Fractals - the illusionary art of infinity

[Randomness]

I got a bit bored at work so I wrote this about the Mandelbrot fractal.

Any comments criticisms most welcome. ----

Fractals - the illusionary art of infinity.

How many numbers are there between zero and one? Are fractals infinite? Two intrinsically linked questions.

Definition: "curve or geometrical figure, each part of which has the same statistical character as the whole." (Oxford English Dictionary). Essentially fractals are self-similar on many levels.

There are many things in nature that fit the definition of a fractal such as ferns or snowflakes but we are going to focus on the Mandelbrot set, a mathematical fractal and one of the most well known.

Mandelbrot set uses a simple equation z = z2 + c to construct an image. This equation can then be applied to plot the course of each pixel using coordinates either to infinity (where it will leave the screen) or to 0 where it will remain trapped in the middle of the screen forever. The number of steps (iterations of the formula) each pixel takes to reach its destination denotes its colour and a complex pattern is generated. Each time you zoom in on the picture you are in effect creating new longer coordinates (adding decimal places) to feed into the equation. For example 0.7 becomes 0.75 when we zoom in further the numbers become increasingly long.

The amount of places (different images that would fill your screen) to visit inside a Mandelbrot fractal increases rapidly in line with the inverse square law. Using an example of a zoom consisting of 50% of the height and 50% of the width of your original image you can see how rapidly these numbers escalate (Z= number of zoom steps P=images available to view) Z 0=P 1, Z 1=P 4, Z 2=P 16, Z 3=P 256, Z 4=P 65,536, Z 5=P 4,294,967,296 and Z 6=P 1.844674407371e+19. Using a regular phone screen the whole set could potentially generate enough different images to pave every surface in the world after zooming in less than a hundred times. Zooming in 5 times generates enough different images that it would take 139.6 years to view them at a speed of one image per second.

Now this fractal some would consider as infinitely deep, the equation can be fed with increasingly long complex numbers (with theoretically an infinite number of decimal places) and still produce a image (as can any equation that can be plotted and allows a variable to be input) but in reality we are bound by our ability to create longer and longer numbers.

We cannot generate infinitely long numbers which would imply that a fractals depth remains governed by our ability to look. This gives rise to the illusion of perceiving infinity "there is more there we just can't see it". We are creating the fractal by generating numbers using the equation as a way to represent this visually. The images inside a fractal do not exist without our looking for them. Although there are an infinite amount of numbers that could potentially be constructed to feed the equation we just lack the ability to express them.

The Mandelbrot set could be used to demonstrate how observation can influence outcome, we are creating the fractal through the act of observation. Nothing is there until we begin to look.

This leads me to my question how many numbers are there between zero and one? An infinite amount remains the correct philosophical answer nothing more nothing less. To how many decimal places can you count however governs the practical answer. I would say that the Mandelbrot set can be no more infinite than ourselves. However it does demonstrate beautifully an example of how simple rules can govern a complex system across vast scales.

Infinity encompasses everything everywhere - but for us seems like a rainbow as the closer we get the further away it becomes. Pure concept which can never be fully expressed, never measured nor quantified as it has no boundaries from which to start.

 

[jbark]

Interesting ideas and well expressed.

The Mandelbrot set could be used to demonstrate how observation can influence outcome, we are creating the fractal through the act of observation. Nothing is there until we begin to look.

This leads me to my question how many numbers are there between zero and one? An infinite amount remains the correct philosophical answer nothing more nothing less. To how many decimal places can you count however governs the practical answer.

But is this not predicated on the idea that zero has an existence beyond theoretical? All math is a mapping of reality, but certain things must remain theoretical (as I understand them, and I am by no means a mathematician). If nothing is there before we look, this includes the concept of nothing itself, or Zero. Except that Zero cannot be actually observed, but rather expressed as merely a theory of "absence". And what of negative numbers? And let's not even open the box on i1, expressed as i^2 = -1 !!

The mandelbrot set is an equation and, from Wikepedia:

"tends towards infinity when a particular mathematical operation is iterated on it."

Implying of course an observer to iterate a mathematical operation on it. Step back one further step to the genesis of the equation and you have a hand, Benoit Mandelbrot's hand, that first scribed the equation that is not "naturally occurring", but rather an interesting "invention", if you will. So on several levels the mandelbrot set, and the concept of zero and the imaginary number line, are not real in any sense beyond the conceptual.

