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Chinese University of Hong Kong CSC 2110 – Discrete Mathematics

Chinese University of Hong Kong CSC 2110 – Discrete Mathematics. Group Project Topic: Golden Ratio Group Member: 李啟端 袁有成 陳雪聰 鄭允邦. Content. History of golden ratio Application of golden ratio Architecture Painting and sculpture Human body Daily life application Investment

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Chinese University of Hong Kong CSC 2110 – Discrete Mathematics

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  1. Chinese University of Hong KongCSC 2110 – Discrete Mathematics Group ProjectTopic: Golden Ratio Group Member: 李啟端 袁有成 陳雪聰 鄭允邦

  2. Content • History of golden ratio • Application of golden ratio • Architecture • Painting and sculpture • Human body • Daily life application • Investment • Properties of golden ratio • Definition • Geometry • Recursion • Relation with Fibonacci Sequence

  3. History of golden ratio Euclid – founder of geometry A proportion derived from a simple division of a line Euclid said,” a line is said to have been cut in “extreme and mean ratio" while the whole line is the greater segment, so is the greater to the lesser” This “extreme and mean ratio” is the first clear definition defined by Euclid that has developed into the Golden Ratio later Born 300 BC Nationality Greeks Field Mathematics

  4. Euclid’s “extreme and mean ratio” • Study the figure: • segment AC is shorter than line AB • segment CB is shorter when compared than AC • if the ratio of AB to AC is the same as the ratio of AC to CBthe line is said to be cut in extreme and mean ratio • in other words: a Golden Ratio * More information on definition will be included in the section “properties of golden ratio”

  5. So, in short, • “Golden Ratio”is a constant of (1+sqrt (5))/2, approximately 1.61803: • Fun corner : • Many have already read the Breath-Taking novel: “The Da Vinci Code“, but there is only few have noticed a blatant mistake in the novel. In an apparent blatant misunderstanding of the difference in meaning between an exact quantity and an approximation, the character Robert Langdon incorrectly claims the value of golden ratio to be exactly1.618 (Brown 2003, pp. 93-95).

  6. 2. Application of golden ratio • “Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.” — Mario Livio, “The Golden Ratio: The Story of Phi, The World's Most Astonishing Number” • Golden ratio (Φ) is specialbecause of its perceived sense of beauty & harmony • Consider the following 3 diagrams: • Fechner, a psychologist, found a preference for rectangle ratios centered on the golden ratio

  7. 2.1 Architecture • Golden ratio as shown: • Height & base width in Φ:

  8. 2.2 Painting and sculpture • Leonardo Da Vinci’s illustration of Φ on human face • “Venus”, showing perceived perfect women figure which is in Φ cont’d …

  9. “The last supper”, Leonardo da Vinci Showing golden squares in the painting cont’d …

  10. Using golden ratio, giving sense of harmony & solemnity A painting from Botero, violating golden ratio in purpose to give totally different feelings

  11. 2.3 Human Body • It is said that Leonardo da Vinci had stolen many dead bodies to study golden ratio since many body parts are in golden ratio! Body's height (red) is Φ with distance from the head to the finger tips (blue) (blue) is Φ with distance from the head to the navel and the elbows(yellow); (yellow) is Φ with distance from the head to the inside top of the arms/ width of the shoulders/ length of the forearm (green); (green) is Φ with distance from the head to the base of the skull/ width of the abdomen (magenta); remaining portions of the magenta line determine the position of the nose and the hairline

  12. 2.4 Daily life application • Perception of beauty favours figures of golden ratio • So, shorter people should avoid wearing long coat, that makes seem like even shorter • For the same logic, heels can help women approach the “golden” figure, that’s why they’re popular even though causing pains

  13. Photography TV broadcasting Situate the main object on one of the golden section points makes the photo more harmonic Anchor not sitting at the centre but the golden bisect point

  14. Plastic Surgery Michelle Pfeiffer, who plastic surgeons regard the most “perfect” face according to Φ rules

  15. 2.5 Investment • Someone uses golden ratio to estimate the magnitude of increment & decrement , claiming that : • When the price is going up, increment is Φ of the following decrement • When the price is going down, decrement is Φ of the following decrement • Remark: !!! We bear no responsibility for any damage or loss of this “theory”

  16. 3.1 Definition of golden ratio • Two quantities are in golden ratio if • the ratio between the sum of • those quantities and • the larger one • is the same as the ratio between • the larger one and • the smaller • i.e. where a >b

  17. since a / b = φ a = b φ substitute the above into

  18. 3.3 Recursion Hence…

  19. 3.4 Relation with Fibonnacci Sequence • The Fibonnacci sequence is defined as: • It is related to the Golden ratio by the way that • http://en.wikipedia.org/wiki/Fibonacci_number

  20. 3.2 Geometry • The golden ratio frequently occurs in area of geometry • It is often encountered when taking the ratios of distances in simple geometric figures such as the • Pentagon • Pentagram • Decagon and • Dodecahedron

  21. References • http://mathworld.wolfram.com/GoldenRatio.html • http://en.wikipedia.org/wiki/Fibonacci_number • http://en.wikipedia.org/wiki/Golden_ratio • http://mathworld.wolfram.com/Pentagon.html • http://mathworld.wolfram.com/FibonacciNumber.html • http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html • http://www.monmouth.com/~chenrich/GoldenRatio/GRTrigonometry.html • http://www.friesian.com/golden.htm • http://mathforum.org/library/drmath/view/52680.html

  22. The End

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