And their relationship with othermathematical conceptsGOLDEN RATIO AND GOLDENRECTANGLE
Prepared By-Name - Meeran Ali AhmadClass - X ARoll No. - 07
CONTENTS What is the golden ratio? History Applications and observations What is the golden rectangle? Applications Objects depicting golden ratio Golden triangle and pentagram Egyptian Pyramids
What is GOLDEN RATIO ?? In mathematics and the arts, two quantities are in the goldenratio if the ratio of the sum of the quantities to the largerquantity is equal to the ratio of the larger quantity to thesmaller one. The figure on the right illustrates the geometricrelationship. Golden Proportion exists between a small and a largesegment: The proportion of the small segment to the large isthe same as the proportion of the large segment to the sum ofboth.
History Some of the greatest mathematical minds of all ages,from Pythagoras and Euclid in ancient Greece, through themedieval Italian mathematician Leonardo of Pisa and theRenaissance astronomer Johannes Kepler, to present-day scientificfigures such as Oxford physicist Roger Penrose, have spent endlesshours over this simple ratio and its properties. But the fascinationwith the Golden Ratio is not confined just to mathematicians.Biologists, artists, musicians, historians, architects, psychologists,and even mystics have pondered and debated the basis of itsubiquity and appeal. In fact, it is probably fair to say that theGolden Ratio has inspired thinkers of all disciplines like no othernumber in the history of mathematics.
Applicationsand observations Aesthetics-De Divina Proportione, a three-volume work by Luca Pacioli,was published in 1509. Pacioli, a Franciscan friar, wasknown mostly as a mathematician, but he was also trainedand keenly interested in art. De Divina Proportione exploredthe mathematics of the golden ratio. Though it is often saidthat Pacioli advocated the golden ratios application to yieldpleasing, harmonious proportions, Livio points out that theinterpretation has been traced to an error in 1799, and thatPacioli actually advocated the Vitruvian system of rationalproportions
ArchitectureThe Parthenons facade as well as elements of its facade and elsewhere aresaid by some to be circumscribed by golden rectangles. Other scholarsdeny that the Greeks had any aesthetic association with golden ratio.For example, Midhat J. Gazalé says, "It was not until Euclid, however,that the golden ratios mathematical properties were studied. Inthe Elements(308 BC) the Greek mathematician merely regarded thatnumber as an interesting irrational number, in connection with themiddle and extreme ratios. Its occurrence in regular pentagons anddecagons was duly observed, as well as in the dodecahedron (a regularpolyhedron whose twelve faces are regular pentagons).”
PaintingLeonardo da Vincis illustrations of polyhedra in De divinaproportione (On the Divine Proportion) and his views thatsome bodily proportions exhibit the golden ratio have led somescholars to speculate that he incorporated the golden ratio inhis paintings.But the suggestion that his Mona Lisa, forexample, employs golden ratio proportions, is not supported byanything in Leonardos own writings.
Approximate and true golden spirals. The green spiral is madefrom quarter-circles tangent to the interior of each square,while thered spiral is a Golden Spiral, a special typeof logarithmic spiral. Overlapping portions appear yellow.The length of the side of one square divided by that of thenext smaller square is the golden ratio.
What is GOLDEN RECTANGLE ?? The Golden Proportion is the basis of the GoldenRectangle, whose sides are in golden proportion toeach other. The Golden Rectangle is considered to bethe most visually pleasing of all rectangles. Adistinctive feature of this shape is that whena square section is removed, the remainder is anothergolden rectangle; that is, with the same aspect ratio asthe first. Square removal can be repeated infinitely, inwhich case corresponding corners of the squares forman infinite sequence of points on the golden spiral, theunique logarithmic spiral with this property.
Applications In all kinds of design, art, architecture, advertising,packaging, and engineering; and can therefore be foundreadily in everyday objects. Golden Rectangles can be found in the shape of playing cards,windows, book covers, file cards, ancient buildings, andmodern skyscrapers. Many artists have incorporated the Golden Rectangle intotheir works because of its aesthetic appeal. It is believed by some researchers that classical Greeksculptures of the human body were proportioned so that theratio of the total height to the height of the navel was theGolden Ratio.
Golden Rectangles can be found in the shape ofplaying cards, windows, book covers, file cards,ancient buildings, and modern skyscrapers. Many artists have incorporated the Golden Rectangleinto their works because of its aesthetic appeal. It is believed by some researchers that classical Greeksculptures of the human body were proportioned sothat the ratio of the total height to the height of thenavel was the Golden Ratio.
A method to construct a golden rectangle. The squareis outlined in red. The resulting dimensions are in thegolden ratio.
Golden Mean Gauge: Invented by Dr. Eddy Levin DDSObjects depicting Golden Ratio
Goldentriangleand pentagram Golden triangleThe golden triangle can be characterized as an isoscelestriangle ABC with the property that bisecting theangle C produces a new triangle CXB which isa similar triangle to the original. Similarly, the ratio of the area of the larger triangleAXC to the smaller CXB is equal to φ, whilethe inverse ratio is φ - 1.
Pentagram The golden ratio plays an important role in the geometryof pentagrams. Each intersection of edges sections other edgesin the golden ratio. Also, the ratio of the length of the shortersegment to the segment bounded by the two intersecting edges(a side of the pentagon in the pentagrams center) is φ, as thefour-color illustration shows. A pentagram colored to distinguish its line segments ofdifferent lengths. The four lengths are in golden ratio to oneanother.
Egyptian pyramids In the mid-nineteenth century, Röber studied various Egyptianpyramids including Khafre, Menkaure and some of the Giza, Sakkara,and Abusir groups, and was interpreted as saying that half the base ofthe side of the pyramid is the middle mean of the side, forming whatother authors identified as the Kepler triangle; many other mathematicaltheories of the shape of the pyramids have also been explored. Adding fuel to controversy over the architectural authorship of theGreat Pyramid, Eric Temple Bell, mathematician and historian, claimedin 1950 that Egyptian mathematics would not have supported theability to calculate the slant height of the pyramids, or the ratio to theheight, except in the case of the 3:4:5 pyramid, since the 3:4:5 trianglewas the only right triangle known to the Egyptians and they did notknow the Pythagorean theorem, nor any way to reason about irrationalssuch as π or φ.
In 1859, the pyramidologist John Taylor claimed that, inthe Great Pyramid of Giza, the golden ratio is represented bythe ratio of the length of the face (the slope height), inclinedat an angle θ to the ground, to half the length of the side ofthe square base, equivalent to the secant of the angle θ. Theabove two lengths were about 186.4 and 115.2 metersrespectively. The ratio of these lengths is the golden ratio,accurate to more digits than either of the originalmeasurements. Similarly, Howard Vyse, according to MatilaGhyka, reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height tohalf-base, again more accurate than the data variability.