I cannot help think of Zeno's paradox, that was best expressed to me as analogy by my father when I was younger: An archer shoots an arrow toward a target, and we can say with great authority that in some elapsing of time the arrow will have travelled half the distance to the target. And in another lapse of time, the arrow will have covered half the remaining distance, and in yet another, half that distance. Following this logic, the arrow must ALWAYS follow this pattern, halving the distances progressively, and will thus NEVER hit the target. But as any archer knows (and any ill fated target, for that matter), the arrow does eventually strike. So the mathematics of the infinite seem at best flawed, if not outright false. (I happen to think that the solution to this is that there may actually be discrete units of time, meaning that to say the arrow travels only through space, neglecting time as a factor beyond a measurement of the time required to travel that distance, is to erroneously accord too much importance to space, and not enough to time: with a discrete unit, in other words, the arrow will never actually be caught in the conundrum of halving over and over, but rather at one point the halving will cease and the arrow will jump a discrete unit of time and hit the target. Have I lost ya? )

What am I getting at? Not quite sure... though to say that mathematics is a reality representing artificial map with a whole slew of fascinating theoretical holes that we build around to construct a cohesive system, might kind of sum it up for me (yes, dumb pun intended, of course!)

Meaning? That it is not surprising that many mathematical concepts are not there until we invent or observe them. THE MAP IS NOT THE TERRITORY, 'n all dat.

And well, not being a mathematician, I am fully expecting to be torn a new hole of my own by those more knowledgeable, a hole that with any luck will fractalize into me and extend to infinity upon close scrutiny and meticulous observation.

Cheers,

JBArk

 

[Randomness]

Zero is like an expression of nothingness just like the concept of cold (lack of energy) as you say it cannot exist outside of our conception but nevertheless it has become essential for us to be able to measure and define the world around us. Kind of like a constant value from which to start. There could be an argument that zero is a state of being or state of rest. Think of binary with all those 0s and 1s. Zero exists to contrast one as without the contrast everything looses it's value and becomes meaningless.

The arrow concept is quite an interesting one. It's not as though it does not hit rather that using that formula of you will never mathematically get to the endpoint. In practicality you would probably arrive at the plank length and then the measurement would have to increase in this fashion at this resolution of scale.

You could in theory plot a sine wave to an infinite resolution and this would behave in a similar fashion to a fractal. It would be incredibly boring to look at though.

I like theoretical mathematics as you have the definable world and then you have this world of esoteric possibilities and paradoxes.

Quantum states are nearly as bizarre. If you think that all the protons and neutrons that make up the atoms we consist of. These are theorised to be in a quantum state neither here nor there until observed and possibly occupying the same space as one another within the atom. Stranger than the arrow paradox? I think so.

Really makes you think about how we define things and as I said it shows you how simple rules can govern complex systems.

 

[HumbleTraveler]

Something else too Ive been thinking about recently. Scale. For us, what we know, we've got the universe itself that contains all we know (and some we dont), then we have the entire run of stars from massive hypergiants down to red dwarves and planets, moons, then on earth we've got gigantic oceans and land masses, trees from huge to small, giant machines, small machines, and then eventually, there is us. Then inside of us, we have blood and tissues made up of endless numbers of molecules containing the fundamental building blocks of matter that can be found everywhere and make up everything in the universe.

Since there is so much scale, scale itself may/must be infinite. Our universe may be contained by something even larger, which is also part of a scaled system as well where there are other things inside it. And Im not really even talking parallel universe theory or the like, Im just talking scale alone. Where does 'scale' itself end and/or begin? Our universe seems endless to us, but to something else out there in the void, the universe itself might be to something what an small insect is to us!

Also, while we're on the topic of some incredibly interesting ideas, heres something Ive always found kind of fascinating to think about.

You will never be able to see your own face. We can see our reflection, or a photograph of ourselves. But we'll never see our whole face with our own eyes even though its right there.

 

[Randomness]

Since there is so much scale, scale itself may/must be infinite. Our universe may be contained by something even larger, which is also part of a scaled system as well where there are other things inside it. And Im not really even talking parallel universe theory or the like, Im just talking scale alone. Where does 'scale' itself end and/or begin? Our universe seems endless to us, but to something else out there in the void, the universe itself might be to something what an small insect is to us!

I like this idea you could almost imaging our solar system turning out to be the nucleus for some massively scaled atom. Bit like a giant fractal.

The reflection idea is quit interesting other than the fact your looking at glass rather than yourself also you are in effect seeing your past self. When you talk to someone they exist in your past light cone and the further away they become the further into your past they slip. The observer appears as the centre of the universe with all light and energy focused either to them or away to infinity (as with a Mandelbrot). We use information from the past to pass judgement on future events - think of catching a ball you see it where it was when the light reflected from its surface not its current location and you use this information to guess its future and hopefully catch it.

