At first thought, nature looks confusing and full of chaos. It is densely wooded, and numerous plants have rooted down in random places. There are inestimable species of insects we don’t know, and animals howling; the world is rowdy and replete with peculiar things that are hard to describe.
People usually say what is not artificial is natural. Unbeknownst to anyone, people regard nature as coincidental, where rules and forms are unsearchable. However, nature has ideal patterns from the smallest atoms, crystals, plants, animals, human bodies, weather, and even to the biggest cluster of galaxy. Though you name anything in nature, it forms a small pattern and the pattern forms a part of another bigger pattern.
Among the phenomenal and various appearances of living creatures, let us take a look at the angle between two leaves in plants. It is called divergence. Most people overlooked them, but Leonardo da Vinci the genius of the Renaissance found this pattern for the first time with his outstanding observation and his intellectual curiosity as a scientist.
He viewed nature as a masterpiece of the Creator and wrote, “For many plants, the sixth leaf always comes out from above the first leaf.” Leaves arise in a spiral shape along a stem, and the arrangement has regularity.
So far, it has been confirmed that plants have a certain divergence according to their species. For example, as for bamboos and grass, a leaf overlaps on the 2nd leaf; as for cyperus microiria and alder trees, on the 3rd leaf; as for cherry trees and apple trees, on the 5th leaf; as for roses, on the 8th leaf; and as for pine trees, on the 13th leaf.
When we consider the 1st leaf as 0, a plant has a rule where its leaves overlap on the 2nd leaf or the 3rd or the 5th or the 8th or the 13th, and the divergence is 180º, 120º, 144º, 135º, and 138.4º. There are very amazing patterns in the number of petals as well as leaves. Almost all the flowers have a peculiar permutation such as 3, 5, 8, 13, 21, 34, . . . in their petals.
For instance, lilies have 3 petals, wild roses have 5 petals, cosmoses have 8 petals, and marigolds have 13 petals. Sometimes, the number of petals does not always match with these numbers, but it is extremely mysterious that only some of the limited numbers are generally observed.
The number of the petals in flowers definitely has a certain pattern; each number can be obtained by adding up the previous two numbers. To give you an idea, it goes 3+5=8, 5+8=13, . . . The reason these numbers of petals look familiar is that they are in accord with the divergences of plants whose leaves overlap on the 2nd leaf, 3rd, 5th, 8th, 13th . . . This is Fibonacci sequence 1.
1. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34 . . . Each subsequent number is the sum of the previous two. It was found by Fibonacci, an Italian mathematician. The number of petals or the number of sunflower seeds follows this sequence, and so does the structure of many other living organisms such as nautilus.
Many products for daily use, such as credit cards and A4 paper, are designed almost in the golden ratio.
This mysterious pattern can be seen in pine cones as well. When you follow along the cones, it makes two different kinds of spirals—clockwise and counter-clockwise, and the numbers of cones are usually 5 and 8, or 8 and 13.
Not only that, but we can also find criss-crossing spirals in the the sunflower head. The spirals of sunflower seeds also vary, depending on the flower’s size; but if it has 21 seeds, then it has 34 seeds to the other direction, and 34 and 55, 55 and 89 . . .; there are always two neighboring Fibonacci sequences.
Fibonacci sequence draws people’s attention because of the golden ratio (1:1.61803…). The golden ratio means that the two quantities’ ratio is the same as the ratio of their sum to the larger of the two quantities. Among the two numbers successively obtained in the Fibonacci sequences, if the latter number is divided by the former number, it comes close to the golden ratio.
And if the numbers get bigger in Fibonacci sequences, the ratio gets closer to the golden ratio. This golden ratio is the most ideal and beautiful ratio in human eyes, and it was used in many works of art such as the ancient Greek Parthenon, Mona Lisa and The Last Supper of Leonardo da Vinci, and Tableau of Mondrian.
From a rectangle which has the golden ratio, if you draw a curve from one corner to the other through the squares that can be drawn infinitely, the curves make a shape of a clam; even a new form of pattern exists, transcending the numerical concept which is easily noticeable.
A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. The structures of clouds, snow crystals, lightening, stems, leaves, veins of leaves display fractals2. No matter how much you zoom in to uncover finer, nothing changes and the same elaborate, organized pattern reappears over and over.
2. Fractal: A small and simple infinite repetitive structure that makes a whole complex, mysterious structure. It has self-similarity and recursion. In nature, a rias coast, distribution form of animal blood vessel, twigs, the way frost gets formed, the shape of mountain ranges are all fractals, and everything in the universe ultimately has the fractal structure.
When it comes to a formulaic form or a certain pattern, people usually think of what humans made, such as an electronic circuit, or the structure of a crowded city seen from the sky, or a geometrical pattern of cloth. Moreover, numerical concepts and geometry have been considered as high dimensional and artificial, so that only humans can do.
However, the world which God created is an elaborate world which humans cannot even fathom with their ability. In comparison with an elaborate, complex structured mosquito, a jumbo jet that requires human cutting-edge technology and expertise is nothing more than a toy airplane for children.
The fine structure of the wing scales of butterflies, waves made on the water or in the atmosphere, stripes of tigers and zebras, movements of insects and animals, movement of celestial bodies . . . They all have regularity. We’ve thought that mathematics and science proves the outstanding ability of humans, but we’ve actually discovered only little rules from what God had made.