Dimensions, Fractals and Consciousness
Dimensions, Fractals and Consciousness
The following is pure speculation and I make no claim that my extrapolation of the basic facts is anything other than fiction. That said, in a world where magic exists might there not be some underlying causal phenomena that might explain the mechanics of its operation. In the paragraphs below I will introduce some concepts. The reader will find much deeper and better expositions of the basics online. If you are interested just search the topic and you will find no end of sites and pages.
Fractals
First consider fractals; there are many examples of fractal objects and there is much written about their visual impact on our senses. However the term fractal was coined as a mathematical concept to explain some peculiar properties of certain objects. One concept that appears again and again in fractals is recursion which is the technique of applying the same transformation or rule again and again to some object.
The original concept, which gave rise to the subject now known as fractals, was to create a mathematical description of roughness. It was found that certain rough objects can appear to have mathematical similarities at different scales e.g. the coastline of Britain can be mathematically evaluated by measurements on maps at different scales to create a single measure of roughness.
Other mathematical patterns that show similarity at different scales can be generated by the solutions to recursive equations such as Z→ Z^2 +C (Mandelbrot). The classic example of this is that the Mandelbrot set contains within itself copies of the initial set. However such sets, although pretty to look at are not very easy to understand.
Understandable fractals
Simpler geometric objects generated by recursion allow us to gain a more intuitive insight. A example of this is the Koch Snowflake shown in the diagram below. This is a fractal pattern based on the equilateral triangle. If you take two triangles and place them on top of each other with one rotated sixty degrees relative to the other then the resultant outline is a six pointed star. Now this star can be ‘unwrapped’ into three sections each rotated sixty degrees. Each segment looks like a straight line with a triangular ‘bump’ in its centre (See diagram). Now if you take the four lines forming this segment and replace them with scaled copies of the segment you get a pattern that starts to resemble a snowflake. You can carry on replacing segments with smaller copies of themselves ad infinitum. The recursive rule is to replace the four parts of the segment with four smaller copies of the segment scaled by a factor of one third. At each iteration there is a consistent ratio between the new segments created and the old segments they replace. (See diagram). Now it is easy to see how this pattern forms the snowflake structure. Hold this thought whilst we digress…
Simpler geometric objects generated by recursion allow us to gain a more intuitive insight. A example of this is the Koch Snowflake shown in the diagram below. This is a fractal pattern based on the equilateral triangle. If you take two triangles and place them on top of each other with one rotated sixty degrees relative to the other then the resultant outline is a six pointed star. Now this star can be ‘unwrapped’ into three sections each rotated sixty degrees. Each segment looks like a straight line with a triangular ‘bump’ in its centre (See diagram). Now if you take the four lines forming this segment and replace them with scaled copies of the segment you get a pattern that starts to resemble a snowflake. You can carry on replacing segments with smaller copies of themselves ad infinitum. The recursive rule is to replace the four parts of the segment with four smaller copies of the segment scaled by a factor of one third. At each iteration there is a consistent ratio between the new segments created and the old segments they replace. (See diagram). Now it is easy to see how this pattern forms the snowflake structure. Hold this thought whilst we digress…
Dimensions
There are three spatial dimensions; Length Breadth, Height… X Y Z… Up/down Left/right Forwards/Backwards … whatever you want to call them. They define the space in which we live. There is also another dimension, known as Time although this one is somewhat peculiar as it doesn’t quite fit in with the others*. So I can argue that I live in a three dimensional universe. I know this because I experience it. However, the only way that it can be defined mathematically is by choosing a frame of reference. There is no physical point in the universe that is defined as zero just as no particular direction is the ‘universal up’. We can however agree that, for the spatial dimensions, some things are evident. Examine the diagrams and think... at first the conclusions seem self evident but they define something. That something is called dimension.
There are three spatial dimensions; Length Breadth, Height… X Y Z… Up/down Left/right Forwards/Backwards … whatever you want to call them. They define the space in which we live. There is also another dimension, known as Time although this one is somewhat peculiar as it doesn’t quite fit in with the others*. So I can argue that I live in a three dimensional universe. I know this because I experience it. However, the only way that it can be defined mathematically is by choosing a frame of reference. There is no physical point in the universe that is defined as zero just as no particular direction is the ‘universal up’. We can however agree that, for the spatial dimensions, some things are evident. Examine the diagrams and think... at first the conclusions seem self evident but they define something. That something is called dimension.
