractal structure (see below).

Like the brain itself the cerebellum has its own white matter inside that connects the different parts of the cerebellum and also connects the cerebellum with the rest of the brain, the brain stem and even the cortex. The white matter also has a detailed fractal branching structure and its stained fractal "skeleton" dimension has been measured and estimated with different optical resolutions down to a scale of 0.25 mm with a resulting fractal dimension of about 2,57 (see below).

Brain Maps

The various functional cognitive regions and areas of the neocortex which are connected with each other and the deeper parts of the brain via the white matter wires can be represented and visualised by brain maps. These maps also show a fractal structure if the resolution is high enough. Constructing such detailed high resolution maps (in 2-D and more recently in full 3-D) of the brain has become a complex science in itself for which for example the Human Brain Project of the EU has spent hundreds of millions of Euro so far already ! The best such maps and brain models have a resolution in 3-D down to a few microns (i.e. about the 10th of the diameter of a single human hair).

Even though the neocortex has a surprisingly homogenous overall cellular structure and cytoarchitecture (densely packed 6-layered vertical columns of neurons), there are minute differences of the cell density, the layers and the cell type and cell body density and distribution and the resulting local thickness of the brain tissue. This results in identifiable areas of the neocortex that can be assigned measurable (via fMRI for example) cognitive functionalities (usually in a distributed manner).

Such areas have been detected, identified and localised in the brain since centuries (often by the autopsy of brains after accidents, strokes etc) and in fine detail already over 110 years ago by the famous neuroanatomist and psychologist Korbinian Brodman who's maps (see below) are still surprisingly accurate and are being used in neuroscience lecture books still today and named after him.

Brodman originally identified around 52 (see simplified map below) distinct functional areas of the cortex by studying the microscopic cytoarchitecture and thickness of the cortex regions. Today however scientists have improved Brodman's original maps by various methods (for example by using fMRI imaging and new high resolution microscopy). The resulting modern brain maps of cognitive areas of the neocortex use somewhere between 500 to 800 different functional areas and each area can be further subdivided into similar functional self-similar sub-regions establishing the fractal structure of such functional brain region maps.

The cortical maps of Brodman and other pioneers of neurscience where static maps. The reason was that they could only inspect the brain once it was dead and removed from the scull and inspected during an autopsy. Therefore the historic maps all showed the cortical functional and cellular areas like a world map shows countries with essentially fixed boarder areas that don't change.

However, neither the cytoarchitecture nor the functional activity areas of our brains are fixed or constant over time. The cytoarchitecture may change due to the plasticity of the brain that changes the micro structure of the brain tissue when we learn and have certain experiences. And the functional areas are usually not bound to only one fixed area but are distributed over many areas in the cortex (connected by white matter). Our brains works like a massively distributed computer system in the cloud. The functions of such a cloud based computing system can usually not be assigned to one single data centre but only to a whole network of servers often spread out over wide distances. The same is true for the functional areas of the cortex. No essential cognitive function can be assigned to only one brain area. A realistic map of the cortex is therefore more like a map of Europe drawn during WW II where the boarders and front lines of the countries constantly changed while the war was going on.

With the modern sophisticated new invasive and non invasive imaging techniques available brain maps are now no longer static but are dynamic maps that can change drastically over time (on a scale of seconds or less). The activities of an individual brain can therefore be visualised like an animated movie of constantly activated and de-activated brain areas even while the living brain performs certain cognitive tasks. We can now literally watch our brain while it is thinking - like a life performance.

The picture below shows a snapshot of such a modern, dynamic fMRI video map of the two hemispheres of the brain. The upper two hemispheres show the active brain regions while the brain is listening to a story read by some person whereas the lower part shows the same two hemispheres and the active areas while the brain is reading the same story. The maps show similarities but at the same time they also show clear and substantial differences in the activated areas of the cortex. Such maps therefore allow us to identify the relevance and importance of certain brain regions for the execution of certain cognitive tasks.

