#An incomplete rant about fractals, systems 

##post-writing preface
What you are about to read is a burst of inspired writing. Once in a blue moon i get motivated enough to sit down and write in a stream of thought kind of way. I am unsure wether or not i should develop this into a full article, but in the meantime instead of rotting away in the hard drive i figure maybe someone will enjoy what ive written down here. Consider this a brain-storm type free form. 

The idea of fractals is a bit nebulous and hard to get across to the normal layperson. My goal is to explain the idea in a more understandable way.

The first thing to understand is that everything that exist, exist because of processes and systems. The idea of systems and processes are themselves abstract, so let me demonstrate with the simplest and most human intuitive system i can think of, walking. Yes, literally putting one foot in front of the other.  We can think of walking as a system of motion, which requires literal steps. In order to get from somewhere to anywhere else, you must perform a iterative process which must be done in a proper sequence. Your legs generally move in the same way, one step is quite like another step after all. However there is some slight variance in the exact motion of each step. There are also ways your legs dont move, constraints imposed by pesky ligaments and joints.

As things move forward in time, all things tend to change and evolve in all sorts of ways.


Another thing to think to really mull over and try to undertand is the nature of relationships. The relationships between ideas and things. The relationship between relationships.  Between what is possible and what is provably not. Even the relationship between things which can be logically and scientifically provable, and things which will forever remain a mystery to the human experience outside of our finite ability to understand. 

	Physically, all systems in nature are built on the relationships between its parts which make up the system. And those parts themselves are governed by another set of different systems one level lower. A person can be thought of as a group of cells which obey a complex cellular automata. Those cells can be thought of as organisms themselves made up of smaller organelles that obey the laws of biology. Those cells are made up of proteins and chemicals which obey the systems of chemistry. Those chemicals and protiens can be thought of as linkings of atoms which obey the systems of quantum mechanical  motion. Those atoms themselves can be thought of as collections of subatomic particles. Those subatomic particles can be thought of as different collections of energy waveforms and fundimental vibrational frequencies in spacetime. Even empty space has stuff going on in it with virtual particles constantly creating and destroying themselves causing slight energy fluxuations even at absolute zero kelvin, and those v
irtual particles too obey laws of conservation and motion. These systems seem unrelated, but in actuality what they imply is that all systems are built upon other systems which nestle and link into eachother, weaving the next level of complexity and a new set of things to create systems. Astronomy tells us that the early stars only contained simple atoms of hydrogen and helium. Just with those two building blocks, the universe found a way to create a more complicated object, a star. That star allowed the creation of more complex and complicated atoms with the help of nuclear fission. Those particles allowed for more complex interactions to take place, giving rise to complex chemistry. Smaller systems creating bigger systems. But to what end? Why and more importantly, how?

	Abstractly, Mathematical ideas such as equations and functions are built upon the relationship (or signage) between the numbers that express them. numbers are like atoms, and equations/functions are like their own unique objects. These ideas are not unrelated, because both follow the same logic, and thus share a deep relationship with each other.

The core of geometry is realizing abstract algebraeic identities can be expressed as shapes, and vice versa. There is a fundimental mutual relationship between quantity and space, even if not obvious. We are visually driven animals, a very large part of our awareness is sight, followed by hearing, smell, and touch. Its easy to reason about shapes because we deal with them every day. So having a way to corrilate algebraic statements to tangable shapes and graphs is invaluable.
But seeing pretty pictures isnt quite enough, you have to understand what statement is being made for the geometric relationships to make sense and be interpretable.

Did WE create those connections between abstractions and shapes, formulas and trajectories, or was that connection always there waiting to be discovered? Do numbers exist in nature or are they creations of the human mind to be used as abstract tools of counting and analysis? If the universe does use numbers (which it does), what does that imply about the 'realness' of abstractions? We like to think the only things that are real are what we can see or detect with instruments, but what if our thoughts exist just as concretely to the universe as a stone. and have as much influence as gravity or heat on how physical systems evolve over time. How do abstractions influence reality, and are some of them more 'real' than others?
If you think about it, the human mind is a construct of the universe, which follows a certain logic in order to maintain a semi-stable reality. Any human reasoning and sense comes from this universal logic. If we concede that we are part of the universes system as one of the ways to express itself, then science and mathematics is the universe trying to understand  its own mechanics.

Serious mathematicians and scientist have an issue with the why? question, because sometimes its more of a philosophical "why do we exist" type question then a useful experemental one. Fortunately, im not a serious mathematician or scientist, just a casual learning enthusiast. While i strongly enjoy science and math as tools for learning about reality, i also see their flaws with the constraints and limits to thought that instututionalized academic thought can bring. By focusing in so heavily on the physical and logical parts of reality, they deny the other half of our existance which cannot be proven or verified. What it means to feel love or hate or anything, what emotions even are, is there a purpose, and what is the nature of the spirit. Because you cant examine happiness under the microscope or smash the soul into a billion particles to be scrutinized they are worthless concepts. Unverifiable.   

Systems and groups are similar ideas. a 'group' is any set of things which have a commonality. whats the connection between one pig, one cat, one pebble, one star, and one atom? One. Singular, they are alone. In a way, quantity is as physical a property as mass. But this physical property is also its own  abstract identity. Oneness, one, the first number on the number line. The state of on, the implication of an existance. One. Its as philosophical as it is concretely logical.

From one, you can reason in two directions:

	Is 'nothing' a property? i.e Does 'nothing' exist? 
	and
	What if you add another thing, or have multiple of them?
Zero is perhaps the most abstract natural number, and probably the longest to be adopted historically across all cultures. Simply put, what does it mean to represent nothing? how can nothing be a number?
 The identity of 2 is really just the group of two ones, 3 is the group of 3 ones, and all natural numbers are really just the amount of ones added up to that point. Yet, we think of numbers not as multiple sets of 1, but as their own identites. This implies that the group of multiple things, is itself also a new thing. This reminds me of how in biology living organisms can be made up of many kinds of smaller organisms which themselves are made up of smaller things. multiple atoms create a protien, multiple protiens make a cell, multiple cells make a person. groups creating groups creating groups. As above so below.