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)
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 cardstock 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 



Check Back Soon For Updates!
NATURAL GEOMETRIC OBJECTS
THIS SITE S SUPPORTED BY SALES FROM MY PERSONAL CRYSTAL & MINERAL COLLECTION
SKIP TO: ARAGONITE, FLUORITE, SELENITE, KYANITE, BARITE, TIBETAN QUARTZ, COMBO SETS, METALLIC, MISC, METEORIC GLASS (MOLDAVITE & TEKTITES)
NEW!!
HYPERDIMENSIONAL
Paul Laffoley Posters: DIMENSIONALITY, THANATON III & KALIYUGA are Available Now at Miqel.com!
CLICK TO VIEW LARGE PICS