Did you know you can see 60% of a neutrino star as it's gravity bends the light emitted allowing you to see the side.

My five year old son got the concept that all the colour we see is just a reflection of what is not being absorbed and that colour comes from light and not the object being perceived. Sadly my other half (his mother) just got confused.

 

[Creo]

Some thoughts.

The Mandelbrot set has an objective existence independent of our ability to compute it, in the same way that the decimal expansion of pi exists independently of our ability to compute it.

We can count the fractions between 0 and 1 and show that there are as many such fractions as there are whole numbers:

1/2 <-> 1
1/3 <-> 2
2/3 <-> 3
1/4 <-> 4
3/4 <-> 5
1/5 <-> 6
2/5 <-> 7
3/5 <-> 8
4/5 <-> 9
1/6 <-> 10
5/6 <-> 11
1/7 <-> 12
2/7 <-> 13
3/7 <-> 14
4/7 <-> 15
5/7 <-> 16
6/7 <-> 17
...

Zeno's paradox can be resolved by adding up the infinite number of lengths (or time periods):

1/2 + 1/4 + 1/8 +1/16 + 1/32 +1/64 + 1/128+ 1/256 + 1/512 + 1/1024 + 1/2048 + 1/4096 + ... = 1

 

[jbark]

Zeno's paradox can be resolved by adding up the infinite number of lengths (or time periods):

1/2 + 1/4 + 1/8 +1/16 + 1/32 +1/64 + 1/128+ 1/256 + 1/512 + 1/1024 + 1/2048 + 1/4096 + ... = 1

But that's a theoretical solution, no? A bit of a workaround? Or rounding, "tout court"? Meaning, "add a bunch of everything it takes to cover a distance and, I guess, you will, necessarily, cover the distance"?

The problem is theoretical, granted, but the halving of each progressive distance to the target is easy enough to conceptualize and difficult to resolve. Except through workaround mathematics that says, well, imagine all the conceivable (even infinite) distances that must be travelled by the arrow and add 'em all up. BIngo: arrow meet target.

It works, but is weak in my opinion. In the same ballpark as saying that Pi must be finite because an infinite number has never actually been expressed to infinity. (ok, maybe not as weak as that, but weak nevertheless )

JBArk.kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk...

 

[Randomness]

To add all the fractions up to and arrive at 1 would imply that you have already reached the endpoint and resolved the paradox. In which case you would not need to bother.

I would say mathematically you can choose how to interpret the event.

One interpretation uses infinity to prevent the arrow striking whilst in a standard interpretation it hits. Neither interpretation is theoretically wrong it's just one can never be fully resolved unless you stop dividing and just add on the remaining distance. To me it seems more of an example of how without a limit on scale and resolution nothing can be measured.

Think of the length of a piece of string. If we measure in cm the length is easy to determine but in mm it becomes harder nm becomes harder still and if we want to define the length of the string at a atomic of quantum level it becomes nearly impossible.

The solution proposed is akin to infinity minus infinity which can be either infinity or zero depending if you are using the same instance of infinity as a starting reference.

My solution would be to stop at the first iteration and use that information to predict the result. 1/2 + 1/2 = 1 Job done. Stay away from the infinite if you would like to see the endpoint. There is also the argument that if you know the value of the fractions then by definition you must have the value of the whole in which case there is no need to work anything out. Paradox becomes an attempt at trying to prevent the inevitable by slowing time into an infinite moment. Unfortunately the cost would outweigh the benefit as to stop the arrow you must sacrifice the future by spending an eternity locked into the equation.

Whichever way you look at it if you are the target the arrow has taken its toll. Your future lies in the arrows path with no way to escape the consequences of it being fired. Either take the impact or reside in an infinitesimally small moment before your demise. Both options have the same limiting effect with regards to any future events you may have wished to partake in.

Not so much a paradox as a choice between being killed or frozen in time. These could be in effect seen as the same thing neither of them involve you having the opportunity to experience future events, make choices or interact with anything.

 

[HumbleTraveler]

I like this idea you could almost imaging our solar system turning out to be the nucleus for some massively scaled atom. Bit like a giant fractal.


My five year old son got the concept that all the colour we see is just a reflection of what is not being absorbed and that colour comes from light and not the object being perceived. Sadly my other half (his mother) just got confused.

Yes! The closest thing Ive found to what I said, kind of, was the concept of eternal inflation. But it doesnt exactly match it, still a ways off.

Just say to your wife, "ohhh man, its a good thing youre pretty! "