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Topic: Golden rectangle

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Golden mean

Mathematics and architecture

Michael Alphonso

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	The Fibonacci Series - The Golden Ratio - The Golden Rectangle
		The Golden Rectangle, alleged to be the most aesthetically pleasing rectangular shape possible, was first constructed by Pythagoras in the 6th century BCE.
		The section labeled "a" is a square drawn in the rectangle with proportions x/x.
		This is the characteristic of a Golden Rectangle.
	library.thinkquest.org /27890/goldenRatio1.html (209 words)

	Golden rectangle - Wikipedia, the free encyclopedia
		A golden rectangle is a rectangle with dimensions which are of the golden ratio, 1 : φ (i.e., 1 : 1.618...
		A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first.
		Since the sixteenth century, rectangles proportioned according to the golden ratio have been considered aesthetically pleasing in Western cultures; the golden ratio is frequently used in art and design.
	en.wikipedia.org /wiki/Golden_rectangle (249 words)

	Golden ratio - Wikipedia, the free encyclopedia
		The philosophy of Summum, a sect of about 200,000 adherents that has incorporated the golden ratio into the design of their Summum Pyramid winery in Utah, maintains that because it is the human mind that interprets the characteristics and qualities of the golden ratio, it should be considered in its relation to the human psyche.
		Nevertheless, some of these ratios are observed to be quite close to the golden ratio in the shape of the organs or parts which closely follow some basic geometrical shape (such as the Nautilus shell, whose construction proceeds in a logarithmic spiral).
		From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem.
	en.wikipedia.org /wiki/Golden_ratio (3564 words)

	Golden rectangle - the free encyclopedia (Site not responding. Last check: 2007-11-01)
		A golden rectangle is a rectangle with dimensions which are of the Golden Ratio,1 : φ (i.e., 1.6180339887498948...).
		That is, sectioned into two shapes: firstly a squarewith one side being one of the lesser sides of the surroundinggolden rectangle; and secondly a rectangle composed of theremainder.
		The new,smaller rectangle is thus a golden rectangle itself, and oneof its longer sides will be the other lesser side of thesurrounding rectangle, the other being one of the sides of the newsquare.
	www.world-knowledge-encyclopedia.com /default.asp?t=Golden_rectangle (111 words)

	Golden rectangle
		A golden rectangle is a rectangle with dimensions which are of the Golden Ratio, 1 :φ (i.e., 1.6180339887498948...).
		That is, sectioned intotwo shapes : firstly a square with one side being one of the lesser sides of the surrounding golden rectangle;and secondly a rectangle composed of the remainder.
		The new, smaller rectangle is thus a golden rectangleitself, and one of its longer sides will be the other lesser side of the surrounding rectangle, the other being one of the sidesof the new square.
	www.therfcc.org /golden-rectangle-19433.html (115 words)

	Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence
		A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio.
		The Golden Ratio is the ratio of BC to AB.
		If you have a Golden Rectangle and you cut a square off it so that what remains is a rectangle, that remaining rectangle will also be a Golden Rectangle.
	mathforum.org /dr.math/faq/faq.golden.ratio.html (497 words)

	Math & Art: The Golden Rectangle
		The Golden Rectangle appears in nature, music, and is also often used in art and architecture.
		The special property of the Golden Rectangle is that the ratio of its length to the width equals to approximately 1.618:
		The Golden Rectangle is considered to be one of the most pleasing and beautiful shapes to look at, which is why many artists have used it in their work.
	educ.queensu.ca /~fmc/october2001/GoldenArt.htm (540 words)

	golden rectangle ratio
		Given a rectangle having sides in the ratio, the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio.
		Constructing the Golden Rectangle With respect to the Golden Ratio by Leanne May The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes.
		The Golden Rectangle and Golden Ratio The Golden Rectangle was one of the discoveries the ancient Greek mathematicians were The sides of a golden rectangle have a certain ratio called (phi), which we are going to discover.
	www.2x3cp.com /trading/84/golden-rectangle-ratio.html (758 words)

	The Fibonacci Series - Applications - Leonardo da Vinci
		If you draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head, you will have a Golden Rectangle.
		Then, if you draw squares inside this Golden Rectangle, as was shown in the Golden Spiral in Action page, you will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth.
		Unlike the Mona Lisa, where all the lines of the Golden Rectangle are assumed by the mathematician, in "The Vetruvian Man", many of the lines of the rectangles are actually drawn into the image, at least in part.
	library.thinkquest.org /27890/applications6.html (650 words)

