Title: Hello Post by: phiaera on March 07, 2012, 01:51:54 AM Hello,
Though I am not necessarily mathematically savvy, my name that I chose contains the greek alphabet, phi, which is (5^(.5*.5+.5))=(x^2=x+1)=1.618... Phi, found in the Fibonacci sequence, has been a main theme in my developing of a systematic mathematical approach in art. While this golden ratio has revealed many hidden properties of art, there is currently a big missing link: the science of developing of efficient networks structures and systems, which I believe to be fractals. I came to this forum since there is no class on fractals offered at my university and learning from the methods taught in books are a bit too abstract for me. I will be asking newbie questions soon. See you then! Title: Re: Hello Post by: asimes on March 07, 2012, 06:57:26 AM Hello and welcome, although I am somewhat new myself. I also like to think about the golden number. So far I have not made too much use of it in generating fractals, but it seems to serve some purpose for the Julia Sets of the Mandelbrot Set.
Here's some images from Wikipedia you might like that include Julia Sets that use the Golden Ratio as their 'c' (where 'c' is part of the equation z=z^2+x): http://en.wikipedia.org/wiki/Julia_set#Quadratic_polynomials Title: Re: Hello Post by: phiaera on March 07, 2012, 07:29:56 PM Hi,
I did not know this, but I think it is interesting that when I first saw the Julia Sets, I actually thought it was a bunch of golden spirals, logarithmic spirals based on the golden ratios, twisting and turning upon one another. One of the attractive things to me about fractals is that I believe structures, that are both fractal, and fractal in ratios of phi, are the most efficient kinds of structures. For example, if we look at squares spiraling around each other infinitively, http://www.stanford.edu/group/dahlia_genetics/2008_reports/lab_3/lab_3_pics/fibonacci_spiral.png We can get the squares to fit together neatly by altering the sizes of the squares. The squares in the picture grow by the golden proportion. The interesting thing is that if we can only choose one ratio by which the squares grow by, we would have to use the golden ratio in order for all the squares to cover the most amount of surface area. For this, I believe that the golden ratio is a quantification of fractals, in terms of efficiency, and this is why many things in nature, such as trees and sunflowers, would use this. For example, trees would use this to cover the maximum amount of area to gather sunlight. Sunflowers would spread their seeds accordingly to the golden ratio in order to get the most efficient seed-spacing method, etc... Title: Re: Hello Post by: asimes on March 07, 2012, 10:43:21 PM I'm not sure if the rate of expansion for the spirals is the golden spiral, but the input for 'c' in a couple of those images make use of the golden number. The second image does have a double spiral that reminds me of the sunflower seed packing. It may be worthwhile to make the measurements and see if those really are golden spirals.
Also, a lot of the fractals that are shown on this forum are made by a method of rotating a point in space by a certain angle * a certain power (exponent). Maybe making use of the golden angle or the golden ratio itself as a fractional dimension would make new / interesting images. Here's a tutorial on Mandelbrot generation by the rotation method: http://www.skytopia.com/project/fractal/mandelbrot.html#pandora Title: Re: Hello Post by: phiaera on March 08, 2012, 03:28:47 AM There is a paper on an approximate golden fractal dimensions found in goose feathers and scientists speculate that it is for this reason that Goose Down is superb material for insulating heat. http://ningpan.net/Publications/101-150/141.pdf I actually speculated upon this a year ago that fractals based on phi would be the most efficient kinds of structure/system/network, even though I did not, and still do not, understand what fractal dimensions really are. I would like to see fractal mountains with various fractal dimensions. Of particular interest, I am interested in the fractal dimensions in between 1.414 to 2.0. If my predictions are correct, the 'tension' in musical scale are due to fractal dimensions and that euclidean geometry is responsible for the octaves in music, and this would reveal a huge missing link that I have been searching for approximately two years now. Is there a program that lets me control the fractal dimensions of something such as a mountain so I can view various selective fractal dimensions? |