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	Golden ratio base - Wikipedia, the free encyclopedia
		Golden ratio base refers to the use of the golden ratio, the irrational number ≈1.61803...
		It is sometimes referred to as the golden mean base, phi-base, or, colloquially, phinary.
		Despite using an irrational number base, all integers have a unique representation as a terminating (finite) base-φ expansion.
	en.wikipedia.org /wiki/Golden_mean_base (1495 words)

	Golden ratio
		This idea that Plato began the study of [the golden ratio] as a subject in itself is not in the least inconsistent with the supposition that the problem of Euclid II, 11 was solved by the Pythagoreans.
		The first known calculation of the golden ratio as a decimal was given in a letter written in 1597 by Michael Maestlin, at the University of Tübingen, to his former student Kepler.
		The result that the quotients of adjacent terms of the Fibonacci sequence tend to the golden ratio is usually attributed to Simson who gave the result in 1753.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Golden_ratio.html (1650 words)

	Proportions: Golden Section or Golden Mean, Modulor, Square Root of Two, Theorie and Construction (Site not responding. Last check: 2007-11-04)
		The Golden Section or Golden Mean is derived with simple geometric constructions, its ratio expressed in numbers is, however, irrational ((Square root of five) - 1) : 2 =.618034 : 1 (= 1 : 1.618034).
		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.
		There is additional significance and deeper meaning to the golden section in combination of pentagons and pentagrams, I recommend reading up on that in Paul von Naredi-Rainer's "Architektur und Harmonie".
	home.att.net /~vmueller/prop/theo.html (1076 words)

	The Bible illustrating phi, the golden section, divine proportion and Fibonacci series
		The Ark of the Covenant is a Golden Rectangle
		The trigonometric relationship of sin 666º to phi is based on an isosceles triangle with a base of phi and sides of 1.
		When this triangle is enclosed in a circle with a radius of 1, we see that the lower line, which has an angle of 306º on the first rotation and 666º on the second rotation, has a sine equal to one-half negative phi.
	evolutionoftruth.com /goldensection/bible.htm (495 words)

	Golden spiral
		Whoever cultivates the golden mean, avoids both the poverty of a hovel and the envy of a palace.
		It is a based on a crop circle shape known as the "Hackpen Hill Formation" and it appeared in an English wheat field in 1999.
		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.
	www.ka-gold-jewelry.com /p-articles/golden-spiral.php (622 words)

	Human Body
		line, defines the distance from the head to the base of the skull and the width of the abdomen.
		The golden section in turn, is also based on 5, as the number phi, or 1.6180339..., is computed using 5's, as follows:
		In this mathematical construction "5 ^.5" means "5 raised to the 1/2 power," which is the square root of 5, which is then multiplied by.5 and to which.5 is then added.
	goldennumber.net /body.htm (307 words)

	goldenratio (Site not responding. Last check: 2007-11-04)
		(The golden ratio is about 1.618033988...) The Golden Ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion.
		Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect.
		The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio.
	www.osogd.org /library/biscuits/goldenratio.html (532 words)

	Welcome to :: The Golden Heart Merkabah of Creation ::
		You can see that if you divide this rectangle into two equal squares, and then move one of those squares into the centre of the rectangle, who's base is created by the diagonal, you are left with two smaller Golden Mean rectangles who's shortest side is proportionally 0.618 in relation to the original.
		It is therefore shown that the value of Phi or the Golden Mean is an inherent quality of the Vesica Pisces.
		It is a matter of debate whether this was "intended" to be the golden section number or not.
	www.magicmerkabahangel.com /goldenmean.html (289 words)

	The Golden Ratio
		The Golden Ratio (and related names and properties: phi ratio, sacred cut, golden mean, divine proportion, golden number, golden section) is an irrational number (expressed by Φ, or Phi) that has some curious properties and is also closely connected to nature as well as mathematics.
		The Golden Ratio is found by dividing a segment into two parts so that the length of the smaller part is to the length of the larger part as the length of the larger part is to the length of the entire segment.
		The Golden Rectangle, as we said earlier, is a rectangle whose sides are in the proportion of the Golden Ratio.
	www.oblivion.net /~ommony/writings/phi.htm (1375 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.
		Then place the point upon the left base line, and marker on the left base line of the 2nd triangle, and draw an arc until one touches the bisection.
	www.akasha.de /~aton/GoldenTriSpiral.html (645 words)

