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	Golden spiral
		From the bone structure of human beings to the seed pattern of a sunflower to the spiral of a sea shell, the phi proportion is there, underlying all biological structures, seeming to be a geometrical blueprint for life itself.
		The other method of determining the Golden Section is by dividing a line segment, AB, at a point C, in such a way that the whole line is longer than the first part in the same proportion as the first part being longer than the remainder.
		This spiral is almost identical to the graph of the logarithmic spiral of the phi ratio, known as the Golden Mean Spiral (The difference is that the Fibonacci spiral is a whole number interpretation of the arithmetically impossible Golden Mean spiral, which has no beginning or end.
	www.ka-gold-jewelry.com /p-articles/golden-spiral.php (617 words)

	Golden spiral - Wikipedia, the free encyclopedia
		In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio.
		A Fibonacci spiral that approximates the Golden Spiral.
		The polar equation for a golden spiral is
	en.wikipedia.org /wiki/Golden_spiral (425 words)

	Levitated \| Recursive Construction of the Golden Spiral \| Macromedia Flash open source
		The beauty of irrationality is easily demonstrated in this recursive construction of three geometric spirals.
		This particular construction uses the irrational number as defined by (1 + Math.sqrt(5)) / 2, also called the Golden Mean.
		Each spiral is generated using recursive attachments of the same geometric shape.
	www.levitated.net /daily/levGoldenSpiral.html (81 words)

	::THE GOLDEN MEAN SPIRAL::
		Both of these two spirals represent the Christ nsciousness grid of the human race that had been destroyed some 13000 years ago; this event caused future generations to perceive reality through low consciousness levels thus creating fear and chaos.
		On the third dimension the grid looks like two spirals (if all the major ancient monuments upon this planet were to be connected we would discover the beginning of the spirals in Egypt).
		This spiral is almost identical to the graph of the logarithmic spiral of the phi ratio, known as the Golden Mean Spiral.
	www.merkaba.co.il /Spiral.htm (587 words)

	logarithmic spiral
		This is the spiral for which the radius grows exponentially with the angle.
		The logarithmic spiral is the curve for which the angle between the tangent and the radius (the polar tangent) is a constant.
		For that proportionality the curve bears the name of the growth spiral: a growth that is proportional to its size.
	www.2dcurves.com /spiral/spirallo.html (652 words)

	The Archdome - Golden Mean Design (Site not responding. Last check: )
		Unlike a regular spiral, the distance between the golden spiral's coils keeps increasing, growing wider as it moves away from the center.
		This is the spiral of constant expansion and growth.
		The golden spiral is the template of growth; the mathematical formula for evolution.
	www.archdome.com /goldenmean.html (619 words)

	Golden ratio - Wikipedia, the free encyclopedia
		The dimensions of the canvas are a golden rectangle.
		The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected.
		The Nautilus shell, whose construction proceeds in a logarithmic spiral, is often cited, usually under the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one.
	en.wikipedia.org /wiki/Golden_ratio (5628 words)

	Golden Section
		The Golden Mean is a ratio that is present in the growth patterns of many things--the spiral formed by a shell or the curve of a fern, for example.
		The beauty of the golden section may be indicated by the fact that a golden section rectangle subdivides into a square and another, smaller golden section rectangle.
		The Golden Triangle is an isoceles triangle with two angles of 72 degrees and one of 36 degrees.
	www.quantumbalancing.com /golden_section.htm (1529 words)

	A MUSEUM DEDICATED TO THE CONCEPT OF HARMONY AND THE GOLDEN SECTION
		The radius-vector and tangent to any point of the equiangular spiral form a constant angle b, that is, the curve intersects all rays coming out of the pole O under the same angle.
		The equiangular spiral is degenerated accordingly to a straight line and circumference with values of the angles b = 0 and b = 90°.
		Note also, that the "golden" spiral is inscribed into the "golden" rectangle in natural mode (Fig.7).
	www.fenkefeng.org /essaysm18004.html (2811 words)