	The Golden Mean, Online Art Instruction
		A rectangle whose sides are related by phi (such as 13 x 8) is said to be a Golden Rectangle.
		It has the interesting property that, if you create a new rectangle by swinging the long side around one of its ends outward from the rectangle, to create a new long side, (in combination with the short side), then that new rectangle is also a golden rectangle.
		Starting with a Golden Rectangle whose short side equals one, and swinging the long side around forms a line (in combination with the short side) made up of two sections having lengths of phi and one, respectively.
	www.makart.com /resources/artclass/golden.html (596 words)

	The Golden Section and the Golden Rectangle
		The golden section was found by the Pythagoreans who used the Pentagram formed, by the diagonals of a regular Pentagon, as a symbol of their school.
		The golden rectangle was considered by the Greeks to be of the most pleasing proportions, and it was used in ancient architecture.
		The golden rectangle is probably the most aesthetic rectangle at least as some tests have shown the result was close to the golden rectangle among different rectangles.
	www.mlahanas.de /Greeks/GoldenSection.htm (1525 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)

	Golden Ratio
		A tall golden triangle is any isosceles triangle with two long sides and one short side, where the ratio of long to short is the golden ratio.
		A short golden triangle is any isosceles triangle with two short sides and one long side, where the ratio of long to short is the golden ratio.
		Prove that ACD is a short golden triangle: that it has two equal short sides, and the ratio of long to short is the golden ratio.
	www.math.csusb.edu /courses/m129/golden/golden_ratio.html (1363 words)

	Fibonacci Numbers and The Golden Section in Art, Architecture and Music
		The golden rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece but there is no original documentary evidence that this was deliberately designed in.
		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.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/fibInArt.html (5226 words)

	Math Forum - Ask Dr. Math
		One thing that is important to remember is that the Golden Rectangle does not refer to one specific size of rectangle.
		The ratio between the sides of your rectangle is 2, and it would be about 1.62 if it were a Golden Rectangle.
		It looks like you took the rectangle, cut it in fourths with cuts parallel to the 3 cm side, and then cut each of the four resulting rectangles along their diagonals.
	www.mathforum.org /library/drmath/view/59032.html (1944 words)

	Golden ratio for a golden rectangle (Site not responding. Last check: 2007-11-01)
		Blank are hint as or golden ratio for a golden rectangle on non-flash.
		Least possible denominator are golden ratio for a golden rectangle wider ratio with two incremental software.
		Dutch nederlands english is golden ratio for a golden rectangle professor michael.
	ratio.naturalcrazy.com /golden-ratio-for-a-golden-rectangle.php (209 words)

	The Golden Section (Site not responding. Last check: 2007-11-01)
		The Golden Section has been referred to as the Divine Proportion, the Golden Rectangle, or the Fibonacci Sequence (after Leonardo Fibonacci of Pisa who pioneered some of the early mathematical phenomena and its connection with nature).
		The Golden Section is a simple tool that may be used to enhance the meaning and beauty of an architectural work.
		Most of all, though, the Golden Section reminds us that proportions are important to all works of art, and that we must always learn from the masters of the past.
	www.ewersarchitecture.com /golden_section.htm (484 words)

	Golden Rectangle In Nature (Site not responding. Last check: 2007-11-01)
		The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms; for years experts have been finding examples in everything from the edifaces of ancient...
		The Golden Ratio is the ratio of the length to the width of what is said to be...
		This rectangle, called the Golden Rectangle, appears in nature and is used...
	www.riverrights.org /golden-rectangle-in-nature.html (306 words)

	Patchpieces.com - Golden Rectangle Quilt Layouts (Site not responding. Last check: 2007-11-01)
		Build one 13" Golden Rectangle in the upper left corner of the quilt using these six squares.
		After I built the one Golden Rectangle out of the squares, use Copy and Paste to create the rest of the layout, rearranging the squares so that they are oriented like the illustration above.
		You can move the rectangles as a group, but to get the squares to snap the grid neatly in the layout you will have to nudge each one separately.
	www.patchpieces.com /goldenrec.htm (336 words)

