Golden mean


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Golden mean

  5. 5. • Phi (Golden Ratio) as a mysterious number hasbeen discovered in many places, such as art,architectures, humans, and plants.• According to the history of mathematics, Phiwas first understood and used by the ancientmathematicians in Egypt, two to threethousand years ago, due to its frequentappearance in Geometry.THE HISTORY OF THE GOLDEN RATIO
  6. 6. CONTD..• The name "Golden Ratio" appears in the formsectio aurea (Golden Section in Greek) byLeonardo da Vinci who used this the Goldenratio in many of his masterpieces, such as TheLast Supper and Mona Lisa.• In 1900s, an American mathematician namedMark Barr, represented the Golden Ratio byusing a greek symbol Φ.
  7. 7. Why are the objects thatcontain theGOLDEN RATIOso pleasing?
  8. 8. THE SECRET OF THE GOLDEN RATIO• Objects that meet the requirements of the GoldenRatio are attractive and pleasing to the human eye.• Throughout history, many experts have tried to findreasonable explanations for this question. Twothousand years ago, ancient Greeks discovered themagic of this ratio from the Golden Rectangle.• The Golden Rectangle is a rectangle that meets theGolden Ratio requirements and contains an infinitenumber of proportional Golden Rectangles withinitself.• The ancient Greeks were attracted by this specialnumber and its unique characteristics.
  9. 9. MATHEMATICAL PROPERTIES OF THEGOLDEN RATIOThe Golden Ratio - Φ is an irrational numberthat has the following unique properties:1. Taking the reciprocal of Φ and adding one yields Φ.phi=(1/phi)+1, or Φ = (1/Φ) + 1.2. Φ squared equals itself plus one.Φ^2=Φ+1. it is the only number in the world that has suchproperties.3. If we convert the equation from 2 into the equation Φ^2-Φ-1=0. Doing this we get x = (1 ± √5)/2. Together, thesetwo solutions are known as Phi (1.618033989) and phi(0.618033989). Phi and phi are reciprocals.
  10. 10. Φ vs. ΠΦ and Π (pi) have this incommon: where Π is the ratio ofthe circumference to its diameter,Φ is the ratio of the length to thewidth of a perfect rectangle.
  11. 11. WHAT IS THE GOLDEN RATIO?The Golden Ratio is a unique number, approximately1.618033989. It is also known as the Divine Ratio, the GoldenMean, the Golden Number, and the Golden Section.WHAT IS THE FIBONACCI SEQUENCE OF NUMBERS?The Fibonacci numbers are a unique sequence of integers,starting with 1, where each element is the sum of the twoprevious numbers. For example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,89, etc.
  12. 12. RELATIONSHIP BETWEEN THE FIBONACCISEQUENCE AND THE GOLDEN RATIO• The Fibonacci Sequence is an infinite sequence,which means it goes on for ever, and as itdevelops, the ratio of the consecutive termsconverges (becomes closer) to the Golden Ratio,~1.618.• For example, to find the ratio of any twosuccessive numbers, take the latter number anddivide by the former. So, we will have: 1/1=1,2/1=2, 3/2=1.5, 5/3=1.66, 8/5=1.6, 13/8=1.625,21/13=1.615.
  13. 13. As we can see, the ratio approaches the Golden Ratio, which isalso known as the Golden Section and Golden Mean. Eventhough we know it approaches this one particular constant,we can see from the graph that it will never reach this exactvalue.
  14. 14. • The Golden Ratio was originally derived from thegolden rectangle. The following is one method toconstruct a golden rectangle:CONSTRUCTING A GOLDEN RECTANGLEGiven: a square ABCD1.Find midpoint on DC.2.Connect MB.3.Draw a circle with the center of M, radius of MB.4.Expand the DC until it meets with the circle. Theintersection is one vertex of the rectangle.5.Complete the rectangle.During the whole process, we have made the two exactly proportional rectangles,each with the side ratio Φ, the Golden Ratio.
  15. 15. CONTD…• The Golden Rectangle is said to be one of the mostvisually satisfying of all geometric forms; whoselength: width ratio is equal to Phi.
  16. 16. THE GOLDEN SPIRAL• When a Golden Rectangle is progressivelysubdivided into smaller and smaller GoldenRectangles the pattern below is obtained.From this, a spiral can be drawn which growslogarithmically, where the radius of thespiral, at any given point, is the length of thecorresponding square to a Golden Rectangle.This is called the Golden Spiral.
  17. 17. The Human Face• Dr. Stephen Marquardt a former plasticsurgeon, used the golden section andsome of its relatives to make a maskthat he claims that is the most beautifulshape a human face can ever have, itused decagons and pentagons as itsfunction that embodies phi in all theirdimensions. Dr. Marquardt has beenstudying on humans facial beauty formany years.THE GOLDEN RATIO AND BEAUTYIN HUMANS
  18. 18. THE HUMAN SMILEA perfect smile: the front two teeth form a golden rectangle(which is said to be one of the most visually satisfying of forms,as it is formed with sides of 1 and 1.618). There is also a GoldenRatio in the height to width of the center two teeth. And theratio of the width of the two center teeth to those next tothem is phi. And, the ratio of the width of the smile to the thirdtooth from the center is also phi.
  19. 19. THE HUMAN LUNGSThe windpipe divides into two main bronchi, one long (theleft) and the other short (the right). This asymmetricaldivision continues into the subsequent subdivisions of thebronchi. It was determined that in all these divisions theproportion of the short bronchus to the long was always1:1.618.
  21. 21. The process of the growing plant followsthe Fibonacci numbers, from the first shoot,to the two shoots, three shoots, and fiveshoots, and eight shoots, and on and on.The branching rates in plants occur in theFibonacci pattern, where the first level hasone "branching" (the trunk), the second hastwo branches, than 3, 5, 8, 13 and so on.Also, the spacing of leaves around eachbranch or stalk spirals with respect to theGolden Ratio.THE GOLDEN RATIO AND BEAUTY IN NATUREPlant Growth
  23. 23. • The Golden Ratio has agreat impact on art,influencing artistsperspectives of a pleasantart piece. Mona Lisas faceis a perfect goldenrectangle, according tothe ratio of the width ofher forehead comparedto the length from the topof her head to her chin.THE GOLDEN RATIO AND BEAUTY IN ART
  24. 24. • The Golden Ratio has appeared in ancientarchitecture. The examples are many, such asthe Great Pyramid in Giza, Egypt which isconsidered one of the Seven World Wondersof the ancient world, and the GreekParthenon that was constructed between 447and 472BC.THE GOLDEN RATIO AND BEAUTY IN ARCHITECTURE
  25. 25. The Great Pyramid at Giza• Half of the base, theslant height, and theheight from thevertex to the centercreate a righttriangle. When thathalf of the baseequal to one, theslant height wouldequal to the value ofPhi and the heightwould equal to thesquare root of Phi.
  26. 26. Fibonacci and phi areused in the design ofviolins and even inthe design of highquality speaker wire.GOLDEN RATIO IN MUSIC
  27. 27. MUSICAL SCALES ARE BASED ONFIBONACCI NUMBERS• There are 13 notes in the span of any notethrough its octave.Note too how the piano keyboard of C to Cabove of 13 keys has 8 white keys and 5black keys, split into groups of 3 and 2
  28. 28. CONCLUSION:• An interesting result has occurred due to ourresearch, we now see the examples of theGolden Ratio everywhere. It is as if our eyeshave been opened to something that existedall around us but to see. For that we arethankful.