	The Golden Mean
		A New Solution for the Parthenon's GM The Golden Mean is a ratio which has fascinated generation after generation, and culture after culture.
		This purpose of this paper is to state in the simplest form problems which relate to the Golden Mean, and pursue a variety of directions which aim to explain the origin of this remarkable ratio and its ultimate meaning in the world of mind and matter.
		The Golden Section is the division of a given unit of length into two parts such that the ratio of the shorter to the longer equals the ratio of the longer part to the whole.
	community.middlebury.edu /~harris/Humanities/TheGoldenMean.html (2544 words)

	Intro into the Golden Proportion (Site not responding. Last check: 2007-11-04)
		First let me say that the Golden Proportion, the Golden Ratio, the Golden Mean or simple Phi all refer to the same magical proportion and for the discussions here are the same thing.
		The Golden Division is the only continuos proportion that yields in progression in which the terms representing the external universe (f squared, f cubed) are in exact, continuos proportion reflection of the internal progression (1 /f squared, 1/f cubed)- the creative dream of God.
		The golden Proportion represents indisputable proportional evidence of the possibility of conscious evolution as well as of an evolution of consciousness.
	www.heartbeat2000.com /phi.htm (1032 words)

	More on Base
		More elegant is a positional system: again working in base 10, we use ten different digits 0,..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4.
		The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.
		In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a1bk + a2bk-1 + a3bk-2 +...
	www.artilifes.com /base.htm (2194 words)

	The 'Golden Mean' in number theory (Site not responding. Last check: 2007-11-04)
		Castro's observation possibly linking the 'Golden String' to a function central to the behaviour of certain eigenvalues in random matrix theory (which in turn appears to be deeply linked to the behaviour of the nontrivial zeros of the Riemann zeta function).
		The choice of the golden mean is dictated by number theory: the golden mean is the irrational number for which it is hardest to give good rational approximants.
		Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of chaos, i.e.
	www.secamlocal.ex.ac.uk /~mwatkins/zeta/goldenmean.htm (1193 words)

	Design and Proportion - The 'Golden Mean'
		The 'Golden Mean' is merely a mathematical ratio usually discerned by the painter as the ratio of the larger side of a rectangle as it relates to the shorter.
		To find out how I designed this particular painting format, using the golden mean, you will need to go to the advanced golden mean section of these lessons as the particular details may cause the odd frown or need for some to visit our site refreshment area (open 24 hours).
		Anyway it does not mention the 'Golden Triangle' (really an isosceles triangle with base angles of 72 degrees and not an unspecified area in SE Asia), and its not for the instinctive painter.
	www.geocities.com /~jlhagan/lessons/design2.htm (423 words)

	Golden ratio - Wikipedia, the free encyclopedia
		The dimensions of the canvas are a golden rectangle.
		The construction of a pentagram is based on the golden ratio.
		The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected.
	en.wikipedia.org /wiki/The_golden_ratio (5541 words)

	::THE GOLDEN MEAN SPIRAL::
		It's structure represents two well known sacred geometry shapes: the golden mean (phi) spiral and fibonacci spiral.
		the golden mean spiral can be found in another constant proportion incorporated into the Pyramid of giza which is that of phi, the golden mean.
		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)

	Golden Ratio
		The golden ratio is an irrational number, therfore it can't be written as a regular fraction.
		Another intersesting way of finding the golden number is by taking the squareroot of a number, add one to the answer, and taking the squareroot of that.
		It's an isoceles triangle with base angles equal to 72 degrees and vertex equal to 36 degrees.
	www.geocities.com /robinhuiscool/Goldenratio.html (759 words)

	Good Math, Bad Math : A Bit About Number Bases
		To write numbers in bases larger than 10, we have one very small problem, which is that we only have numbers to write digits from 0 to 9, but hex has digits from 0 to 15.
		The smallest in base 4 is 77 (1031=13*23).
		A friend told me years ago that the most natural base is base e (Euler's number), and that the closes integer base, that is 3, would have been much more efficient for computations than binary.
	scienceblogs.com /goodmath/2006/10/a_bit_about_number_bases.php (5187 words)

	Photography Techniques
		Let's start with an introduction of a technique that is well known for many centuries now: The "Golden Mean" (sometimes called "Golden Section") is a geometric formula by the ancient Greeks.
		The Golden Mean was a major guideline for many artists/painters so it is certainly worth to have in mind for modern day photographers as well.
		Thereafter we expand the base of the square till it hits the circle at point z.
	www.photozone.de /4Technique/compose/goldenmean.htm (310 words)