	ACD Systems
		The Golden Mean (aka Golden Section or Golden Ratio) is an ancient and uncanny ratio that has brought us the Parthenon, the pyramids, and the Mona Lisa, and which continues to confound with its inexplicable aesthetic appeal.
		The Golden Mean is a ratio that underlies numerous growth patterns throughout nature - from the spiral of a Nautilus shell to the petals of a sunflower, it has an uncanny way of showing up in all manner of things we tend to deem beautiful.
		Remember that the Golden Spiral can be rotated in the frame or transposed from top to bottom to give your pictures more variety in their composition.
	www.acdsee.com /newsletters/article?id=2005-10-30 (799 words)

	Divine proportion the easy way - golden section Photoshop plugin
		One interesting consequence of this ratio is that if you have a rectangle where the sides have the golden ratio, then you can divide the rectangle into a square and a rectangle, where the new rectangle also has the golden ratio between its sides.
		The ratio between the length of your nose and the distance from the bottom of the chin to the bottom of the nose is the golden ratio.
		Here the golden triangles balance the diagonal and the golden spiral enforces the motion upwards right by placing focus on the dramatic point: the index finger pushing up the chin while at the same time taking the motion from the hand entering at the lower right corner.
	www.powerretouche.com /Divine_proportion_tutorial.htm (1358 words)

	Golden Section and Rule of Thirds (Golden Mean, Golden Ratio, Golden Spiral, Golden Proportion, Golden Triangles).
		Golden Section and Rule of Thirds (Golden Mean, Golden Ratio, Golden Spiral, Golden Proportion, Golden Triangles).
		Probably, you are here because you would like to know how to improve your photo technique, what is the Golden Mean, look at examples, read some articles or maybe use our on-line photo-adjuster (I am still working on this, though).
		And one more rule is a "Golden Spiral" or "Golden Rectangle" (you'll see why it's a rectangle in the tools section).
	photoinf.com /Golden_Mean/Eugene_Ilchenko/GoldenSection.html (373 words)

	the golden ratio
		The vertices of three golden rectangles two by two orthogonal are the vertices of a regular icosahedron; more generally two opposite edges of a regular icosahedron define a golden rectangle (thus there are 15).
		A rhombus whose diagonal's ratio is the golden ratio is a golden rhombus (its vertices are the midpoints of the sides of a golden rectangle).
		An ellipse inscribed in a golden rectangle is a golden ellipse (ratio of the axis equal to φ).
	www.ac-noumea.nc /maths/amc/polyhedr/gold_.htm (658 words)

	Golden ratio in arts
		The Golden Triangle is an isoceles triangle with two angles of 72 degrees and one of 36 degrees.
		A golden triangle is an isosceles triangle with base angles of 72º and a vertex angle of 36º.
		The ratio of tile A to tile B is the Golden Ratio.
	eduwww.mikkeli.fi /opetus/myk/kv/comenius/kultainen.htm (837 words)

	Sacred Geometry Exercise-Nautilus Shell Spiral
		The nautilus shell spiral is a logarithmic spiral similar to other spirals such as the Golden Mean or phi spiral, but with slightly different proportions.
		The spiral of the chambered nautilus as well as other logarithmic spirals can be found throughout the human body and nature.
		The drawing of the inner ear and the photo image of a star cluster nebula shown to the right are just two examples which can be seen with a microscope or a telescope.
	www.sacredarch.com /sacred_geo_exer_shell.htm (377 words)

	Phi-A.K.A. the Golden Ratio
		The Golden Ratio, along with Fibonacci's Sequence is present in almost all ares of life.
		The Golden Ratio is in effect, the "ultimate" proportion to which beauty in almost all eyes is seen, and in almost all ears is heard, etc. It is present everywhere becuase of the way we percieve the universe in which we live in.
		This spiral is also known as the golden spiral.
	members.tripod.com /Golden_Ratio_Web/startframe.htm (442 words)

