Response to any redefining of fractals

[Thunderwave]

I rented that and loved it.  I plan to take it a step further.

I think I understand the Mandelbrot, z = z^2+c.  C is a complex number a+bi.  It is all about the fact that a negative number times a negative number makes a positive number and when you add itself to it and square it, it bounces all over the place at times(somewhere between -2 and +2 real or imaginary) or it goes to infinity.  The color is generated on how long until it goes to infinity. if it doesn't go after so many times, it is in the Mandelbrot set and therefore is usually black.

Is that about right?

[Sockratease]

I rented that and loved it.  I plan to take it a step further.

I think I understand the Mandelbrot, z = z^2+c.  C is a complex number a+bi.  It is all about the fact that a negative number times a negative number makes a positive number and when you add itself to it and square it, it bounces all over the place at times(somewhere between -2 and +2 real or imaginary) or it goes to infinity.  The color is generated on how long until it goes to infinity. if it doesn't go after so many times, it is in the Mandelbrot set and therefore is usually black.

Is that about right?

Sounds convincing to me...

But I thought you were looking for a non-mathematical definition.  One that uses words like Self-Similar, Variations, Chaos, Order, Lissa, Repeating, Scale, Zoom, and even etc...

[Thunderwave]

Oh, yes, I do.

I just need to understand what some of the equations mean to me.

z=z^2+c means to me that Mandelbrot is playing with the rules of math we (humanity) created.  It shows a paradox in math.  When the system of math refers to itself it seems to sometimes go chaotic(crazy infinite loop) (GEB:Hofstadter).  Just like any of us may go crazy, the more time we question our own existence. WHY?  I wonder if it's because a system can't easily define its own system.  Thus, systems generating systems to define their system is in my opinion a fractal, which is math, science, religion, philosophy, etc...  Anything that uses itself to better understand itself creates a loop that may be infinite and paradoxical.  Now, yes, we have things outside of ourselves to investigate using ourselves to understand something outside of ourselves.  But!  We are still using ourselves to understand what is out there and then we reference it to ourselves with knowledge of ourselves using theories we generated.

Most of what I said here could be investigated further by reading Godel, Escher, Bach: Hofstadter.  Although, add the fact that it is coming through my strong opinions and my reading is perhaps unfortunately skewed by these opinions. (That's why I reread things to see if I have done this).

Also,  I do not mean to judge any of these systems.  I am only speaking on the amazement of chaos, fractals, infinite loops, and self-referencing that occurs within systems.

Yes I do plan to speak about them using those terms you mentioned:

But I thought you were looking for a non-mathematical definition.  One that uses words like Self-Similar, Variations, Chaos, Order, (Lissa:)), Repeating, Scale, Zoom, and even etc...

Ideas on the essay:
1. Self-similarity is like to daily repeating actions - the mundane (maybe)  Still working it out.  Broccoli is a common explanation.  I want something different.
2. Fractal dimensions on family span and relation of children who are uncles of children about the same age. (Shrinking, and multiple Copies) More ideas are coming up about this.
3. Chaos will be talked about in some form.  Not sure of much now that my head is hurting.
4. plus more to come.

I'll continue more later.  Some ideas are jotted down but not sure any of them are worth mentioning.

I don't want to steal any ideas here, unless you want to give them to me and I will regurgitate them mixed with all of my ideas. 

[Tglad]

Another take.. the mandelbrot inside is negative feedback, the outside is positive feedback, just like most systems the border between positive and negative feedback is rough/chaotic/fractal.
Its what makes earthquakes and volcanoes so hard to predict, and landslides, avalanches, cave-ins, stock market crashes. They all occur when negative feedback tips over into positive feedback.

[Sockratease]

I don't want to steal any ideas here, unless you want to give them to me and I will regurgitate them mixed with all of my ideas.

Feel free to use anything I said, if any of it is worth stealing.

Yes I do plan to speak about them using those terms you mentioned:

But I thought you were looking for a non-mathematical definition.  One that uses words like Self-Similar, Variations, Chaos, Order, (Lissa:)), Repeating, Scale, Zoom, and even etc...

Well of course I had to include Lissa in such a list!  I found her inside a 3D Fractal, after all...

[Thunderwave]

Another take.. the mandelbrot inside is negative feedback, the outside is positive feedback, just like most systems the border between positive and negative feedback is rough/chaotic/fractal.
Its what makes earthquakes and volcanoes so hard to predict, and landslides, avalanches, cave-ins, stock market crashes. They all occur when negative feedback tips over into positive feedback.

I like that idea.  I would have to research it enough to make sure I get it full and then I can make references.

Feel free to use anything I said, if any of it is worth stealing.

Thank you!

Well of course I had to include Lissa in such a list!  I found her inside a 3D Fractal, after all...

Oooooo.... this makes me wonder if I can find my one and only inside my fractals as I research?   

[Calcyman]

I think that the following definition encapsulates fractals in an unambiguous way:

Define a pathological point to be a point on a curve where, at arbitrarily small scales, it does not asymptotically approach a straight line. Examples include the vertices of polygons, or the centre of a logarithmic spiral.

A fractal curve is any curve that contains an infinite number of these pathological points in a finite volume.

[jehovajah]

Pathology implies abnormality. 

Although i would say that fractal points exhibit new and interesting behaviours to those we think we understand in Euclidean geometry, i would not go so far as to call them pathological.

I know very little is meant by it, but from the days when fractals were called mathematical monsters and horrors, i think we have moved on to appreciate they are perhaps the norm, and our cosy exact "rational" take on the world is just one model among many.

It is interesting how influential greek geometry still is, and a thing to note that Euclid and his school never actually defined a point, basing the axioms on line segments generalised to infinite lines, and the intersection of line segments. There is therefore a epistemological difficulty with the "euclidean" notion of point, and it may not even be a Euclidean apriori Axiom.

Therefore i am not disparaging of points no matter how non euclidean they are! All are welcome if they exist at all.

For me Fractal is a buzz word, a bit like digital. Benoit designed it as a buzz word, and each of us has to put a meaning to it from the basic examples of what Benoit was studying at the time. I like his later focus on the notion of roughness, as the notion of dimensionality has really dropped away. Levy says that the only defining characteristic seems to be self similarity at differing scales.
 I personally feel that the method of construction defines a fractal, and the method must be iterative. Thus a jigsaw puzzle requires an iterative construction technique, but the result may not look like a fractal! However if i focus on the boundary lines it looks like a jig saw puzzle, and thus is fractal, but can i find self similarity at scales? Well roughly speaking, yes. And there is the nub of the question: should this self similarity be exact or "roughly" similar"

Making it exact excludes an awful lot natural forms, and Benoit was dealing with natural and irregular forms, So it does not seem appropriate to be exact about anything in a fractal. Roughness and approximation seem more in keeping with the notion.

Argh! i can hear the perfectionist cry! And argh! back to them for ever thinking that perfection Exists! The best use of exactitude is to approximate reality, to give a reference frame to adjust our manipulations to reality. Now that is a pathological point!

[Thunderwave]

I think in my further studies and readings, I perceive that fractals tell us this concept...

There is an infinite amount of space between any "point" (whatever point means to you).  Quantum Physics shows us that paths to points are not predictable.
Well we see from our own mental concept of how to get from one point to the next by skipping an infinite amount of points in between.  Yet to maybe some "foreign' being we are acting unpredictable because they are measuring movement with a different mental ruler then we are.  Since there's an infinite amount of ways to get from A to B, then no one measurement can satisfy them all.