Fractal Geometry of Nature

[Guest chendoh]

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

 

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."

 

Anyone have any ideas on Geometric function found in Nature?

How leaves form a uniform pattern, usually??

Racoon in reference to your quote here's something you might find interesting:

 

There's an article in the April 2003 Issue of Astronomy pg 52 describing the "Golden Ratio" phi Ф 'Discovered or created'? by Euclid…..It is a special number like pi ,in that it has a never ending, never repeating sequence.

It's beauty is " This ratio describes such divergent systems as seashells, galaxies, leaves."

I wanted to put a link in, but astronomy doesn't have one for that article. Here's a few others.

The first should be just what your looking for.

 

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

 

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/whatsnew.html#multimedia

 

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

 

http://www.google.com/search?num=50&hl=en&lr=&q=+the+golden+ratio&btnG=Search

[Turtle]

There's an article in the April 2003 Issue of Astronomy pg 52 describing the "Golden Ratio" phi Ф 'Discovered or created'? by Euclid…..It is a special number like pi ,in that it has a never ending, never repeating sequence.

It's beauty is " This ratio describes such divergent systems as seashells, galaxies, leaves."[/u][/color][/url]

 

Good call Chendoh on the fractal features of phi!:eek:

We have some Hypography threads on the topic as well:

http://hypography.com/forums/physics-mathematics/1902-vesica-piscis-real-sacred-geometry.html?highlight=phi

http://hypography.com/forums/physics-mathematics/4908-phi.html?highlight=phi

http://hypography.com/forums/science-projects-homework/2440-watching-phi-grow.html?highlight=phi

 

 

 

 

:eek:

[TheBigDog]

So, Rac,

did you download the Mandelbrot Explorer and actually check it out?

Does anyone need instructions for using the MX?

And I am curious as to how BigDog is doing. Has he gotten his MX to work, yet?

Has anyone ELSE actually played with MX?

 

[...the sound of crickets in the distance...]:)

Behold! The Fractalator!!! The battle is over. I am victorious over math. Thank you Pyrotex for your wise council.

 

The Fractalator needs .Net Framework. That is all. You can explore fractals, and save your results.

 

Click around on the thing to see how it works. It is much easier to use than to write.

 

Good night now. I am exhausted.

 

Bill

[Racoon]

Behold! The Fractalator!!! The battle is over. I am victorious over math. Thank you Pyrotex for your wise council.

 

The Fractalator needs .Net Framework. That is all. You can explore fractals, and save your results.

 

Click around on the thing to see how it works. It is much easier to use than to write.

 

Good night now. I am exhausted.

 

Bill

 

[sounds of uproarious applause!] :)

BigDog takes a bow

[crowd chants Encore!]

 

Rac can't seem to upload the Frigger. err the file.

having computer quirkiness ;) ;) :)

 

Good Job BD!;)

[Guest chendoh]

Thank You, Turtle...Racoon

[TheBigDog]

[sounds of uproarious applause!] :cup:

BigDog takes a bow

[crowd chants Encore!]

 

Rac can't seem to upload the Frigger. err the file.

having computer quirkiness :hihi: :cup: :cup:

 

Good Job BD!:(

Thanks Racoon. I am going to put out a 2nd version of the Fractalator. There are three things that I want to accomplish with the second release.

 

• Adding a feature to allow customization of the colors that are used. This will also set the number of iterations. So the function of the iterations field will change a bit.
  
• Add the ability to use more fractal formulas. Currently Mandelbrot only. Any other sets that people would like to see? I am open to suggestions!
  
• Tinkering with the idea of being able to save sets of Fractals in sequence so you can play them back like a movie.
  

 

I am open to suggestions and eager for feedback.

 

Bill

[Pyrotex]

Behold! The Fractalator!!! The battle is over. I am victorious over math. Thank you Pyrotex for your wise council....

Congratulations!!

:hihi: :eek: :hihi: :

:cup: :( :cup: :

:cup: :cup: :cup: :

You are most welcome BigDog!! I can't wait to open this puppy and see what it will do. :(

[Pyrotex]

...I want to accomplish with the second release.

-...to allow customization of the colors that are used. This will also set the number of iterations. ...

-...use more fractal formulas. Currently Mandelbrot only....

