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Golden Ratio

Jingyang Yu Xiangning He Yingxi Wu Ying Luo Siru Wen. Golden Ratio.

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Golden Ratio

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  1. Jingyang Yu Xiangning He Yingxi Wu Ying Luo Siru Wen Golden Ratio

  2. Other names frequently used:golden section& golden meanOther terms encountered include:extreme and mean ratio, divine proportion, divine section, golden proportion, golden number, and mean of Phidias.

  3. History of the Golden Ratio------- How they found the Golden Ratio ?

  4. Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere are said to be circumscribed by golden rectangles.

  5. Euclid(c. 325–c. 265 BC) in his Elements, saying the line AB is divided in extreme and mean ratio by C if AB:AC = AC:CB. This is the first recorded definition of the golden ratio: extreme and mean ratio. Euclid also gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron. First construct an isosceles triangle whose base angles are double the vertex angle. This is done by taking a line AB and marking C on the line in the golden ratio. Then draw a circle with center A radius AB. Mark D on the circle so that AC = CD = BD. The triangle ABD has the property that its base angles are double its vertex angle. Then start with such a triangle ABD draw a circle through A, B and D. Then bisect the angle ADB with the line DE meeting the circle at E. Note that the line passes through C, the point dividing AB in the golden ratio. Similarly construct F and draw the pentagon AEBDF.

  6. Nobody believes that Euclid's Elements represents original.Now some historians believe that Book II of The Elements covers material originally studied by Theodorus of Cyrene while others attribute the material to Pythagoras, or at least to the Pythagoreans. Also, some say is Plato firstly began the work. Euclid

  7. Fibonacci (1170–1250) mentioned the numerical series(Liber Abac). The ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically. Luca Pacioli(1445–1517) defines the golden ratio as the "divine proportion“: (... just like God cannot be properly defined, nor can be understood through words, likewise this proportion of ours cannot ever be designated through intelligible numbers, nor can it be expressed through any rational quantity, but always remains occult and secret, and is called irrational by the mathematicians.) Johannes Kepler(1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers and describes the golden ratio as a "precious jewel“.

  8. Illustration from Luca Pacioli's Divine Proportion applies geometric proportions to the human face.

  9. Michael Maestlin First person to publish a decimal approximation of the golden ratio, in 1597.

  10. Mathematician Mark Barr, in 20th century, proposed to denote golden ratio by the Greek lowercase letter phi (φ) , while its inverse, (1/phi), or 0.6180339887... is denoted by the uppercase variant Phi (Φ). This is named for the Greek sculptor Phidias.

  11. Concept “In mathematics and the arts, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.” The golden ratio is an irrational mathematical constant, approximately 1.6180339887 (1:0.618 or 1.618:1). This proportion is aesthetically pleasing.

  12. Two quantities a and b are said to be in the golden ratio φ if: The fraction on the left can be converted to Multiplying through by φ produces The only positive solution to this quadratic equation is

  13. Construction of a golden rectangle:1. Construct a unit square (red).2. Draw a line from the midpoint of one side to an opposite corner.3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

  14. A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship:

  15. If you pay close attention to your surroundings, you may be able to spot in them the many things that contain the Golden Ratio, for instance, in architecture, art, music, astronomy, the human body or in nature. How the Golden Ratio impacts society?

  16. The Golden Ratio & Beauty in Architecture The Golden Ratiohas appeared in many architecture. The examples are many, such as the Great Pyramid in Egypt, which is considered one of the Seven World Wonders of the ancient world, and the Greek Parthenon constructed between 447 and 472BC. Not only did the ancient Egyptians and Greeks know about the magic of Golden Ratio, so did the Renaissance artists, who used the Golden Ratio in the design of Notre Dame. Some modern architecture are also influenced by Golden Ratio as well, such as the United Nations Building.

  17. Golden Pyramids in Egypt Half of the base, the slant height, and the height from the vertex to the center create a right triangle. When that half of the base equal to one, the slant height would equal to the value of Phi and the height would equal to the square root of Phi.

  18. The Pentagon in US

  19. Louvre in France & Winding Staircase

  20. The UN Building In the United Nations building, the width of the building compared with the height of every ten floors is a Golden Ratio.

  21. The Golden Ratio & Beauty in Art The Golden Ratiohas a great impact on art, influencing artists' perspectives of a pleasant art piece. Have you ever wondered why Da Vinci's Mona Lisa looks so beautiful? Da Vinci covers the Golden Ratio in his work. Scientifically, Mona Lisa’s face appears in a golden rectangle, which makes her face appear more beautiful to human eyes. Also another masterpiece, the Last Supper, contains Golden Ratios. The French Impressionist painter George Seurat is famous by his new technique of drawing - Pointilism, which "attacked every canvas by the golden section."

  22. Mona Lisa Her face is a perfect golden rectangle, according to the ratio of the width of her forehead compared to the length from the top of her head to her chin.

  23. The Last SupperThe Golden Ratio appears in both the ceiling and the position where the people sit.

  24. The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.

  25. Statue of Athena the first Golden Ratio is the length from the front head to the ear opening compared with the length from the forehead to the chin. The second one appears in the ratio of the length from the nostril to the earlobe compare with the length from the nostril to the chin.

  26. The Golden Ratio and Beauty in Humans Phi is a mysterious number which has some related quantities and shapes, and it appears in the proportions of the human body, and other animals', in plants, in DNA, in solar system, in music, etc. Nevertheless, it had been debated on whether or not it is a natural beauty-maker. German physicist and psychologist Gustav Theodor Fechner conducted in the 1860s. His experience is simple: a person needs to choose the most pleasing rectangle among ten rectangles that are placed in front of him, and all have different ratios of length to width. The result shows that 76% of all choices are the three rectangles having ratio of 1.75, 1.62, and 1.50, which are really close to 1.618.

  27. Dr. Stephen Marquardt is a former plastic surgeon, has used the golden section and some of its relatives to make a mask that he claims that is the most beautiful shape a human face can ever have.

  28. Leonardo da Vinci's drawings of the human body emphasized its proportion. The ratio of the following distances is the Golden Ratio: (foot to navel) : (navel to head)

  29. Currently, we also use the Golden Ratio to measure if one has a “pleasing” face or good body shape.

  30. Work Cited http://library.thinkquest.org/trio/TTQ05063/ http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html http://www.google.com/#q=history+golden+ratio&hl=en&prmd=ivns&tbs=tl:1&tbo=u&ei=vt1UTar7Icmr8AaTju3yBg&sa=X&oi=timeline_result&ct=title&resnum=11&sqi=2&ved=0CF4Q5wIwCg&fp=f306bccb705f8a44 http://en.wikipedia.org/wiki/Golden_ratio http://www.facialbeauty.org/divineproportion.html

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