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Permalink Reply by Darren on December 1, 2011 at 1:18pm

http://www.miqel.com/fractals_math_patterns/visual-math-wada-basin-...

 

ø PHI: The Golden Ratio or Golden Section (In Nature, Art, Science and Religion) The Golden Section is a unique Ratio (or relationship between parts) that seems to be preferred by Nature as the best geometry for growth, energy conservation, elegance and has some fundamental relationships to the platonic solids and the Mandelbrot set. It was formally discovered by the Greeks and incorporated into their art and architecture, but it has been shown to occur even in prehistoric art, possibly as a function of Man's natural affinity for it's beauty. The rectangle at left has a vertical edge length of 1 the horizontal or width of the rectangle is 1.618 If we make a line inside and form a square (far left) it creates another 'golden rectangle' (at right) this subdivision continues inward in a spiral fashion tracing the form of a perfect PHI spiral seen in galaxies, seashells .... Even your hand, arm, ear, teeth, etc are in PHI Proportions! This is more clearly illustrated in the next 10 images which I downloaded from the great PHI resource site http://goldennumber.net (image credit: Gary Meisner) Phi Proportions in Finger-bone Lengths ... pattern carries on out to hand and arm Phi Spiral of the Ear Phi proportions in teeth and lips Phi Proportions in Faces are used for creating realistic results in cosmetic surgery ... Potential Relationships in Penta-Phi Matrix Reveal A Face Phi in plant growth Phi proportions in Moth wings and pattern Images above are © http://goldennumber.net (image credit: Gary Meisner)

Below LEFT is a fractal made of Phi sized circles in a symmetrical relationship The image at RIGHT is the elegant and natural Phi Spiral of the Nautilus shell - the Five Sided Pentagon has a huge variety of Golden Section Possibilities, due to the fact that it's basic shape has a phi edge/width ratio. The somewhat famous Penrose Tiles are also Phi Proportional - discovered by mathematician Roger Penrose in the 1960s - they are one of the only ways to tile a plane with pentagonal symmetry the patterns above are composed of 2 complimentary tiles, each is an expression of Phi / the golden ratio more Penrose Tile possibilities from the fractal page of Jos Leys - - I printed out a few pages of tiles on heavy card-stock paper and here's my first arrangement

Another Perfect Phi Rectangle - this time cleverly embedded in a Crop Circle Image Credit: Allan Brown - www.roundhillpress.com

PHI and FRACTALS Phi Web, a nested pentagonal wave vortex Phi and the MANDELBROT SET Masonic Mural In California Grand Lodge, based on Golden Proportion Master Artist Paul Laffoley's Paintings (see gallery) often incorporate PHI on many levels - More PHI in art and design Dan Winter's Golden Section / DNA Theories (His theory is that the PHI Geometry of DNA implies a wave-form preference for Phi-fractal self-embeddedness. He relates this semi-proven phenomenon to the human properties of consciousness and love) above: Icosa-dodeca phi-ratio recursive solid.

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