We can now even generate semantic maps of the neocortex where each coloured area represents a brain region of a living human brain that is active when a human is processing specific semantic information. It shows again the two hemispheres (right and left) of a person on the bottom in a 3-D rendering and above that the flattened 2-D maps of each 3-D hemisphere with the related corresponding semantic content colour coded and listed at the top of the picture. Such maps are used now to study the language processing regions in the brain and how we handle and use semantic verbal, acoustic and visual information (like when we process pictures or watch videos):

The abstract map below gives an additional very condensed and highly informative view of our brain and shows as well how much progress neuroscience has made since the days of Brodman. The chart shows all the relevant brain regions in a circular arrangement, their white matter connections as curved lines as well as further detailed information about the interconnections of the regions and the relevant cytoarchitectural parameters - all in one chart. This is a new brain map type that can be used in combination with the above described dynamic maps. This kind of map also re-confirms the fractal nature of the brain because certain brain areas can be removed or added (which models a zoom in or zoom out) without essentially changing the essence or structure of the maps.

The Fractal Structure of the Brain Cells

Our brains, especially the neocortex, is a gigantic web of billions of two classes of tiny cells: neurons and glial cells which are arranged in various structures, layers, tilings and columns.

The fractal structure of neurons and especially their sometimes massive dendritic branching receptive fields (the end of the neurons from where they receive signals from other neurons) as well as their axonal branching endings that allows them to send signals out to other neurons has been known and studied already since the end of the 19th century due to the masterful work of the famous Spanish researcher Ramon Y. Cajal and the Italian Camillo Golgi who developed the chemical staining methods to make the tiny cells visible under a microscope. They jointly won the Nobel Prize of 1906 for their groundbreaking work.

Cajal drew very detailed pictures of what he saw under the microscope by hand. These drawings (see below) are now famous and they have build the foundation of modern neuroscience:

Cajal also drew many detailed network pictures of neuronal assemblies already and noticed among many other details that the cell bodies of the neurons in the neocortex are arranged in dense horizontal layers, usually 6 layers well visible in the second drawing below, especially in the left column:

Using modern imaging techniques combined with computer vision algorithms, the current pictures of neurons have been dramatically improved with resolutions down to the micron meter scale combined with new methods using genetic modifications and auto bio luminescence staining. Here a recent example showing neurons with their many dendrides and thousands of synapses (the connection area for the axons of other neurons):

The layered structure of the cortex is now usually depicted in images like the one below which a focus more on the vertical columns. These have captured more attention from researchers now as they seem to be the real major functional units of the cortex:

The cortex therefore has two interwoven fractal sub-structures, a horizontal layered structure of dense cell bodies with massively branching dendritic connections to locally neighbouring neurons, and a vertical column self-similar fractal structure of neurons making up the major processing units and assemblies of the neocortex. These self-similar vertical columns are densely packed and are placed next to each other in the millions all over the neocortex as depicted below (using a different staining method than above to not show the horizontal layers but only the functional vertical fractal columns and how they bundle together in the cortex):

The next picture below shows a zoomed in high res version of just a single column and its clear fractal branching nature:

Whereas the horizontal layers of nerve cells are mostly connected over short distances and the dendritic branches of the neurons and their local horizontal axonal connections, the vertical columns are more often connected in groups that project their combined signals over longer distances via a hierarchical projection mechanism needed for the communication between different groups of columns via the white matter. The overall network connection structure of these columns is often organised in a so called "small world" architectures as characterised by and studied in mathematical graph theory.

According to certain cognitive theories this allows the cortex to simultaneously execute massive parallel as well as sequential, step wise hierarchical computations when needed (for example for visual classification tasks or conscious processes). Such concurrent sequential and parallel processing is needed for most higher level cognitive functions and (sub-) conscious decisions.

The hierarchical projections between groups of vertical columns is visualised in the graphic below which shows the hierarchical mapping of areas of bundled vertical columns onto other such areas in different regions of the cortex. This projection gives rise to another fractal process: the hierarchical mapping of columns to other groups of columns which then themselves may project to further groups of columns and so on - all in a self-similar repetitive process. Please note that the word "hierarchical" here does not mean a physical hierarchy of layers stacked on top of each other in a hierarchy but rather refers to a logical hierarchy. The cell groups mapped onto each other can be placed directly physically next to each other but belong to a different logical hierarchy level of signal processing.

The hierarchical mapping between brain areas is especially useful for pattern recognition and pattern classification purposes and the reason why such projection hierarchies are commonly used in Deep Learning artificial neural networks in AI.