One dimensional universe
If we only have a single spatial dimension we have a line. Take any line in this one dimensional universe and scale it down by two. We now need two of the smaller lines to re-create the original.
Two dimensional Universe
Now, let’s take two dimensions and represent a simple object like a square. ( it could be argued that a circle or triangle are simpler but the illustration is easier to visualise with a square.). Do the scaling down by a factor of two as before. We now need four of the smaller squares to make up the original. In simplistic terms this says that squares are different to lines.
Three dimensional Universe
The same goes for three dimensions. Take a cube (easiest to visualise) and scale it down by two. We now need eight of the smaller cubes to make one of the larger cube. So; Cubes are different to squares and lines.
Where N is the number, S the scaling factor, and D the dimension. Since raising things to powers is awkward mathematicians like to simplify this by taking logarithms and it becomes log(N)= -D times log(S) and then we can separate out the dimension to give D = log(N)/ log(S) (the minus sign disappears because we are scaling down so scaling by 1/3 is the same as scaling up by -log(3))
So, going back to the lines, squares and cubes
If we only have a single spatial dimension we have a line. Take any line in this one dimensional universe and scale it down by two. We now need two of the smaller lines to re-create the original.
Two dimensional Universe
Now, let’s take two dimensions and represent a simple object like a square. ( it could be argued that a circle or triangle are simpler but the illustration is easier to visualise with a square.). Do the scaling down by a factor of two as before. We now need four of the smaller squares to make up the original. In simplistic terms this says that squares are different to lines.
Three dimensional Universe
The same goes for three dimensions. Take a cube (easiest to visualise) and scale it down by two. We now need eight of the smaller cubes to make one of the larger cube. So; Cubes are different to squares and lines.
We can express this as a formula
N is proportional to S to the power of -D
Where N is the number, S the scaling factor, and D the dimension. Since raising things to powers is awkward mathematicians like to simplify this by taking logarithms and it becomes log(N)= -D times log(S) and then we can separate out the dimension to give D = log(N)/ log(S) (the minus sign disappears because we are scaling down so scaling by 1/3 is the same as scaling up by -log(3))
So, going back to the lines, squares and cubes
for the line (one dimension) we have log(2)/log(2) = 1
for the square (two dimensions) we have log(4)/log(2) =2
and for the cube (three dimensions) we have (log(8)/log(2) = 3
Now, if you use this formula to calculate the dimension of some shapes (an example is the Koch snowflake that we talked about above) D turns out to be something other than a whole number. This is then known as a fractal dimension; so for the Koch snowflake it has a dimension of
And now another digression.
log(4)/log(3) = 1.2619.
However the snowflake itself is clearly a two dimensional object. Mathematically this is resolved by stating that apparent physical dimensions aren’t the same as fractal dimensions.And now another digression.
Organic fractals.
There are numerous objects in the natural world that display the property of seeming similarity at different scales. Trees, plants and lightning are often quoted as examples but there are three that are more relevant to this speculation.
The human circulatory system; Major blood vessels, minor blood vessels and capillaries form a fractal set; this is how the human circulatory system can have an effective length of sixty thousand miles but which can still be contained in a body only six feet tall.
The human respititory system; The lungs in particular are fractal in nature, again nature has evolved the ability to create an organ with an area of 80 square metres and contain it within a volume of approximately three litres.
The Human nervous system and the brain; Scanned images of the human nervous systems show a similar branching structure to the circulatory system and the respiratory system. The structure of the cortex which contains a hundred billion neurons and one hundred trillion connections is all contained within an active volume of seventy cubic centimetres ( about the same as a double espresso).
Therefore, many aspects of our physical bodies are fractals. Might this also extend to the way that nature has evolved the higher functions of our brain.
There are numerous objects in the natural world that display the property of seeming similarity at different scales. Trees, plants and lightning are often quoted as examples but there are three that are more relevant to this speculation.
The human circulatory system; Major blood vessels, minor blood vessels and capillaries form a fractal set; this is how the human circulatory system can have an effective length of sixty thousand miles but which can still be contained in a body only six feet tall.
The human respititory system; The lungs in particular are fractal in nature, again nature has evolved the ability to create an organ with an area of 80 square metres and contain it within a volume of approximately three litres.
The Human nervous system and the brain; Scanned images of the human nervous systems show a similar branching structure to the circulatory system and the respiratory system. The structure of the cortex which contains a hundred billion neurons and one hundred trillion connections is all contained within an active volume of seventy cubic centimetres ( about the same as a double espresso).
