Fractal Geometry of Nature

[Racoon]

Turtle did something stupid, and gave Racoon a book;

It just sat on the shelf, until Geometry overtook!!..............

 

The Fractal Geometry of Nature.

 

What is a Koch Pyramid?

 

Poincares remark that " there are questions that one chooses to ask and other questions that ask themselves "....

 

FRACTAL is a word invented by Mandelbrot to bring together under on heading a large class of objects that have [played]... an historical role... in the development of pure Mathmatics"....

 

Heres the discussion point: Dimension, Symmetry, and Dirvegence found in nature to have mathmatical principles behind them...

 

It talks about Euclidean Stuff...

 

Says " The homogenous distribution on a line, plane, or space has two very desirable properties. It is invariant under displacement, and it is invariant under change of scale. When we move on to Fractals, either invariance must be modified and/or restricted in its scope. Hence the best fractals are those that exhibit the Maximum of invariance."

 

Lots of formulas my keyboard won't type.

 

Anyone have any ideas on Geometric function found in Nature?

How leaves form a uniform pattern, usually??

 

The Book I am Referring to is called " The Fractal Geometry of Nature" by Mandelbrot; Published by Freeman and co. New York, originally in 1977 and this copy is 1983...

 

Talks about Math related to:

>Galaxies and Eddies

> Scaling

> Self- mapping

> Randomness

> Stratified random

> logic of Fractals in Statistical Lattice Physics

> Random midpoint Displacement curves

 

and a whole lot more...... :hyper: :phones: :lol:

 

My Math level is College Algebra. I haven't had any Calculus..

So I am asking the boundry-less brains of this forum... :hyper:

[Racoon]

30 views and no reply?

 

I guess nobody gives a rip.

This book could be bogus.

Its all Turtle's fault!

 

OK, heres a little more to Slake your Thirst...:hihi:

 

The Koch curve:for coastlines and such

the Dimension D = log^4 / log^3 ~1.2618>1

 

ah screw it! :shrug:

 

If my keyboard could type up some of these equations, then it could be beaten up by the Math Whizzes we got running around this site!

 

Do any of these things ring a bell??

 

>Self-Inverse Fractals, Apollonian Nets?

> Chance as a tool in Model Making

> Conditional Stationarity and Cosmographic Principles

> Random Chains and Squigs

> Brownian Motion

> Interval Tremas; Linear Levy Dusts

> Subordination; Spatial Levy Dusts; Ordered Galaxies

> yada yada yada...

 

I hear crickets :eek2:

[InfiniteNow]

Sorry Rac... I don't know too much about the question personally. Your post did remind me of a thin little book I read several years ago called "The Geometry of Art and Life." That was cool. Spoke of the Golden spiral/rectangle a lot...

[Racoon]

Sorry Rac... I don't know too much about the question personally. Your post did remind me of a thin little book I read several years ago called "The Geometry of Art and Life." That was cool. Spoke of the Golden spiral/rectangle a lot...

 

I guess thats why I stick to 'softer' science...:shrug:

Except Chemistry. Although I am a little soft in It.

 

heres one I found I could type:

 

Sample standard deviation of X(t),

S^2(d) = X^2*(d) / d - X*^2(d) / d^2

 

This book isn't really making sense to me. :eek2: :eek2:

 

ps. Nice HStaff you're sportin' there IN!

[InfiniteNow]

Don't be too hard on yourself. Often it's not the concept that eludes you, but the inability of the author to explain it adequately. Maybe try a book on the same topic by another author. That might just turn things around as far as your understanding is concerned...

 

 

Cheers. :shrug:

[Turtle]

30 views and no reply?

 

I guess nobody gives a rip.

This book could be bogus.

Its all Turtle's fault!

 

OK, heres a little more to Slake your Thirst...:shrug:

 

The Koch curve:for coastlines and such

the Dimension D = log^4 / log^3 ~1.2618>1

 

I hear crickets :eek2:

 

___Few give a rip, not nobody gives a rip.:eek2:

___Any book could be bogus; reader beware.

___It is not my fault, I am the fault; cracked Earth, get it.:Guns: :hihi: :eek2:

___The book is the seminal work describing and literally defining 'fractal'. Never mind if you understand the text & formulae, just read it, front to back while enjoying the pretty pictures & drawings. Let the general sense of the work soak in like the steady new waves lapping a sandy shore.

___Because with this new understanding of dimension, that is we have not just 2-dimensional thingys, & 3-dimensional thingys, but 1.2618 dimensional thingys & so on in between, and because they pervade nature, and because they share recursion as an operative element, I think any real grand unified theory must include recursion as an operative element.

___Pretty pictures...pretty pictures...:hihi:

[Racoon]

___The book is the seminal work describing and literally defining 'fractal'. Never mind if you understand the text & formulae, just read it, front to back while enjoying the pretty pictures & drawings.

 

Oh :shrug:

I've been reading it back to front! :eek2: :eek2:

 

Ok, mostly looking at the Pretty Pictures :eek2:

[TheBigDog]

Didn't Mandelbrot develop the theory from studying details in the charts of sheep futures? I have fractal screen saves on all my computers. I think they are beautiful.

 

Bill

[Racoon]

What can you say about Mandelbrot??

 

Dude knew his Fractals.........??

 

:steering:

[Qfwfq]

Haven't you folk learnt to delete posts properly?

