golden ratio

1..We will call the Golden Ratio (or Golden number) after a greek letter,Phi () here, although some writers and mathematicians use another Greek letter, tau (). Also, we shall use phi (note the lower case p) for a closely related value. The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes . The designations "phi" (for ) and "Phi" (for the golden ratio conjugate ) are sometimes also used (Knott).
2..
The arrangements of leaves is the same as for seeds and petals. All are placed at 0·618034.. leaves, (seeds, petals) per turn. In terms of degrees this is 0·618034 of 360° which is 222·492...°. However, we tend to "see" the smaller angle which is (1-0·618034)x360 = 0·381966x360 = 137·50776..°. When we look at properties of Phi and phi on a later page, we shall see that
1-phi = phi2 = Phi-2

If there are Phi (1·618...) leaves per turn (or, equivalently, phi=0·618... turns per leaf ), then we have the best packing so that each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible area exposed to falling rain so the rain is directed back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination.
The whole of the plant seems to produce its leaves, flowerhead petals and then seeds based upon the golden number.

And why do the Fibonacci numbers appear as leaf arrangements and as the number of spirals on seedheads?
3...
Botanists have shown that plants grow from a single tiny group of cells right at the tip of any growing plant, called the meristem. There is a separate meristem at the end of each branch or twig where new cells are formed. Once formed, they grow in size, but new cells are only formed at such growing points. Cells earlier down the stem expand and so the growing point rises.
Also, these cells grow in a spiral fashion, as if the stem turns by an angle and then a new cell appears, turning again and then another new cell is formed and so on.

These cells may then become a new branch, or perhaps on a flower become petals and stamens.

The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. So, once an angle is fixed for a leaf, say, that leaf will least obscure the leaves below and be least obscured by any future leaves above it. Similarly, once a seed is positioned on a seedhead, the seed continues out in a straight line pushed out by other new seeds, but retaining the original angle on the seedhead. No matter how large the seedhead, the seeds will always be packed uniformly on the seedhead.

And all this can be done with a single fixed angle of rotation between new cells?
Yes! This was suspected by people as early as the last century. The principle that a single angle produces uniform packings no matter how much growth appears after it was only proved mathematically in 1993 by Douady and Couder, two french mathematicians.
4...The Rhind Papyrus of about 1650 BC is one of the oldest mathematical works in existence, giving methods and problems used by the ancient Babylonians and Egyptians. It includes the solution to some problems about pyramids but it does not mention anything about the golden ratio Phi.
The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1·6. It is a matter of debate whether this was "intended" to be the golden section number or not.
According to Elmer Robinson
using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi.