1. Definition
1.1. Euclid's Elements
1.1.1. "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."
1.1.1.1. "extreme and mean ratios"
1.2. The larger value 1.6180339887498948482…
1.3. irrational
1.3.1. Φ = (√5 + 1) / 2
2. way to find Phi using calculator
2.1. Calculator Method 1: Invert and Add 1
2.1.1. Enter 1 to start the process. Take its reciprocal (the 1/x button or else x^y and power y:-1). Add 1. Take its reciprocal. Add 1. Take its reciprocal. Add 1. Keep repeating these two operations take the reciprocal, add 1) and you will find that soon the display does not alter and settles down ("converging" as mathematicians call it) to a particular value, namely 1.61803... .
3. Other names
3.1. divine proportion
3.2. golden mean
3.3. golden section
4. History
4.1. First studied by Ancient Greek mathematicians
4.1.1. Phildas(490-430BC) made the Pathenon statues that seems to embody Golden ratio
4.1.2. Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio
4.2. 1875 - first known use of this term in English
4.2.1. James Sulley's article on aesthetics in the 9th edition of the Encyclopedia Britannica
5. Application
5.1. Aesthetics
5.1.1. 1509
5.1.1.1. Luca Pacioli
5.1.1.1.1. De Divina Proportione
5.2. Architecture
5.2.1. Parthenon
5.2.2. the Egyptian Pyramids
5.2.2.1. length of a face of the Great Pyramid : the distance from the same point : the exact centre of the pyramid's base square = 1·6
5.3. Painting
5.3.1. Heinrich Agrippa(16th-century philosopher )
5.3.1.1. a man over a pentagram inside a circle
5.3.2. Leonardo da Vinci
5.3.2.1. De Divina Proportione (On the Divine Proportion)
5.3.2.1.1. some bodily proportions exhibit the golden ratio
5.4. Book design
5.4.1. truly beautiful page proportions 2:3, 1:√3
5.5. Perceptual studies
5.5.1. Fechner
5.5.1.1. test the idea that the golden ratio plays a role in human perception of beauty
5.6. Music
5.6.1. James Tenney
5.6.1.1. For Ann (rising)
5.6.1.1.1. upwardly glissandoing tones
5.7. Industrial design
5.7.1. Example
5.7.1.1. shapes of postcards
5.7.1.2. playing cards
5.8. Nature
5.8.1. Adolf Zeising
5.8.1.1. Animals
5.8.1.1.1. the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals
5.8.1.2. Plants
5.8.1.2.1. branches along the stems of plants and of veins in leaves
5.9. Optimization
5.9.1. key to the golden section search
5.10. Finance
5.10.1. used in trading algorithms, applications and strategies
5.10.1.1. typical forms: the Fibonacci fan, the Fibonacci arc, Fibonacci retracement and the Fibonacci time extension