Group ProjectTopic: Golden Ratio

Group Member: 李啟端袁有成陳雪聰鄭允邦

Chinese University of Hong Kong

CSC 2110 – Discrete Mathematics

Content1. History of golden ratio

2. Application of golden ratio Architecture Painting and sculpture Human body Daily life application Investment

1. Properties of golden ratio Definition Geometry Recursion Relation with Fibonacci Sequence

1. 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 BCNationality GreeksField Mathematics

• Euclid’s “extreme and mean ratio”

Study the figure:

1. segment AC is shorter than line AB

2. segment CB is shorter when compared than AC

3. if the ratio of AB to AC is the same as the ratio of AC to CB

the line is said to be cut in extreme and mean ratio

4. in other words: a Golden Ratio* More information on definition will be included in the section “properties of golden ratio”

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 exactly 1.618 (Brown 2003, pp. 93-95).

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 special because of its

perceived sense of beauty & harmonyConsider the following 3 diagrams:

Fechner, a psychologist, found a preference for rectangle ratios centered on the golden ratio

2.1 Architecture

Golden ratio as shown:

Height & base width in Φ:

2.2 Painting and sculptureLeonardo Da Vinci’s

illustration of Φ on human face

“Venus”, showing perceived perfect women figure which is in Φ

cont’d …

“The last supper”, Leonardo da Vinci

•Showing golden squares in the painting

cont’d …

Using golden ratio, giving sense of harmony & solemnity

A painting from Botero, violating golden ratio in purpose to give totally different feelings

2.3 Human BodyIt is said that Leonardo da Vinci had stolen

many dead bodies to study golden ratio since many body parts are in golden ratio!1. Body's height (red) is Φ with distance from the

head to the finger tips (blue)

2. (blue) is Φ with distance from the head to the navel and the elbows (yellow);

3. (yellow) is Φ with distance from the head to the inside top of the arms/ width of the shoulders/ length of the forearm (green);

4. (green) is Φ with distance from the head to the base of the skull/ width of the abdomen (magenta);

5. remaining portions of the magenta line determine the position of the nose and the hairline

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

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

Plastic Surgery

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

2.5 InvestmentSomeone 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”

3.1 Definition of golden ratioTwo 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 >bb

a

a

ba

since a / b = φa = b φ

substitute the above into b

a

a

ba

3.3 Recursion

Hence…

3.4 Relation with Fibonnacci SequenceThe Fibonnacci sequence is defined as:

It is related to the Golden ratio by the way that

http://en.wikipedia.org/wiki/Fibonacci_number

3.2 GeometryThe golden ratio frequently occurs in area

of geometryIt is often encountered when taking the

ratios of distances in simple geometric figures such as the PentagonPentagramDecagon and Dodecahedron

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