Sunday, October 23, 2011

Fractals, Magic, Math, Beauty and You

I always love it when something propels itself into your sphere of consciousness by proclaiming the words "...and you" and that for whatever reason it assumes it has some relevance in your life. For example, "Fashion And You" or "Home and You" or "Your Child and You" or "Cariboo and You" are some of the examples that came up in a google search. Anyway...

[It is at this point that Ann suggests that I should tell everyone to smoke a big fat joint. So if you have one light it up now]
We could begin by asking ourselves; "Is there magic in our world?" When I speak of magic, I mean actual magic, not illusion. Well, is there math? The short answer to that is yes, pretty much. Lets begin with Fractals.


The Magical Dr. Bunsen Honeydew, not a fractal.
(Not pictured: Beaker, my personal fave and the star of my desktop background entitled; "Beaker And You")


The fractal Benoit Mandelbrot
Father of fractal geometry and Bunsen Honeydew look alike
(Really, who hasn't been compared to Honeydew? That would make a good post to phone in one day)


This is a Classic Mandelbrot Fractal. Isn't it pretty? No! It's geometry! Get a grip.


Mandelbrot Double Spiral. OK, maybe a little pretty.

The Brian Mayndalbrot Fractal. Damn it, just when I thought it was getting pretty! 
~

Hands up who knew this post was heading in the direction of Queen's awesome guitarist? Shoot, I didn't even see that coming! Dr. Brian May, astrophysicist, would probably be able to explain the math behind fractals and create a killer guitar riff to go with it. An author at The Spine puts forth the theory that Brian May himself is a fractal. I might get back to that...but probably not.

Fractals are interesting in that they are formed from a pattern that continues to repeat itself in smaller and smaller ratios. What does this mean for you? I have no idea because unfortunately for me, this involves math...

... which is not one of my better subjects. Try to hide your shock. 

The self repeating pattern is so exact, a mathematical formula can be written to recreate them. All this fractal stuff is dubiously interesting you might say but when will I ever come across a fractal? Do not scoff, they are here on earth living among us, right now, and they don't even try to disguise themselves.


Real Fern

Fractal Fern

With a fractal you have a geometric shape that is identical to itself through infinite repeats in a recursive pattern and through infinite detail. This brings us to the Magic...and math, and beauty.

Fibonacci numbers and the Golden Ratio


Sunflower seeds arranged in a spiral of decreasingly
sized seeds following a precise formula.

Notice the precise incremental increase/decrease,
depending on which way you're looking, in the proportions of the hand and arm.


A closer look at the bones in the hand reveals symmetry.



Not only are the florets arranged in decreasing sizes, the florets themselves are composed of tinier florets also decreasing in size in a spiral.
This a fractal lovers dream within a dream.


The Chambered Nautilus gives us a really good idea of how the individual chambers are decreasing in size according to a precise ratio.





Buildings showing the incorporation of the Golden Ratio

What is the numerical value of the ratio since clearly there is one? It's rounded off at 1.618 and is denoted by the Greek letter Phi or φ. (One of my game names was Phi 1618. I'm a total loser geek poseur and I know it) Please keep in mind that I'm simplifying this in a way that would make a scientist weep.

Hand in hand with the Golden Ratio is Fibonacci numbers. Stay with me....it's easy. If I can figure it out you can. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...I didn't even have to copy and paste it. Each number is the sum of the two numbers before it. What is the ratio between the numbers? 1.618 aka The Golden Ratio.

See? Magical.








...and beautiful. This Fractal is by Roger Johnston.

Roger Johnson is a physicist, aerospace engineer, and artist. He creates fractal art with a freeware Windows app called Apophysis. The ware is free but getting the physicist and aerospace engineer designation, not so much.

More of his stuff here



Extra ice cream if you saw a Fractal/1.618/Fibonacci connection. A lot of fractals also have spirals. This one above clearly has self imitating recursive examples of the Golden Ratio. Fractals...Fibonacci...Golden Ratio... all form a web full of examples of mathematical beauty

There were several articles from which I gleaned information. This is one, Mathematical, yet Magical Beauty of Nature here which tries to explain it in terms I can grasp.

A good article on Fibonacci numbers is found here at How Stuff Works.

Fractals can be found in abundance doing a google image search. Try it!


9 comments:

Angie said...

I read your posts and realize you are way too smart to be hanging with the likes of me.

Frimmy said...

No, I certainly am not. Don't forget, I'm a poseur AND I like tater tots! While I like fractals, I had to look up a lot of info to be able to explain them and even then it doesn't read clearly to me.

Noelle said...

It reads perfectly clear. You come off as quite clever. Your a good poseur. :)

Frimmy said...

Well, sadly I have quite an ordinary mind. I just like weird stuff.

Angie said...

They are very beautiful and interesting to look at :)

Anonymous said...

You should have warned us all to smoke a big fat joint before reading this post.

Ann

Frimmy said...

I will do that now...

Aggie said...

I don't know how I missed this post. I am able to follow along with what you're saying as the golden spiral, etc. were explained to me in my Art classes back when I started college. Unfortunately, I don't remember much about them other than that my teacher was mad for showing us pictures of how all of that stuff creates itself in nature. I always felt he could teach a separate class on that.

Andrew Spain said...

Something else that is interesting about fractals is how computers arrive at plotting them 2 and 3 dimensionally. fundamentally how they even discovered what a fractal looks like is plugging numbers in a formula hundreds of times and having a computer put the solutions on a graph, like so:

Lets take a simple formula (not realistically a fractal)


y = (x * 2) - 2

And plug 3 as X
when x = 4
y = 6

you take the solution (y = 6) abd plug that back in as X

x = 6
y = 10

If you continue to do this you arrive at a table


_x_[]_y_
4 [] 6
6 [] 10
10 [] 18
18 [] 34


You now have a set of points you can plot on a graph. you plug the right formula in

z_{n+1}=z_n^2+c

and you get a mandelbrot set.

I studied them a bit before i started designing them.

cheers :)
http://answerth.deviantart.com/