Botany

  Fibonacci sequence might as well be the overlooked backbone of patterns in nature. In the branches of trees, arrangements of leaves, the lineage of bees, the fruitlets of pineapples, in chamomiles,  and daisies. 


Botany: 

Pentagonal symmetry is one of the structural symmetries. It includes 5 parts around a central axis (72 degrees apart). It is seen in phyla, in the likes of green algae and the water net, "Hydrodyctyaceae Hydrodictyon" 
Moreover, the Fibonacci Sequence may also be observed in families of vascular plants, compositae. It includes the likes of sunflowers, daisies and asters. The have a disc-shape arrangement, composed of pentamerous (arranged in groups of five) florets centripetal to an involucrate head, surrounded by fanning petals on the outside. 

The sunflower displaying an equiangular spirals running clockwise and counterclockwise


It is observed that these bidirectional spirals intersect each other in a manner that the numerator and the denominator follow the Fibonacci Sequence. For example 2/3, 3/5, 5/8, 8/13, 13/21, and so on, successively. 

According to Mathai's and Davis's  1974 research findings, "individual flowers emerge at a uniform speed at fixed intervals of time along a logarithmic spiral, with an initial angle at 137.5"

More about The Fibonacci Sequence in Nature and Botany:

Wilhelm Hofmeister suggested (in 1868) that new cells always tend to form in the least crowded/filled spots on the meristem. 
As stated by Seewald, each sequential structure of a plant still in growing process forms at the meristem, and then branches radially outwards, at a rate proportional to the stem's growth.  The second structure/primordium the farthest from the first, and the third grows as far as possible from the first and second primordia. Seewald said, "As the number of primordia increases, the divergence angle eventually converges to a constant value" (of 137.5). 
 Akhtaruzzaman and Shafie, two scientist brothers from the 1830s, discovered that the rotation of these tends to be at an angle derived from a fraction of two sequential Fibonacci Numbers, such as 1/2, 2/3, 2/5, 3/8, 5/13 etc.

the spiral aloe. 










The mathematical growth pattern of plants was first noticed by Theophrastus (ca. 372 B.C.- ca. 287 B.C.), however, the phenomenon was called "phyllotaxis" (leaf arrangement in Greek) by Charles Bonnet (a Swiss Naturalist) in 1754. 

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