Fibonacci spirals

PHI is the ratio of any two sequential numbers in the Fibonacci (V sequence. If you take the numbers 0 and 1, then create each subsequent number in the sequence by adding the previous two numbers, you get the Fibonacci sequence. For example, 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Ifyou sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times (10) the next Fibonacci number. This property results in the Fibonacci spiral seen in everything from seashells to galaxies, and is written mathematically as l2 + l2 + 22 + 32 + 52 = 5 x 8. 

Figure 17.10 The head of an ox-eye daisy, with the Fibonacci spirals drawn in to show the arrangement... 

Fig. 1 Points generated in a golden rectangle by a Fibonacci spiral with a variable convergence angle of 4jt/ 2n — 1). Numbers indicate the distance to the spiral center in units of oq...

By the same argument, the ratio of Z/A = e 0.58 should be interpreted as characteristic of free space. It is of interest that also the ubiquitous mysterious ratio [18] of 0.1115 2e. Maybe it is no accident that the approximate formula for a Fibonacci spiral [19]... 

Figure 8-12. (a) The scattered leaf arrangement (phyllotaxis) of Plantago media (drawing by Ferenc Lantos, Pecs). Fibonacci numbers of spirals in the patterns of (b) Scales of a pinecone and (c) A cactus in Hawaii (b and c, photographs by the authors). 

In the next morning s sun, I emerged from the womb of my tent to find hundreds of mushrooms of a variety I had never before encountered, arrayed in a Fibonacci-like spiral around me. Later, the young fellow who was to pack me out was not the one to arrive, but it was his grandfather, the eldest surviving of the Esselens. 

Some of the most beautiful shapes in nature, on close examination, are found to be governed by a simple series of numbers. For example, a pineapple has eight spirals of bumps going in the clockwise direction and 13 in the anticlockwise direction. Pine cones are built up in a similar pattern five spirals go clockwise and eight anticlockwise. If we start with the number 1, and create a sequence of numbers built up so that each number is the sum of the previous two numbers, we obtain the Fibonacci sequence ... 

Fig. 3 A golden spind inscribed in a golden rectangle defined by a Fibonacci sequence. The increasing size of successive squares is defined by the Fibonacci labels ( ), and the ratio of their side lengths approaches r as —> oo. The inscribed spiral approximates an equiangular logeirithmic spiral [4]...

Fig. 6 A sequence of Fibonacci squares on a scale of 1 2 serves to generate the 21 cm X 13 cm golden rectangle with its inscribed spiral. Direcily measurable radii of n a at convergence angles of 4jr/(2n — 1) terminate at the labeled points...

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