...

We are the Champions!

We are the Fractalizers!

Our balls are made of brass!

We can do Complex Math! :hihi: :eek: :cup:

 

Well done, BigDog. It took me an HOUR to find where in God's hell MS kept their downloads of .Net. The THIRD thing I downloaded was apparantly the correct thing.

 

I like your interface very much. Being able to select how much zoom and how much move is a nice touch. Creating thumbnails until you get to where you want to go saves a bunch of time.

 

A word of advice about colors. Do not slave colors to the number of iterations. You will NEED iterations as high as 10,000 or more to achieve fine border detail. If you slave colors to that, then the colors around the MS [black] will all look alike, their shades will be so close. You will essentially have two-tone pix. You need to be able to select the number of colors independently. I find 64, 128 and 256 colors give the prettiest rainbow effects. I wish I had 512 as well. You may also want to select the color palatte independently.

 

The JAVA code I sent you does this. I defined six color palattes by building a 6 by 5 array. Each element of this array was a 1 by 3 array, which defined a color in RGB fashion. Each palatte was therefore a sequence of 5 colors, spanning four spectrums. All I had to do was pick the 5 colors I wanted in each palatte. The application then automatically does linear interpolation between those 5 colors to give me 4 smooth spectrums in each palatte. The application then just cycles through each of the sub-colors in the palatte chosen by menu.

 

Again...RAT ON, DUDE, RAT ON!!!!

[Racoon]

Hey!

What about all the work I did by mentioning all of this!?? :cup:

I can't let you two take home ALL the glory... :cup:

 

Good Job Boys! :( :cup: :cup: :hihi:

Drinks Are on Racoon :eek:

[Pyrotex]

What about all the work I did by mentioning all of this!?? ...

Drinks Are on Racoon

Okay, let's not forget Racoon!

It was he who started this thread.

Let's drink all the booze that he can buy.

And then merrily go to bed! :hihi: :eek: :cup:

[TheBigDog]

A word of advice about colors. Do not slave colors to the number of iterations. You will NEED iterations as high as 10,000 or more to achieve fine border detail. If you slave colors to that, then the colors around the MS [black] will all look alike, their shades will be so close. You will essentially have two-tone pix. You need to be able to select the number of colors independently. I find 64, 128 and 256 colors give the prettiest rainbow effects. I wish I had 512 as well. You may also want to select the color palatte independently.

 

The JAVA code I sent you does this. I defined six color palattes by building a 6 by 5 array. Each element of this array was a 1 by 3 array, which defined a color in RGB fashion. Each palatte was therefore a sequence of 5 colors, spanning four spectrums. All I had to do was pick the 5 colors I wanted in each palatte. The application then automatically does linear interpolation between those 5 colors to give me 4 smooth spectrums in each palatte. The application then just cycles through each of the sub-colors in the palatte chosen by menu.

I am writing an interface to completely customize the colors that are used. I have a set hardcoded in right now. Once you have defined them you will indicate how many times you want to cycle through all of them as your number of iterations. So if you define a sequence of 57 colors with iterations set at 1 it will end at 57, with 58 being MS. If you set interations at 2 then it will look at 114 with 115 beign MS, and so on. I want to make it so the user can save color schemes too. That will do it for me for right now.

 

Bill

[TheBigDog]

Fractalator 2.0 is here. New feaures abound! 100% customizable colors. Save color schemes. Unlimited lists of saved fractals. Map indicates where you are on the fractal. Warning when you zoom near the point of distortion (limitation of double data type near 10^15 zoom). Unlimited "go back" feature. If you use it, please send me some feedback. The only additional feature I am contemplating is a movie maker for flying through fractals. But that is going to wait.

 

Now, on to the "Strange Numberator 1.0"! You will want tight socks for that one Turtle.

 

Bill

[Turtle]

Fractalator 2.0 is here. New feaures abound! 100% customizable colors. Save color schemes. Unlimited lists of saved fractals. Map indicates where you are on the fractal. Warning when you zoom near the point of distortion (limitation of double data type near 10^15 zoom). Unlimited "go back" feature. If you use it, please send me some feedback. The only additional feature I am contemplating is a movie maker for flying through fractals. But that is going to wait.