	Divine proportion the easy way - golden section Photoshop plugin
		the ratio between the length of limbs of your fingers is the golden ratio.
		The ratio between the length of your nose and the distance from the bottom of the chin to the bottom of the nose is the golden ratio.
		Here the golden triangles balance the diagonal and the golden spiral enforces the motion upwards right by placing focus on the dramatic point: the index finger pushing up the chin while at the same time taking the motion from the hand entering at the lower right corner.
	powerretouche.com /Divine_proportion_tutorial.htm (1013 words)

	Spirals
		Note: The Fibonacci series spiral on the left is slightly different from the perfect spiral generated by Phi (1.61804...) because of the approximations early in the series leading to Phi.
		Alternate spirals in plants occur in Fibonacci numbers
		Plants illustrate the Fibonacci series in the numbers of leaves, the arrangement of leaves around the stem and in the positioning of leaves, sections and seeds.
	goldennumber.net /spirals.htm (229 words)

	Activity 2
		The ratio of the length of the longest side to the length of the shortest side of a, golden rectangle is the golden ratio.
		In fact, a spiral can be drawn from the diagram above by dividing BCDE into a square and another golden rectangle, and by continuing to separate the golden rectangles into squares and new golden rectangles.
		The first part of the spiral is drawn with a compass point at B and an arc from A to E; next, the compass point at G and an arc from E to H. Continue in this way until the spiral is completed to your satisfaction.
	homepage.mac.com /efithian/Geometry/Activity-02.html (585 words)

	Golden Ratio
		The golden ratio is an irrational number, therfore it can't be written as a regular fraction.
		A Fibonacci Spiral is almost the same as a Golden Spiral, except for one difference.
		To construct the spiral, bisect one of the 72 angles, and draw a circle centered at the 108 angle formed.
	www.geocities.com /robinhuiscool/Goldenratio.html (759 words)

	Math Trek: Sea Shell Spirals, Science News Online, April 2, 2005
		A logarithmic spiral follows the rule that, for a given rotation angle (such as one revolution), the distance from the pole (spiral origin) is multiplied by a fixed amount.
		Such a logarithmic spiral can be inscribed in a rectangle whose sides have lengths defined by the golden ratio.
		Starting with the observation that shell spirals are logarithmic spirals, many people automatically assume that, because the golden ratio can be used to draw a logarithmic spiral, all shell spirals are related to the golden ratio, when, in fact, they are not.
	www.sciencenews.org /articles/20050402/mathtrek.asp (746 words)

	Golden Spiral - TeacherVision.com
		The Golden Spiral manifests itself in such familiar forms of nature as sunflowers, pinecones, and shells, but it may also appear in the structure of such large-scale phenomena as hurricanes and spiral galaxies.
		A Golden Spiral can be derived from a Golden Rectangle, a specific type of rectangle whose ratio of long side to short side is approximately 1.618034 to 1.
		To create this special type of spiral, simply partition off a square from the Golden Rectangle in such a way that its sides are equal to the short side of the rectangle.
	www.teachervision.fen.com /mathematics/resource/5989.html (605 words)

	"Golden" spirals
		The spiral is a plane line derivated by a driving point, which moves away according to the definite law from the beginning of the ray uniformly rotated around of the beginning.
		Note also, that the "golden" spiral by a natural mode is connected to the "golden" rectangle (Fig.2).
		If as the beginning of the spiral we select the point, to which "golden" rectangles sequentially converge, the "golden" spiral will pass through three of four topics of each "golden" rectangles sequentially constructed on Fig.
	www.goldenmuseum.com /0212Spirals_engl.html (359 words)