The fractal Nature of the Glial Cells

So far we have only discussed the neuronal nerve cells and their networks, layers and columns in the brain. However, there is the vast second group of brain cells called Glial Cells that have historically mostly been neglected and have only just in the recent years received more attention from neuroscientists (by the way, the term "neuroscientist" says it all - they should actually be called "neuro-glial scientists").

This fact is quite astonishing as it is still not known how many glial cells there are in the brain compared to the number of neurons. The scientific estimates vary widely and range from a ratio of about 10:1 for the glial cells versus the neurons down to a ratio of just around 1:1 i.e. about an equal number of neurons and glial cells. But whatever the actual ratio turns out to be at the end, even the lowest estimate puts the glial at about par with the number of neurons and should suffice to take the glial cells serious.

The main reason for the historical neglect of the glial cells seems to have been that the glials were mostly thought of as just (structural) support cells (like the scaffolding of the brain that holds the nerve cells in place) or as care-taker and brain immune cells that protect the neurons and clean up after them and get rid of the debris and excretions of the nerve cells and to feed them and provide blood and nutrition for them. The picture below shows how the glial cells have historically mostly been visualised and seen: like a greyish kind of web or glue (hence the name "glia") that holds the more important nerve cells in place:

It turned out however, that the glial cells play a crucial role for our brain development and even for our cognition. Although they do not communicate with each other like the neurons via electric/chemical signals they still have their own communication system and communicate with each other via calcium ion signalling waves and they also communicate with the neurons via neuro transmitters (again, they should actually be called "neuro-glia transmitters"). They can also sense and react to the electrical activity of the neurons.

We now know that the glial cells have a massive and crucial impact on how neurons find their final location in the brain during the embryogenesis and how they connect with each other. They also modulate the communication between neurons at their synapses and thereby influence strongly the adaptation and learning processes in the brain. Glials also guide the axonal growth of the neurons and hence are responsible for the overall connection patterns of the neurons which in the end defines all our cognitive abilities. For these reasons glial cells are now taken much more serious by neuroscientsts (and recently AI researchers as well) as their role and function may be crucial for understanding how intelligence and consciousness in our brains emerge.

There are probably many hundreds of different types of glial cells as there are also many hundreds of different types of neurons known today. Most of them however are still unknown mostly due to their small scale and size and the lack of appropriate staining mechanisms. In recent years especially the so called astrocytes have caught the attention of the scientists. Among other important functions for the brain, these glial cells modulate and maintain the correct functioning of the synapses of the neurons and they prune the neurons from synapses that are not efficient and no longer needed (synaptic pruning).

The astrocytes are even more fractal in their structure than neurons because they have a more radial shape and many more branches. Neurons have usually a directed lengthy shape with the dendrites on one end and the axons on the other. Astrocytes have their branches reaching out all around the cell body. They can connect to hundreds of thousands of nerve cells and other glial cells at the same time, some even to several millions ! Compared to this, neurons are very simple cell types that on average "only" connect to a few thousand other cells and never to hundred thousands or even millions of other neurons. The pictures below show the beautiful and complex fractal branching structure (using different staining techniques) of astrocytes:

Without some specialised glial cells our bodies and brains could not function properly or not at all. The so called oligodendrocytes (see below) for example are responsible for the speed of transmission of the electrical action potentials inside the axons of the neurons. They wrap the axons in a sheet of fat (myelin) which isolates axons from electrical interference with the signals of other neurons and speeds up the electrical signal inside the axon by a factor of 10 or more (depending on the diameter of the nerve bundles). Without these glial cells our intellectual performance would fail and degrade and/or slow down dramatically. For example the currently untreatable brain disease multiple sclerosis is caused by a malfunctioning and lack of the oligodendrocytes in the brain.

As a matter of fact, many brain diseases and especially most forms of brain cancer are caused by glial cells, not by neurons. The glial cells can divide continuously and usually do so all over their lifetime and hence they can cause tumours when their growth gets out of control. Mature neurons cannot self-divide and self-replicate anymore. They lose this ability due to their specialisation as signalling neurons. Because of this inability once they are mature they usually cannot divide anymore and hence cannot cause cancers in a mature brain (only immature neurons can still self-divide and cause cancers but that usually only in young and immature brains like in young children). This is an example of a fractal cell type and process in our body with good but also potential bad consequences for our health.

