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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

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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
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I printed out a few pages of tiles on heavy card-stock paper and here's my first arrangement



Another Perfect Phi Rectangle
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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

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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 that most of the rant above is based on.

I don't really trust Dan Winter as a person, *same goes with 'Drunvalo Melchizedek', Dan is a plagiarist who got sued and lost (by Stan "the man" Tenen of MERU Foundation) for using copyrighted ideas and claiming them as his own, with no credit. but I do think despite his personal shortcomings, *some* of his ideas have merit and deserve investigation.




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An Introduction to the Fascinating Patterns of Chaos & Visual Math
Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals
Multi Dimensional Forms The Famous Mandelbrot Set Fractal Types & Categories
PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections
Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns



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