[InfiniteNow]

Haven't you folk learnt to delete posts properly?

<shyly... kicking dirt with foot a bit sheepishly...>

 

Ummm... actually... no. Just tried using EDIT, but saw no delete buttons or options. I missing something?

[Pyrotex]

the[/u] seminal work describing and literally defining 'fractal'. Never mind if you understand the text & formulae, just read it, front to back while enjoying the pretty pictures & drawings. Let the general sense of the work soak in like the steady new waves lapping a sandy shore....I think any real grand unified theory must include recursion as an operative element...

I just bumped into this thread. Shiny!!!

I also have Mandelbrot's book. Read it some years ago. When I go home I will look for it somewhere on my 7 six-foot bookshelves.

 

I just happen to have a Mandelbrot Generator that I wrote in Perl, that is easy to use--IF you have a Perl 5 Interpreter on your PC. I will also change my avatar to be one of the outputs of that program.

 

So, I can't promise to answer anything fractal, but do you have any questions?

[Racoon]

<shyly... kicking dirt with foot a bit sheepishly...>

 

Ummm... actually... no. Just tried using EDIT, but saw no delete buttons or options. I missing something?

 

I heard Infinite, that after a 1,000 posts, you get special abilities like deleting posts complterly and invisibility...:steering:

[Racoon]

I just bumped into this thread. Shiny!!!

I also have Mandelbrot's book. Read it some years ago. When I go home I will look for it somewhere on my 7 six-foot bookshelves.

 

I just happen to have a Mandelbrot Generator that I wrote in Perl, that is easy to use--IF you have a Perl 5 Interpreter on your PC. I will also change my avatar to be one of the outputs of that program.

 

So, I can't promise to answer anything fractal, but do you have any questions?

 

Woot! Pyrotex has the book too!?

Good.

It would be easier to discuss if our/my keyboard could type out the formulas contained within.

 

The Basic tenet is that Nature forms with a Geometrical Math pattern.

Leaf shape. Coastlines, snowflakes..

Nature is fundamentally geometric math sequences!! :hyper: :steering:

And there are formulas that become recurrent theme!

 

Do tell a little more Pyro....

ps. I don't think, am pretty sure actually, that I don't have Perl 5.

cool Avatar.

Thats Fractal Folks!!

[Pyrotex]

Woot! Pyrotex has the book too!?

Good....The Basic tenet is that Nature forms with a Geometrical Math pattern.

Leaf shape. Coastlines, snowflakes.....Do tell a little more Pyro....

ps. ...!!

Well, I'm going to try to upload the executable. Maybe (:confused: ) that doesn't require having Perl. We'll see.

 

The fractal nature of nature is really simpler than it looks at first glance (maybe). In biology, for example, DNA doesn't have to code for the entire shape of a leaf and the twigs and the branches and the trunk. DNA only has to code for a repetitive algorithm of putting proteins here... here... there and there... and then repeating. So instead of several megabytes of storage for a description, it takes only a few hundred bytes for a cyclical sequence of operations. The real beauty of this is that each cycle does not have to be perfect or exactly like the one before. Variation adds to the beauty and complexity.

 

The Mandelbrot Space is of infinite complexity, but the algorith is adsurdly simple. In the region around 0, 0i in the complex plane, pick any point C.

C = a, bi. Where a and b are real numbers.

Now square C then add C -- to get D.

Now square D then add D -- to get E.

Now square E then add E -- to get F... etc.

 

Each time you do this, compute the distance of the point from the origin. Mandelbrot proved that if the sequence C, D, E, F, ever gets further than 2 from the origin, then the sequence will go "outward" towards infinity. Bye-bye. However, there are an infinite number of starting points (C') where the sequence C', D', E', F' always stays in the vicinity of the origin. These points make up (define) the Mandelbrot Space.

 

In the algorithm to "draw" the Mandelbrot Space, we count the number of times we repeat the cycle (square and add) until the distance of the resulting point exceeds 2 from the origin. We do this up to some maximum, say 512 iterations of the cycle.

 

If the 512th point in the sequence is still < 2 from the origin, then we color that point BLACK. These BLACK points define the Mandelbrot Space in the program. If the distance from the origin exceeds 2 at the m-th iteration, then we color that point a color associated with the number m. Since we typically have only k colors to choose from (say, 64) we actually use m' = mod(m, k). So if m = 68, then m' = 4 and we use the 4th color for that point.

 

The beauty of the Mandelbrot Space is NOT the black areas which are IN the Mandelbrot Space but the rainbow-colorful region that is ALMOST BUT NOT QUITE IN the Mandelbrot Space--the points that eventually diverge to infinity after several HUNDRED or THOUSAND iterations of the algorithm.

 

More later if you want. Now to see if I can upload the executable. To run it, just double-click it. -- DANG!! INVALID FILE!! Tormod may have to permit ".exe" files to upload.

[Pyrotex]

We will try again.

This is a zip file. You will have to unzip it to use it.

It should produce a folder named "MandelFiles" (I think).

Extract the folder to your hard drive.

Inside should be the executable Manbrot2.exe.

Double click on that.

 

And it is written in Java, not Perl. :confused:

 

My avatar was made with this program!!!

Every time you click anywhere in the program's display, it will redisplay with 2X magnification, with your click point at the center of the new display.

[InfiniteNow]

http://en.wikipedia.org/wiki/Fractal#Fractals_in_nature

 

Cool...