Quantum Fractals in the Brain

We have seen and discussed so far a wide variety of fractals in our bodies and brains spanning many scales reaching from the brain as an organ itself to the smaller sub-structures of the brain (the cerebellum as an example) down to the microscopic and sub-microscopic cellular level of neurons and glial cells. We will now make a final "quantum leap" scale jump downwards and will zoom into the sub-cellular molecular and quantum levels of the inner structures of the nerve cells.

It was the Nobel Laureate and theoretical physicist and mathematician Sir Roger Penrose and his collaborator and anaesthesiologist Stuart Hameroff who first proposed a quantum level theory of intelligence and human consciousness since the 1990's.

Stuart Hameroff?is an emeritus professor for psychology and anaesthesiology and current director of the Center for Consciousness Studies at the University of Arizona proposed to Penrose to study the microtubules to explain the emergence of consciousness and the place where quantum effects may take place in the brain. As a practising anaesthesiologist he is also an expert on human consciousness as he took patients in and out of consciousness every day during his professional work.

Penrose and Hameroff argue that to explain intelligence and especially consciousness it is not enough to study nerve cells and their synaptic interactions but one would have to go one essential level deeper and consider and take into account certain quantum processes inside the nerve cells in our brains. With this approach they triggered the emergence of a whole new science called Quantum Biology.

Hameroff and Penrose were convinced - contrary to the popular opinion of the time and even still today - that consciousness does not emerge from the signalling between the neurons and synaptic plasticity in our brains, but that consciousness emerges from a deeper level layer inside the neuron cells. In his first book called?Ultimate Computing?published in 1987 (before he met Penrose) Hameroff already discussed the potential computational abilities of microtubules.

Hameroff argued that the microtubules could be the key information processing units in the brain rather than the superstructures of neurons with their electrical potential signalling. When he then later read Penrose's book "The Emperor's new Mind", he saw a possible combination of his approach with Penrose's quantum approach. This collaboration between Penrose and Hameroff led Penrose to publish his book:?Shadow of the Mind?in 1994. They have cooperated and refined their combined theory ever since until today.

Hameroff proposed and suggested to Penrose that the key elements within the neurons that may explain the emergence of intelligence and our consciousness are the so called microtubules (see pic below) inside the cells in our bodies and our nerve cells. They are part of the so called cytoskeleton of our cells because they make up a dynamic inner skeleton of the cells which allows them to have a stable shape, change their shape when needed and to grow and mature over time. The microtubules also allow the cells to move.

On the molecular level microtubules are tiny nano size self-assembling and self-replicating fractal cylinders.

Microtubules self assemble and disassemble within our cells from two tubulin molecules (alpha and beta tubulin). They attach to each other in alternating sequences to form cylindric 3-D tubules. The cylindric forms can change their length from a few nanometers to 50 micrometers in very short timeframes allowing cells to contract and expand and to be mobile. They grow and shrink by a fractal self assembly polymerisation process by adding or removing tubuline dipole pairs of peanut shaped molecules of alpha and beta tubulin (see pictures below) .

Besides these complex functions in the cells the microtubules also play a crucial role in the fractal self-replication of the cells and the self-division as they attach to the chromosomes inside the cell and pull them apart during the cell division to make sure each new offspring cell receives exactly one complete copy of the chromosomes of the original cell after the division.

On top of that the microtubules function also as a kind of transportation and data highway infrastructure inside the cells that is used by the nerve cells to move certain proteins around especially the vesicles of neurotransmitters along the inside of the axons towards the synaptic cleft.

The microtubules in the dendrites and the cell body are organised as a fractal network connected by so called microtubule associated proteins (MAPs).

In the axons of the neurons however they are usually aligned like highways in long, continuous, non fractured tubule chains.

According to Hameroff many proteins used in anaesthetic drugs bind to the tubulin proteins. These drugs then interfere with the dynamics (see picture above showing the build up and deconstruction processes of the tubules) and hence can strongly effect the cell functions and growth and even lead to apoptosis (the genetically programmed cell death). Several drugs can stabilise and de-stabilise the tubule dynamics and can be used also for cancer treatments. Microtubules are polymers and very sensitive to extra cellular environmental effects. They can also influence and control gene expressions within the cells.

Microtubules as fractal Quantum Computing Devices

Hameroff had suggested in his ultimate computing book that the Microtubules can be seen and understood as powerful self-similar 3-D nano scale computing devices. He showed how they can model and execute basic classical computational operations (logical gates) with tubulin molecules as the cells of a cellular automata model (see Part 2 of this article and the chapter about cellular automata as fractals; see also my post : "From the Game of Life to Consciousness").