 

Now, on to the "Strange Numberator 1.0"! You will want tight socks for that one Turtle.

 

Bill

 

Fexplicative WOW! :singer:

I may need more than tight socks; no...I definately need more than tight socks. I'll dress down & meet you on the strange court coach.:singer:

The Fractalator is simply fantastic so far. I haven't tried all the features, but I found some interesting curves. If I'm not mistaken, people still make new pattern discoveries in the Mandlebrot set, so saving those unusual variations has merit.

Thanks Bill for your work & a tip o' the hat to your support crew.

:D

[Buffy]

Way cool Mr. Bill!

 

I built something like this a long time ago both as an app like you have as well as a screen saver. I've been meaning to drag it out and get it running again. Some recommendations:

• Use a mouse drag on the main panel to zoom: its not that hard to code and its way easier than the paradigm you've got.
  
• You can do really fun things with cycling colors and different algorithms for generating them. When I last worked on mine you had rgb only, but direct access to color pallette cycling.
  
• If you're gonna do interesting screen savers, you've got to do a loop of selecting random rectangles and then testing the current zoom level for "interestingness". Zooming into the black areas or the white areas are painfully boring!
  
• It takes a minor amount of fiddling, but you can come up with more interesting designs by fiddling with basic Mandelbrot function: Benoit was trying to make the function as simple as possible, but there's *endless* numbers of interesting fractals that you can generate!
  

What did you code this in?

 

Math is pretty!

Buffy

[TheBigDog]

Way cool Mr. Bill!

 

I built something like this a long time ago both as an app like you have as well as a screen saver. I've been meaning to drag it out and get it running again. Some recommendations:

• Use a mouse drag on the main panel to zoom: its not that hard to code and its way easier than the paradigm you've got.
  
• You can do really fun things with cycling colors and different algorithms for generating them. When I last worked on mine you had rgb only, but direct access to color pallette cycling.
  
• If you're gonna do interesting screen savers, you've got to do a loop of selecting random rectangles and then testing the current zoom level for "interestingness". Zooming into the black areas or the white areas are painfully boring!
  
• It takes a minor amount of fiddling, but you can come up with more interesting designs by fiddling with basic Mandelbrot function: Benoit was trying to make the function as simple as possible, but there's *endless* numbers of interesting fractals that you can generate!
  

What did you code this in?

 

Math is pretty!

Buffy

Thanks Buff!

 

I used vb.net. And I did some stretching in my knowledge for this little guzzie. I had thought about the drag to zoom (two applications that I have managed at work recently have that feature), but I don't know how to do it. I tried once before and gave up. I have a screen saver that I downloaded which I really enjoy, so I didn't go down that road. I am using ARGB colors. I understand what you are saying about rotating the color pallets, but I will leave that to the pros. I am just a humble hobbyist.

 

I am going to comment up the code as is and post it in my VB.Net thread. I welcome any feedback. You will probably cringe when you see how I beat this thing into working order. Thank goodness for fast computers, I don't need to make efficient code - although I try.

 

I plan on going back to this in a year or so and adding function for adjusting the fractal calculations and saving them. That was just too much for me right now. I limited myself to just so much screen space for controls on purpose to give myself some development scope. It had to fit to be included.

 

Bill

[Pyrotex]

Fractalator 2.0 is here. New feaures abound! 100% customizable colors. Save color schemes. Unlimited lists of saved fractals. Map indicates where you are on the fractal. ...

BigDog,

I have to hand it to you. Excellent work.

Lotsa fun playing with it.

Well done. :cup: :) :love:

 

Can't wait to see the source code.:hihi:

[Turtle]

I chanced on a PBS show last night entitled Colours of Infinity. Hosted by Arthur C. Clarke and featuring Mandelbrot himself, Steven Hawking, and Prof. Ian Stewart, it presented some breakthrough applications of fractals. Specifically, a new technique in image data compression that is phenominal! Basically, (as if!), any image is scanned or otherwise digitized and then a

a fractal equation is formulated on that image. Then taking that new fractal equation, it is applied to the zoomed in/pixelated view and the detail is 'filled' in! :beer: :beer:

The show will air again on February 4th; here's a couple of links:

 

Program Information | PBS 06:00:00&use_gmt=t

PBS | TV Schedules