Therefore, many aspects of our physical bodies are fractals. Might this also extend to the way that nature has evolved the higher functions of our brain.
Emergent properties
It is often assumed that we can understand things by breaking systems down into smaller and smaller divisions. However, in practice this reduction also necessitates a loss of the higher level features that exist in the greater system. Consider an ocean; it is composed of water. If you take a glass of it then you still have water but you no longer have waves, sounds, storms or even just the magnificent sight and sense of space that an ocean creates. So we have lost something. Now take the glass of water and take a small collection of molecules known as a micellar group. This is still water but it has lost certain properties; wetness, surface tension, flow. Now break the micellar group down into component water molecules H2O but we have lost the subtle crystalline structure that the micellar groups have. Examining the H2O molecule tells us nothing about the beauty of an ocean sunset. The concept of emergent properties is just that. It basically states that, although they are composed of the same building blocks, there are properties that only exist in larger systems.
It is often assumed that we can understand things by breaking systems down into smaller and smaller divisions. However, in practice this reduction also necessitates a loss of the higher level features that exist in the greater system. Consider an ocean; it is composed of water. If you take a glass of it then you still have water but you no longer have waves, sounds, storms or even just the magnificent sight and sense of space that an ocean creates. So we have lost something. Now take the glass of water and take a small collection of molecules known as a micellar group. This is still water but it has lost certain properties; wetness, surface tension, flow. Now break the micellar group down into component water molecules H2O but we have lost the subtle crystalline structure that the micellar groups have. Examining the H2O molecule tells us nothing about the beauty of an ocean sunset. The concept of emergent properties is just that. It basically states that, although they are composed of the same building blocks, there are properties that only exist in larger systems.
Consider consciousness, many believe that it is an emergent property of a basic component; information. Take a raw fact; a number. In itself it is pretty useless, it is only when it is joined by other numbers that its properties grow; there are relationships between numbers that create the next level of complexity; mathematics. This inter-connectivity of facts is what we can use to make computers that can perform tasks like making a web page. By allowing numbers to connect together in other ways computers can display the ability to learn and even infer. This is the next level of complexity which is called learning. However we do know that there is a thing above this level called consciousness and it appears to be an emergent property that is at the next level of information processing after learning.
So what is the difference between living systems and computers? Perhaps it is the fractal nature of the ability to process information that is given to us ( and other animals) by the fractal nature of our brain. Hence we can ask the question; Is consciousness an emergent phenomena associated with the fractal organisation of information?
Thus we have fractals as a measure of dimensions in space, fractals as physical manifestations of the way nature organises things to create complexity and life and fractals as creating emergent properties and maybe even consciousness.
Magic is Fractal?
So much for facts… Now a little more speculation.
In a world where magic works might not it be that magic exists in a fractal dimension. Let us suppose that this fractal dimension is tangible and it can interact with the normal three dimensions and the energies they encompass. Thus magic can affect things and certain things can affect magic.
This explains why certain humans, whose internal fractal geometries are in tune with the fractal dimension where magic exists may be able to interact with it and, by controlling the magical energy, interact with the material world. Most humans, whilst still being fractal cannot interact in this particular dimension. They are not ‘lesser’ they are ‘other’. So magic is controlled by the fractal mind.
So much for facts… Now a little more speculation.
In a world where magic works might not it be that magic exists in a fractal dimension. Let us suppose that this fractal dimension is tangible and it can interact with the normal three dimensions and the energies they encompass. Thus magic can affect things and certain things can affect magic.
This explains why certain humans, whose internal fractal geometries are in tune with the fractal dimension where magic exists may be able to interact with it and, by controlling the magical energy, interact with the material world. Most humans, whilst still being fractal cannot interact in this particular dimension. They are not ‘lesser’ they are ‘other’. So magic is controlled by the fractal mind.
Now it is a property of some objects to display resonance. Energy from one object can be transferred to another object... this is most efficient in systems that have similar geometries. Could one fractal mind interact with another? Could a mind which can control magical energy influence any fractal, including the minds of others both animal and human? Thus the natural fractal control could explain the mental abilities of some wizards.
Fractals also exhibit in crystalline forms, indeed there are some relatives of the Koch snowflake that display a remarkable resemblance to the basic crystalline structures in crystals. Hence there may be some crystals that can interact with both the fractal minds of wizards and with the intrinsic fractal nature of magical energy.