Omnipresent phyllotaxis

All in the Nature is subordinated to the stringent mathematical laws. It appears, that the arrangement of leafs on stems of plants also has a stringent mathematical nature and this phenomenon is called in botanic by "phyllotaxis". The essence of phyllotaxis consists in screw arrangement of leafs on plant stems (branches on trees, petals in racemes etc.).

In the phyllotaxis phenomenon the more complicated concepts of symmetry, in particular, the concept of the "screw axis of a symmetry", is used. Let's consider, for example, arrangement of leafs on the plant stem (Fig.1). We see, that the leafs are at different altitudes of the stem along the screw curve winded around of its stem. To pass from the underlying leaf to the next one it is necessary mentally to turn the leaf on some angle around of the vertical axis and then to raise it on a definite distance up. In it the essence of the "screw symmetry" consists.

The 'screw symmetry'
Figure 1. The "screw symmetry".

And now let's consider characteristic "screw axes" arisen on plant stems (Fig.2). In Fig.2- the stem of plant with the symmetry screw axis of the third order is shown. Let's observe the line of leaf-arrangement in this figure. To pass from the leaf of 1 to the leaf of 2, it is necessary to turn the leaf of 1 around of the stem axis on 120° counter-clockwise (if to look from below) and then to move the leaf of 1 along the stem in vertical direction so long as it will be combined with the leaf of 2. Repeating similar operation we can pass from the leaf of 2 to the leaf of 3 and then to the leaf of 4. It is necessary to attract attention to the fact that the leaf of 4 lies above of the leaf of 1 (as though repeats it, but its level is higher). Note that moving from the leaf of 1 to the leaf of 4 we made turn triply on the angle 120°, i.e. we executed the full revolution around of the stem axis (120° ´ 3 = 360°).

The screw stem axis of symmetry
Figure 2. The screw stem axis of symmetry.

Botanists are called the turn angle of the screw axis as the "leaf divergence angle". The vertical straight line connecting two leafs arranged one the stem one above another is named the "ortho-line". The line segment 1-4 of the "ortho-line" corresponds to the full translation of the screw axis. As we will see further a number of the revolutions around of the stem axis for transition from the lower leaf to the upper one arranged exactly above lower (on the "ortho-line") can be equal not only 1, but also 2, 3 and so on. This number of the revolutions is called the "leaf cycle". In botanic it is custom to characterize the screw leaf-arrangement with the help of some fraction; the numerator of the fraction is equal to the "leaf cycle" and the denominator to a number of leafs in this "leaf cycle". In the case considered above we have the screw axis of the kind 1/3.

Fig.2- demonstrates the "pentagonal" symmetry screw axis with the "leaf cycle" of 2 (for transition from the leaf of 1 to the leaf of 6 it is necessary to make two full revolutions). The fraction describing the given axis is expressed by 2/5; the leaf divergence angle is equal to 144° (360° : 5 = 72°; 72° ´ 2 = 144°). Note that there are also more intricate axes, for example, of the kind of 3/8, 5/13 etc.

There is a question: what can be numbers a and b describing the screw axis of the kind of a/b. And here the Nature presents us the next surprise by the way of the so-called "Law of phyllotaxis".

Botanists assert that the fractions describing the plant screw axes form the stringent mathematical sequence consisting of the adjacent Fibonacci numbers ratios, that is:

1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, ... .(1)

Let's remind that the Fibonacci series is the following number sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... .(2)

Comparing (1) and (2) it is easy to see that the fractions in the sequence of (1) will be derivated by the Fibonacci numbers taken through one number.

Botanists established that the phyllotaxis fraction from the sequence of (1) are characteristic for different plants. For example, the fraction of 1/2 is peculiar to cereals, birch, grapes; 1/3 to sedge, tulip, alder; 2/5 to pear, currants, plum; 3/8 to cabbage, radish, flax; 5/13 to spruce, jasmine etc.

Gray paperomia
Figure 3. Gray paperomia.

What is the "physical" cause underlying the "Phyllotaxis Law"? The answer is very simple. It appears that just at such arrangement of leafs on the plant stem the maximum of the solar energy inflow to the plant is reached.

Taking into consideration this remark you will be not surprised also with that fact that practically all racemes and densely packaged botanic structures (pine and cedar cones, pineapples, cactuses, heads of sunflowers and many others) also strictly follow to Fibonacci numbers regularity.

The seeds in the sunflower head
Figure 4. The seeds in the sunflower head are placed on spirals, thus the number of the left-hand spirals to the right-hand spirals is equal to the ratio of the next Fibonacci numbers.

The echmeia raceme
Figure 5. The echmeia raceme fits to the stringent mathematical law based on Fibonacci numbers.

But not only the plants but also some animals, for example, the snakes use the same principles in organization of their exterior forms (see below the snake).

Thus, the stringent mathematics we can see and in arrangement of the rose petals and in the apple cut ("pentagram") and in the pinecone, and in the sunflower head. And we again and again are convinced that all in the Nature is submitted to the unified laws and to discover and to explain these laws is the main problem of human science.