The Fibonacci Sequence & Golden Ratio

by haikuhouse

Although this book was very interesting, it was very slow reading at certain points. However there’s some wonderful information that is also relevant to the creative & inspiring content you expect here at Haiku House. 

 The best way for me to extend this delightful enlightenment to you is to just share a page or two worth of reading straight from the book. 

Regarding the Fibonacci numbers…

 “A lot of fascination with them is due to the surprising frequency with which they arise in nature.” 

For example… 

“The number of petals on flowers is a Fibonacci number more often than would be expected from pure chance: an iris has 3 petals; primroses, buttercups, wild roses, larkspur, and columbine have 5; delphiniums have 8; ragwort, corn marigold, and cineria 13; asters, black-eyed Susan, and chicory 21; daisies 13, 21, or 34; and Michaelmas daisies 55 or 89. 

Sunflower heads, and the base of pinecones, exhibit spirals going in opposite directions. The sunflower has 21, 34, 55, 89, or 144 clockwise, paired respectively with 34, 55, 89, 144, or 233 counterclockwise; a pinecone has 8 clockwise spirals and 13 counterclockwise. All Fibonacci numbers.

A third example arises in phyllotaxis, which studies the arrangement of leaves on plant stems. As they go up the stem, they spiral round. Start at one leaf and let p be the number of complete turns of the spiral before you find a second leaf directly above the first. Let q be the number of leaves you encounter going from that first one to the last in the process (excluding the first one). The ratio p/q is called the divergence of the plant. Common divergences are: elm, linden, lime, and some common grasses 1/2; beech, hazel, blackberry, sedges, and some grasses 1/3; oak, cherry, apple, holly, plum, and common groundsel 2/5; poplar, rose, pear, and willow 3/8; almonds, pussywillow, and leeks 5/13, all ratios of Fibonacci numbers. 

The key mathematical fact underlying nature’s seeming preference for Fibonacci numbers is their close connection to an equally famous mathematical constant known as the Golden Ratio. Often denoted by the Greek letter Phi, the Golden Ratio is, like mathematical constant Pi, an irrational number-a number whose decimal expansion continues forever, without ever settling into a regular, repeating pattern. The decimal expansion of Pi begins 3.14159; Phi starts 1.61803. 

The number Phi first appeared in Euclid’s ‘Elements’ (written around 350 BCE), where it is defined as the ratio into which you should divide a line so the ratio of the entire line to the longer division equals that of the longer division  to the shorter. Euclid gave it the name “extreme and mean ratio”. In the fifteenth century, the Italian mathematician Luca Pacioli gave it the more evocative name “Divine Proportion”, publishing a three-volume work by that title. It acquired the alternative name “Golden Ratio” in 1835, in a book written by the mathematician Martin Ohm. 

With two suggestive names, one hinting at God, the other at wealth, various false beliefs have attached themselves to the number….However, the Golden Ratio is genuinely exhibited by the growth of plants. Nature’s inevitable preference for efficiency leads it to place petals on flowers, seeds in flowerheads, and leaves on plant stems in a fashion that depends on the Golden Ratio, which has a mathematical property that results in optimized structure.

The connection between the Fibonacci numbers and the Golden Ratio was first verified  in the nineteenth century: If each Fibonacci number is divided by the one that precedes it, the answers you get grow steadily closer to the Golden Ratio-in mathematical terms, the limit of those ratios is the Golden Ratio. (The first few values work out: 2/1=2; 3/2=1.5; 5/3=1.666; 8/5=1.6; 13/8=1.625; 21/13=1.615; 34/21=1.619; 55/34=1.618.) Since phi is an irrational number, whereas the number of petals, spirals or stamens in any plant or flower has to be a whole number, Nature “rounds off” to the nearest whole number, and because of the above limit property, this will tend to be a Fibonacci number.” 

 So that’s all I wanted to share from The Man Of Numbers: Fibonacci’s Arithmetic Revolution by Keith Devlin. 

What are your thoughts on all this? 

Peace & many blessings,