History of the Fibonacci Sequence and the Golden Ratio

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In this video I go over the history of the Fibonacci sequence and Fibonacci numbers as well as their relation to the Golden Ratio and Golden angle, rectangle, and spiral. Although the Fibonacci sequence was named after the 12 century Italian mathematician Leonardo Bonacci, the Fibonacci numbers themselves appear to have first risen in 200 BC. Fibonacci never connected the sequence to the golden ratio, and it wasn't until the 17th century astronomer and mathematician Johannes Kepler that discovered that the limit of the ratio of consecutive Fibonacci numbers approaches the Golden Ratio. The golden ratio and Fibonacci sequence is seen throughout nature and in human art, design, and manufacturing. The Golden spiral, and its approximate Fibonacci spiral, is of particular prominence throughout nature.

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Timestamps:

Life of Leonardo Bonacci aka Fibonacci (1175 to 1240-50 AD): 0:00
Fibonacci wrote Liber Abaci (Book of Abacus or Book of Calculation) in 1202 and popularized Hindu-Arabic numerals in Europe: 2:50
Fibonacci detailed how the Hindu-Arabic numerals can be used in accounting, finance, converting different currencies, and accounting: 4:48
Fibonacci's rabbits problem gave rise to the Fibonacci sequence: 5:55
Fibonacci sequence and its relationship with the Golden Ratio: 7:35
Fibonacci square tiling is an approximation to the Golden Spiral: 9:11
Fibonacci numbers first arose in 200 BC: 10:20
Fibonacci numbers are the sums of the shallow diagonals of Pascal's triangle: 12:17
Golden ratio = 1.6180339887... = ratio of two numbers is the same as ratio of their sum to the larger quantity: 13:09
Line segments in the golden ratio, and golden rectangle: 15:57
Golden ratio is the limit of the ratio of successive terms of the Fibonacci sequence, as originally shown by Kepler: 17:29
Ratios of Fibonacci numbers alternate greater and lesser than Golden Ratio phi: 19:30
Golden angle = 137.5077 degrees = 2.39996 radians = is the smaller angle of a golden circle: 22:47
Golden angle in some flowers: 25:00
Golden Spiral gets wider every quarter by a factor of phi: 25:27
Golden and Fibonacci spirals in nature, art, design, corporations, and advertising: 27:05
- Golden spiral in pineapples, eggs, continents, Mona Lisa painting, Sonic the Hedgehog, website designs, movie posters, Apple logo, hurricanes, galaxies.
- Fibonacci numbers in tree branches.
- Trump golden spiral hair LOL: 31:53