Tuesday, 28 June 2011

She loves me, she loves me not.

Pulling the ray florets off a daisy head one by one, while chanting the rhyme above, is supposed to help lovers to know if their love is reciprocated.  Clearly, odd numbers of rays will yield "she loves me", while even numbers will yield "she loves me not".  But ray number on daisies isn't random.  If you count the rays on heads of different species in the daisy family, you'll find some numbers keep cropping up again and again: 5, 8, 13, 21, 34, 55 (although individual heads might vary a bit around a mean that is one of these numbers).  Pick your daisy carefully and you'll get the answer you want.
She loves me, she loves me not, she loves me, ...
Leonardo of Pisa, a.k.a. Leonardo Fibonacci, was an Italian mathematician who introduced Arabic numerals to Europe in the early 13th century.  Every time you do arithmetic, remember it's because of Leonardo that you don't have to use Roman numerals (e.g., CXCVII + MDMLXVIII = MDCCXV).  He also introduced the number series that's named after him, originally demonstrated to predict the reproductive output of a pair of rabbits.  The Fibonacci series crops up again and again in biology, as in our daisy heads.  
In botany, one common place where we see it is in the arrangement of leaves on a stem, referred to as phyllotaxis. Leaves may be attached one at a node (alternate or spiral), two at a node (opposite), or three or more at a node (whorled).
Hebes, like this Veronica benthamii from Campbell Island, have opposite leaves, with each pair at right angles to the one below, an arrangement known as decussate.

Alternate leaves vary in how steeply they're spiralled.  When each leaf is directly above the one below (one complete turn between leaves), it is called monostichous, if it is at 180° to the one below (half a turn between leaves), so that they form 2 rows of alternate leaves, they are called distichous, and if they form an angle less than 180° they are referred to as spiral.
You can describe the spiral by finding two leaves that are one above the other. Now count the number of spirals around the stem between the two leaves, then the number of leaves in the spiral(s) (count the lower leaf of the pair as 0 and the number you’re after is the number of the upper leaf). You can then describe the phyllotaxis as a fraction, where the numerator is the number of turns of the spiral and the denominator is the number of leaves, eg, 2/5, 3/8.  
If you examine enough of these spirals, you’ll find the same numbers crop up again and again. These numbers are the Fibonacci series: 1,1,2,3,5,8,13,21,34...., which is made by adding two adjacent numbers to get the next one in the series. In plants, the phyllotaxis fractions that are found have the denominator as the next number but one in the series from the numerator, e.g., 2/5, 3/8, 5/13, 8/21, etc.

2/5 phyllotaxis
 Looking down on the stem from above, the angle between two adjacent leaves is given by the phyllotaxis fraction (1/1 gives 360° for monostichous leaves; 1/2 gives 180° for distichous leaves; 144° for 2/5 phyllotaxis. These fractions approach, but never exactly reach, 137° 30’ 28” of arc, and that’s the angle that results in the least amount of leaf shading along a stem. Finally, the really cool thing about these angles is they’re related to the Golden Ratio, widely used in Greek and Roman architecture and modern paper shapes[1]. If the angle between one leaf and the next is A, and B is 360–A, then the ratio of A:B is the same as the ratio of B:360.

Spiral phyllotaxis on a female cone of kauri (Agathis australis)
Many other spiral structures in nature follow the Fibonacci series, e.g., the arrangement of eyes on a potato or scales on a pine cone; it even describes the spiral of a gastropod shell. Note that a mathematical description, however elegant, is not an explanation. So why is the Fibonacci series so widespread in biology?  Spiral phyllotaxis according to the Fibonacci series allows these arrangements to keep the same shape regardless of their size, and this means the arrangement of the primordia at the tiny stem apex is just the same as the arrangement of the mature leaves on a thickened stem.

[1] The A series of paper sizes (e.g., A4) is organized on the golden rectangle.  Thus A5 is half the area of A4, but exactly the same shape, and is obtained by cutting A4 in half cross-wise.  A0 has an area of 1m2.

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