	Golden Rectangle (Site not responding. Last check: 2007-11-01)
		The golden rectangle is found in some art, especially 20th Century art.
		The Parthenon is the most famous example of the use of the golden rectangle.
		The golden rectangle and the golden ratio sometimes pop up in nature.
	www.aquilaartglass.com /goldenrectangle.htm (124 words)

	"Golden" Rectangle (Site not responding. Last check: 2007-11-01)
		The "golden" rectangle is called such rectangle, in which the ratio of the larger side to the smaller one is equal to the golden proportion, that is:
		As all sides of the rectangle AEFD are equal among themselves, this rectangle is a square.
		It is clear that the "golden" line GH divides the "golden" rectangle EBCF on the square GHCF and the new "golden" rectangle EBHG.
	www.goldenmuseum.com /0209Rectangle_engl.html (401 words)

	Fotogenetic - 35mm Film and the Golden Rectangle
		In Figure 7, we see that the aspect ratio of 35mm film is in fact a very close approximation of the Golden Rectangle.
		However, I would venture to say that the Rule of Thirds is merely a specific application or simplification of the Golden Rectangle.
		In this overlay, the four points located at the intersections of the lines dividing the image into thirds, considered the sweet spots of composition, fall approximately where the Golden Rectangle converges if allowed to repeat inside itself.
	fotogenetic.dearingfilm.com /golden_rectangle_2.html (802 words)

	The Golden Mean/Rectangle (Site not responding. Last check: 2007-11-01)
		he Golden Rectangle is a rectangle that is based upon the Golden Mean, which is a number that is represented by the Greek Letter phi (F) or represented decimally 1.6180339887499 etc. The dimensions of a Golden Rectangle are 1.618, therefore a rectangle made using the Golden Mean for example be 13 feet by 8 feet.
		The reason why the Golden Mean was used for architecture was that the ratio was very easy to reproduce accurately without using highly technical methods of calculation due to the fact the Golden Mean ratios all differ from the number representing the Golden Mean by less than 0.003.
		After the fall of Rome, knowledge of the Golden Mean was lost until the Renaissance, when many Italian painters rediscovered the ratio, using it to create perspective in their paintings, to construct buildings and decorate rooms.
	www.themystica.com /mystica/articles/g/golden_mean_rectangle.html (434 words)

	Golden Rectangle
		The Golden Rectangle is one of the most visually pleasing forms to the human eye.
		In a Golden Rectangle, the ratio of the length to the width bears this proportional relationship.
		The Golden Rectangle is used in the Egyption pyramids and in Greek architecture, such as the Parthenon, and it pervades natural and architectural proportion.
	www.discount-credit-cards.com /goldenrectangle.html (237 words)

	Center for Technology and Teacher Education: Mathematics Activities
		In this activity, students will construct a golden section and a golden rectangle and study their characteristics and connections with the Fibonacci series.
		Using their "golden" construction, they will discover the use of the golden rectangle within famous works of art.
		This exploration was adapted from an activity developed by Elizabeth Boiardi while she was a graduate student at the University of Virginia in the summer of 1998.
	www.teacherlink.org /content/math/activities/skp-goldenrec (266 words)

	golden rectangle the golden rectangle golden rectangle history (Site not responding. Last check: 2007-11-01)
		Golden Rectangle" "The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms; for years experts have been finding examples in everything from the edifaces of ancient Greece to art "
		Golden Rectangle" "The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms; for years experts have been finding examples in everything from the edifaces of a "
		The Golden Rectangle" "Look at the following rectangles and think about which one most appeals to you: You probably chose rectangle C because it is what is called a golden rectangle.
	www.meanses.com /dawssons.asp (1081 words)

	The Golden Mean
		The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos.
		When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively.
		This entire 'Golden Mean' site is dedicated to Liane Hansen of National Public Radio, whose honesty about her unfamiliarity with the 'Golden Section' during a 1995 interview inspired the original version of this page, and to Ken Beattie, who first got me thinking seriously about the signifcance of phi.
	www.vashti.net /mceinc/golden.htm (831 words)

	Golden rectangle
		A rectangle with proportions that from classical Greek times has been thought optically pleasing.
		The relation between the shorter and longer side of a golden rectangle is the same as the relation between the larger side to the sum of both lengthes: a : b = b : (a+b)
		Inscribing a square in a golden rectangle leaves another golden rectangle.
	www.daube.ch /docu/glossary/goldenrect.html (260 words)

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