Such a microtubule based cellular automata ("microtubule automata") would use and act upon the tubulin neighbours of the dimers on the cylindric 3-D structure of the microtubules by simple rules based on dipole coupling strengths.

Penrose and Hameroff have since extended and refined this idea and postulated and suggested the quantum superposition of the high frequent vibrations of the tubulin protein molecule as the basic quantum computing operations of the microtubules . They now argue that the microtubule qbits (the quantum version of "bits") are based on the oscillating tubulin dipole chains forming superposed resonance rings in the helical pathways of the lattice like surface structures of the microtubules (see below). This means that the superposed quantum states of the microtubules are not composed of the tubulin molecules or parts thereof but rather are the spiralling pathways that make up the computation processes and interaction between the tubulin dipole chains winding around the microtubule helical cylinders.

Orchestrated Objective Reduction (Orch OR) and Microtubules

With trillions of microtubules in our brains doing all their individual computations according to Penrose and Hameroff, it is hard to see how all these potential quantum processes going on in the microtubules can cause a stable and unified consciousness and experiences and can survive the quantum coherence necessary to have a real effect on the functioning of the neurons. Somehow the activities of the quantum processes in the microtubules have to be synchronised and "orchestrated" to build up a critical mass and effect and to build a unified experience of consciousness in our brains.

Hameroff and Penrose have argued that this synchronisation or orchestration of the quantum activities of the microtubules is linked to the microtubule associated protein (MAP) attachments sites on the microtubules. They "tune" the quantum oscillations in lattice-like sub-structures of the microtubules and thereby regulate the objective reduction or self collapse of the wave function in certain co-ordinated frequencies that can be measured.

Each time such a collapse occurs, a very short "conscious now" event happens. Sequences of such individual events create an experienced "stream of consciousness". Our actual experience of a continuous stream of consciousness is an illusion caused by an effect similar to when we watch a movie. A movie is not a continuous stream but a fast display of a sequence of still images just displayed at rapid speed of 25 frames per second. The same happens with our "conscious movie" that we call experience. The many conscious "now moments" just happen so often that our brains experience them as a continuous flow without noticing the non conscious moments in between.

Predictions and consequences of the Orch-OR model

The Orch-OR theory allowed Penrose and Hameroff to derive several very interesting consequences and predictions. Hameroff proposed that quantum particles can "tunnel" back and forth between neurons and other brain cells using so called gap junctions like astrocytes are know to do (direct connections of cells with no myelin barriers between - see chart below). They further theorised that microtubules are using cross gap junctions quantum entanglements of the microtubule vibrations to form dynamically larger groups of synchronised oscillating microtubules.

In this way a complex and powerful communication network between all brain cells can be established that does not even need or require the firing of neurons and their communication via synapses - it allows a kind of base level high bandwidth wireless communication between our brain cells. This consequence of their theory was confirmed experimentally in 1999.

Hameroff and Penrose further suggested that EEG signals (which we will discuss in more detail in the second part of this article below) are caused by the underlying interference and synchronisation of the quantum vibrations of the microtubules of the pyramidal neurons which project their axons to the top of our brains right underneath our scull where they can be measured as EEG signals from the outside. The microtubules vibrate in a wide range of higher frequencies and the interference patterns of these higher quantum vibration frequencies produce the lower frequency signals of the measurable EEG recordings.

Hameroff and Penrose therefore suggest that their model explains how consciousness is emerging and acting in the brain. According to their theory consciousness arises as an orchestrated self-similar and iterative resonance of self-replicating patterns of quantum entangled and superimposed vibrations of dipole chains of the microtubules. These vibrations "move around" in the brain as dynamic fields of synchronised and resonating microtubule networks combining the activities of neurons and glial cells.

Consciousness according to this theory stems from a synchronised resonance of quantum vibrations of many different scales and locations in the brain and is comparable to an orchestrated "music" generated by the microtubules in the brain rather than a computation.

I will discuss the influence of music and its processing in the brain and an alternative fractal adaptive resonance theory of our cognitive processes that does not require quantum mechanisms in Part 2 of this article.

E.Schoneburg

Berlin, July 2021