By: Ranilo Abando (Manila, November 14, 2020) ****
Benoit Mandelbrot is a mathematician who coined the word “fractal” in 1975 to refer to a shape that appears similar at different levels of magnification. One of the simplest mathematical expressions of Mandelbrot’s set is: Z = z2 + c, in which the equals sign actually has small arrows with the upper and lower pointing in opposite directions. I will not go into the details of the equation because the main point here is: by using a simple and elegant mathematical expression, you can generate a fractal image which when zoomed in, you descend into beautiful infinity. Below is the image generated by a Mandelbrot set.
Many fractals possess the property of self-similarity, at least approximately if not exactly. A self-similar object is one whose component parts resemble the whole. Fractals occur in nature and the following are some examples of them.
A fern leaf shown above exhibits fractal properties. The leaf pattern is self-similar across smaller scales.
The gas bubbles above exhibit different degrees of order. They are not only separated from the water, which is a very basic degree of order, they also exhibit a more complex order by having fractal characteristics in the sizes of the bubbles. Fractals indicate a higher degree of order. If fractals can be expressed by a mathematical formula, then it means that they follow rules. Rules are indications of order. Total chaos does not follow any rule.
Above, a mountain range exhibits fractal pattern. The whole peak shown above is actually composed of smaller peaks that look similar to it.
Above, the natural pebbles in a beach exhibit fractal composition. The smaller and smaller pebbles look similar to the bigger ones.
Above, the tree branches exhibit fractal patterns. The branches are self-similar even at smaller and smaller-scale.
Above, the veins of this fallen autumn leaf exhibit fractal patterns. The bronchial tree of the human lung shows similar patterns.
Rules are expressions of order in this physical reality. For example, the Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … where the next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1). The 3 is found by adding the two numbers before it (1+2), and the 5 is (2+3), and so on. This simple rule is expressed in many natural things that occur in nature, and is often used by gifted artists in their works of art. The Fibonacci appears in some of the largest, to the smallest objects in nature — from a spiral galaxy to the microscopic ovary of an Anglerfish. Many scientists are puzzled why is that so. The answer is: the Fibonacci Rule is an expression of order. In fact, all rules are expressions of order. And all expressions of order originate from the most fundamental level of reality – the spiritual level.
Above, the succulent Aloe Plant exhibits the Fibonacci Spiral.
Above, the Nautilus shell exhibits the Fibonacci spiral, a relatively higher degree of order and, therefore, an expression of beauty.
Above, the Sunflower exhibiting Fibonacci spirals. Higher degree of order expresses beauty. In this physical world, rules such as that of gravity produce large-scale order like the formation of beautiful planets, stars and spiral galaxies.
Discussion:
The occurrence of fractal patterns in Nature means that everyday things that look disordered and random actually have hidden order or pattern in them. It means they can be mathematically expressed. That intrinsic order within ordinary things in this world is just a proof of influence from the spiritual level, the ultimate source of all expressions of order. Although physical things are virtual in nature, they are often fractal expressions of much deeper realities. Thus, by simply studying the patterns in everyday life, it will give us some basis or idea of the truths that lie beyond this physical world.
In my long study of the nature of reality, I have found out that it is not only the physical things that are fractal but also non-material or abstract concepts. For example, music is apparently a fractal of the ebbs, crests, and flow of human life. It seems that the human cycle of day and night is a fractal of the human cycles of death and rebirth, which in turn are fractals of the mega-cycles of evolution – the metaphorical Day and Night of Brahma. Apparently, Edgar Allan Poe, the famous American writer, had stumbled upon the fractal relationship between nightly sleep and physical death when he wrote – “Sleep, those little slices of death — how I loathe them.”
Shown above is a drawing that reproduces an actual ancient petroglyph found in Arizona, U.S.A. The Double Spiral symbol is interpreted here as representing the connected human spirit and his main spirit above the mind column. The smaller spiral inside the head of the human figure represents the human spirit, a smaller-scale fractal of his main spirit which is represented by the bigger spiral shown above the human figure.
In the spiritual level, it seems that the human spirit is a smaller-scale fractal of more evolved spirits dwelling in the realities high up the mind column, while the animal spirit is a smaller-scale fractal of the human spirit. Likewise, the human mind is a fractal of more evolved minds, giving hope to everyone of us that we have the potential to be much much better than we are now.
“The universe is in a grain of sand”. This common saying can be construed as an expression of the fractal nature of existence.
(Thanks to pixabay.com for the photos above, except the fern leaf photo which is thanks to pexels.com – Tausif Hossain)