	The Math Forum - Math Library - Golden Ratio/Fibonacci (Site not responding. Last check: 2007-11-04)
		Fibonacci numbers are closely related to the golden ratio (also known as the golden mean, golden number, golden section) and golden string.
		The Golden Mean or Golden Section was derived by the ancient Greeks.
		An introduction to the golden ratio, also known as phi: its connection to the Fibonacci sequence, related constructions (star, line segment, rectangle, and spiral), and connections to biology, history, and aesthetics.
	mathforum.org /library/topics/golden_ratio (2460 words)

	GOLDEN SECTION
		NOTE IN THE FOLLOWING ILLUSTRATIONS THE WORDS: PHI, GOLDEN SECTION, GOLDEN MEAN, DIVINE PROPORTION AND GOLDEN RECTANGLE are all used to describe the rectangle with a 1.618 ratio or that proportion.
		To divide a piece of paper into a GOLDEN SECTION you must first draw a square as in 'a/b' (in the illustration the light blue area) divide the line by half 'c' illus.
		Now by extending the base of the square to point e and squareing the area (the gold area) you get an extended rectangle made from your square.
	www.cyberspaceorbit.com /ironhart/goldensec.html (588 words)

	The Golden Mean & Penrose Tilings , Towards Hyperdimensional Intergeometry (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-11-04)
		This great British physicist and mathematician, has developed a series of aperiodic tilings, which themselves incorporates the Golden Mean and the five-fold pentagonal symmetry lies inherent in it, and hence is infinitely recursive in its connectivity.
		These are in fact two Golden Triangles, base-to-base), and the other one has the angles of 72° and 108° degrees, which again are part of the Unified Field Cartography system, as well as the Unified Field Universal Map, which we have been mathematically generating in our recent work (see ATON).
		And to illustrate the coherent path of the Golden Middle Way, the ratio of the decagon populations are, quite naturally, the ratio of the Golden Mean.
	www.akasha.de.cob-web.org:8888 /~aton/PENROSEtile.html (796 words)

	Sacred Geometry Home Page (Site not responding. Last check: 2007-11-04)
		(The golden ratio is about 1.618033988749894848204586834365638117720309180...) The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion.
		The ratio of the axes of the form is the square root of 3, which alludes to the deepest nature of the triune which cannot be adequately expressed by rational language alone.
		This spiral generated by a recursive nest of Golden Triangles (triangles with relative side lengths of 1, phi and phi) is the classic shape of the Chambered Nautilus shell.
	www.intent.com /sg (1931 words)

	A Few Things About Penrose Tilings (Site not responding. Last check: 2007-11-04)
		The British physicist and mathematician, Roger Penrose, has developed an aperiodic tiling which incorporates the golden section and the five-fold symmetry inherent in it.
		The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B).
		Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean.
	britton.disted.camosun.bc.ca /goldengeom/goldenpenrose.html (246 words)

	Golden Ratio in Geometry
		The classical shape based on φ is the golden rectangle where φ appears alongside the perfect (unit) square:
		The golden rectangle has dimensions 1×φ such that removing the unit square one is left with the rectangle
		A derivation based on the presence of similar triangles was posted by Jan van de Craats as a solution to Coxeter's problem and included by R. Nelsen in his collection Proofs Without Words II.
	www.cut-the-knot.org /do_you_know/GoldenRatio.shtml (967 words)

	Fibonacci Numbers and The Golden Section in Art, Architecture and Music
		The dividing partition in the inner temple seems to be on the golden section both of the main temple and the inner temple.
		At the golden section point vertically is the navel indicated at the narrowest part of the waist and also the lower edge of the girdle (belt or waist-band), shown by blue arrows.
		The pitches of his notes are often based on powers of Phi and their order is fixed by a number sequence, such as the Fibonacci numbers, or R(n) - the number of Fibonacci representations of n or on many other sequences that are described here on my Fibonacci site.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/fibInArt.html (5226 words)

	Golden Mean
		Giza is arguably the greatest structure on the earth, and is clearly related to Phi, the Golden Mean.
		If one drops a plum line from the apex to the center of the base and measure from that point to the base of the pyramid's face, the distance is found to be 374.3 feet.
		The base line of 374.3 feet divided by 605.8 feet equals ~0.618, which is the Golden Mean, Phi.
	www.halexandria.org /dward824.htm (996 words)

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