	Golden Ratio Spiral the Breath of Brahma
		The mathematics that describe the radiation of energy throughout the vast spectrum of vibration of the universe is the Golden Spiral.
		Spiraling through all vibration are a balance of centrifugal and centripetal currents fractally emanating from an infinite core.
		The Golden Spiral is a key to looking beyond the physical to the unified field that is the substratum underlying manifestation.
	www.weare1.us /Golden_Ratio.html (8574 words)

	The "Golden" spirals and "pentagonal" symmetry in the alive Nature
		The "golden" spirals are widespread widely in the biological world.
		It was proved that among different kinds of spirals showing in horns of rams, goats, antelopes and other horned animals the "golden" spirals meets most often.
		The plant tendrils are became twisted by spirals, the growth of tissues in tree's trunks is realized by spiral there, the sunflower seeds are arranged on the spirals, the helical motions are watched at growth of the roots and sprouts.
	www.goldenmuseum.com /0602PentagonSpirals_engl.html (522 words)

	Flower of Life - The Golden Mean Spiral and The Merkaba - Article by Ronald L. Holt -
		Although the Golden Mean Spiral is principally derived by utilizing mathematics, it is equally mystifying and intriguing to note that this mathematical spiral has additional properties that can be experienced by humans on a profound level that does not require an intellectual understanding of the mathematical principles.
		Up until now, in depth material about the significance of the Golden Mean Spiral and how it relates to the merkaba hasn’t been deeply explored even though their interdependence is intricately entwined.
		In order to simplify the initial steps to understanding how interlocked and interdependent the golden mean and the merkaba are, we need to have a general understanding and overview of several models that serve as building blocks.
	www.floweroflife.org /spiral01.htm (409 words)

	The Equiangular Spiral: Spira Mirabilis
		Certainly, most will have witnessed the sublime geometry of a spiral in the vortex of water rushing down a drain, the tempest of a hurricane, the hypnotic pattern of a magician’s spiral, the shape of a snail’s shell, the inscrutable movement of the arms of a spiral galaxy like our own...
		The "wonderful" properties of the spiral are further enhanced by its connection to Golden mean and Fibonacci numbers, themselves objects of mystery and intrigue.
		Following this pattern, the Golden rectangles are inscribed on the interior of the spiral.
	online.redwoods.cc.ca.us /instruct/darnold/CalcProj/Sp98/GabeP/Spiral.htm (755 words)

	Golden Triangle and Spiral
		The Golden Mean Spiral, the seven toroid cascade from the heart's cardio-rythm at coherent moments, known as Compassionate Love, which establishes the recursive fractal or mirror-into-mirror cascade, that we call the Pyramid Vortex of the Phoenix and Dove, can be made by the simplest Universal Keys of Trinity, that comprises this Universe's university for co-creatorship.
		To then generate the Golden Mean Spiral, conscious coherence, from heart to mind, Love as the Ultimate Logic, then place your point on the left hand base, and marker on the top, and draw an arc stretching from the top vertice of the obtuse angle to the bottom right of the isoceles triangles.
		There secret symbol of identification was the Pnetagramme inscribed in a circle, which was known to them as the "pentagram star", variations with a point in the middle and an infinity sign.
	www.akasha.de /~aton/GoldenTriSpiral.html (645 words)

	Math Forum - Ask Dr. Math
		The Golden Spiral is a logarithmic spiral; that is, its radius grows exponentially.
		To derive the spiral formula, we also need to remember that the ratio of the sides of a golden rectangle is equal to (1 + sqrt(5))/2, often represented by the Greek symbol phi.
		If the spiral above is a Golden spiral, then we know the following: The lengths of the parts of the axes cut off by the spiral fit the Golden ratio.
	mathforum.org /library/drmath/view/51760.html (403 words)

	Sacred Geometry - Crystalinks
		The golden ratio, geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture.
		The golden ratio, also known as the god ratio, golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina, is an irrational number, approximately 1.618 033 988 749 894 848, that possesses many interesting properties.
		Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry.
	www.crystalinks.com /sg.html (1135 words)

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