Mathematics is not just numbers and equations—it’s a language that nature uses in various aspects to make herself attractive and beautiful. To further increase the enchanting beauty a mathematical number ‘Golden Ratio’ is often used by the nature. This ratio also known as the golden number, golden proportion, or the divine proportion, is a mathematical constant equals to approximately 1.6180339887 and is expressed by the Greek letter φ. It is a real but irrational number that cannot be expressed as a ratio of two integers and has an infinite decimal expansion. It is strongly associated with the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1.
It can be explained in such a way that if a line is split into two parts – a longer and a shorter segment, the ratio of the whole line to the longer segment is equal to the ratio of the longer to the shorter segment. Similarly, if 360 degrees of a circle is divided into a 137.5 and a 222.5 piece, the ratio between the longer to the shorter arc is about the same as the golden ratio. Here, 137.507764°, is called the golden angle. In the same way, a golden rectangle satisfies that length/width = (length + width)/length. One interesting property of the golden rectangle is that if a square is chopped off from one end, the remaining (smaller) rectangle is also a golden rectangle, i.e., the ratios of the sides remain the same. Continuing this process of chopping off squares, a “nested” sequence of golden rectangles is obtained with the “golden spiral” fitting perfectly.
The golden ratio is the most artistically pleasing of all proportions. Its fantastic association with the aesthetic beauty and harmony, ability to provide visual balance, and universal acceptability have made many of the famous mathematical scholars, artists, musicians, historians, and architects of all ages think about the number, and inspired them to do a lot of work on it. From Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figure Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. They have investigated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines so much that no other number in the history of mathematics could do.
The inherent efficiency of consistency, sense of proportionality, and symmetry of this number influences human perception, making it naturally pleasing and aesthetically attractive to the human eye, thus making it acceptable and usable in various fields like science, art, beauty, architecture, and other design-related fields.
Use of the Golden ratio In the structural and functional design of many astronomical and natural processes is absolutely striking. The spiraling pattern of the galaxies, the distribution of the planets in the solar system, some of the complex processes inside the atomic world, the destructive pattern of the tornadoes, waves breaking on the beach, the tail of a comet as it winds around the sun, whirlpools etc., exhibit astonishing use of this number. It is observed that based on the golden number, these spirals follow a precise mathematical pattern.
The use of the golden number and the spiral pattern is visible in the structural design of many animals and plants. The spiraling pattern of leaves around the stem assures each leaf to receive maximum exposure to sunlight and air without shading or crowding from other leaves. Use of this number is seen in the spiral seed patterns in sunflowers and daisy flowers. Looking carefully at a sunflower, it will be observed that two sets of spirals (rows of seeds or florets) are spiraling in opposite directions. When these spiral rows are counted in their opposite directions, it is seen that in the overwhelming number of the cases these numbers, depending upon the size of the flower, will consistently be of the following ratios: 42 if small, 34 and 55; if medium 55 and 89; if large 89 and 144 if small, 34 and 55; if medium 55 and 89; if large 89 and 144. Similarly, this number is seen in the spiral rows of the oval-shaped pine tree fruits.
Not only in plants and animals; it is woven into our own anatomy in ways we’ve never imagined. The Golden Ratio could be inherently efficient or stable, making it a preferred form for self-replicating molecules, which often makes it associated with natural growth patterns and may represent a fundamental principle in the formation of complex biological structures. This principle has extended the golden ratio’s influence even to the blueprint of human life itself, DNA. In the structure of the DNA molecule, each complete cycle of its double helix spans 34 angstroms in length and 21 angstroms in width, following the divine proportion.
It is a general belief that a person with numerous golden proportions in their facial features is more beautiful. Some measurements of the human body, such as the height between the navel and the foot, overall height and the length of the forearm, the distances between forearm to the hand, the shoulder to the elbow, elbow to the wrist, chin to lips, lips to tip of nose, tip of nose to pupils, pupils to the top of head, the spiral-shaped cavity within the inner ear, and even the bones in each finger exhibit the golden ratio in their relative lengths.
Golden ratio is seen among insects and animals too. The golden spiral is most commonly observed in shells, ram’s horn, sea-horse tail, spiral patterns of webs built by spiders, the placement of the fins and tail relative to the body length of the dolphins, in butterfly wings, and also in the spiral coiling of their proboscis (feeding tube), five points of a pentagon-shaped starfish, penguin’s body, and even on the face of a tiger and in the construction of the ears of apparently all mammals.
As golden proportion is remarkably attractive and pleasing to the human eye this ratio has been used by the ancient artists and architects of Greece and Egypt, even the famous artists like Leonardo da Vinci, Van Gogh, Vermeer, John Singer Sargent, Monet, Whistler, Renoir, Mary Cassatt, Giotto, Durer, and others employed this number in their works. The ratio is also evident in Great Pyramid of Egypt, the Parthenon in Greece, the aesthetic design of the Taj Mahal in Agra, and as the basic shape for many of the ancient pictures, vases, doorways, windows, and statues. This ratio has also been used in the design of the United Nations building in New York, USA.
Artists of the modern world are also prone to using this ratio in their artistic creations. Many things of our daily use are (approximately) patterned after the golden rectangle—credit cards, playing cards, postcards, light switch plates, writing pads, 3-by-5 and 5-by-8 index cards, etc., are good examples. Recently graphic designers are becoming inspired to use this divine proportion in their artistic works to make them more attractive.
Many expert musicians believe that if musical time is divided into periods based on the golden proportion to determine the beginnings and endings of themes, moods, texture, and so on, they will be more melodious and euphonic. The famous musicians like Beethoven, Bartok, and Garland have given importance to this golden ratio in their compositions.
Though nature uses the golden ratio as one of her most indispensable measuring rods, absolutely reliable, yet never without variety, producing perfect stability of purpose without the slightest risk of monotony.
Dr. Zakia Begum, Nuclear Scientist and Professor, CSE Department, Eastern University, Savar, Dhaka.
