Assistant Professor - Department of Physics Kalinga University, New Raipur
An early mediaeval mathematician made a sarcastic allusion to the rapidity with which rabbits multiply in his Fibonacci series. However, it has evolved into a valuable resource for learning about the arts, architecture, the natural world, and how things work. When it comes to the structures of leaves and lungs, this mathematical game explains it all. It’s also used as a foundation for the Parthenon and the pyramids. Find out how and why the Fibonacci sequence keeps appearing.
The Origin of the Series
Leonardo Fibonacci, a twelfth-century Italian mathematician, is the originator of the Fibonacci Series. He was trying to figure out the appropriate growth rate for a pair of rabbits over the course of a year. After one month of maturation, he anticipated each couple would generate another pair. bunnies were born in January, and they would achieve maturity by the next February, giving rise to a new litter of rabbits by the following March (2). Finally, they would mate once more in April and May and produce two more sets of offspring each. The rabbits born in March would mature in April, resulting in two fresh pairs of bunnies being born in May, making it a total of five pairs. Finally, all of the rabbits born between January and April would be able to contribute to June’s total of eight pairs.
With each new pair of Fibonacci Series, the expansion would continue, with each new pair maturing and starting their own. If no fatalities occurred during this time period, the rabbit pair expansion would take the following form:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . .
Everybody can see that by December, he or she will be overrun with rabbits. Each new number in the sequence is a combination of the two numbers before it, as can be seen by the attentive reader. Thirteen is the sum of five and eight. The sum of eight and thirteen is equal to twenty-one, and so on and so forth.
Fibonacci Goes Gold in Art and Architecture
As a result of this, many people would shrug and make mental notes to never allow Fibonacci near their rabbits again. However, it turns out that he was right on the money. This series of numbers was coated in gold by mathematicians and painters. To begin, I divided each number in the series by the one before it. The outcomes don’t appear noteworthy at first glance. A number split by a number equals a number. Two is equal to one divided by one. 1.5 is the result of dividing three by two. What a read. Odd things start to happen as the number of steps grows. Three times five equals 1.666. 1.6 is the result of dividing 8 by 5. The answer is 1.625 when you divide 13 by 8. Thirteen divided by twenty-one equals 1.615.
As the series progresses, the ratio of the most recent number to the most recent number approaches 1.618 as a whole. It gets closer and closer to 1.618, but it never quite reaches that ratio. There are numerous examples of what has been dubbed “The Divine Proportion” or “The Golden Mean” in art and architecture.
The Golden Rectangle was built by the Greeks using the 1.618 ratio. a rectangle with sides of one and 1.618 was formed (or with side measuring to consecutive Fibonacci Numbers). This was widely employed in architecture because it was seen to be the most mathematically beautiful structure. It’s no secret that the Parthenon features a plethora of Golden Rectangles throughout its design and construction. In addition, the Pyramids’ Divine Proportions are unique. The sloping sides of the Pyramids are 1.618 units long, and the height is equal to the square root of 1.618 units long if the base of the Pyramids is used.
The use of golden proportions has become common in modern photography and painting. The two sides of a Golden Rectangle are the Fibonacci Numbers. The smaller the rectangle, the more Fibonacci-proportioned the smaller the rectangles and squares will be. Objects of this size and shape are common in many works of art.
Curves, on the other hand, are a different storey. To me, that’s where the Fibonacci sequence really comes into its own! From one of the stacked squares to the other, draw an arc in the middle. The Golden Spiral is formed when the number of squares in each row and column is increased to an ever-increasing number. Outside of a gallery, you’ll notice spirals all over the place..
Fibonacci Spirals in Nature:
Until now, the Fibonacci Series has appeared only because people are obsessed with a particular set of numbers. Although the original problem was demonstrated by rabbits, everyone knows that the population doesn’t grow that way. Rabbits don’t always give birth in pairs, and despite their reputation as prolific breeders, they don’t always succeed in getting pregnant.
Real-world Fibonacci sequences can be found in the plant kingdom, where they can be seen in abundance Fibonacci numbers appear in many plants that branch out toward the sun. The first sprig of a plant is born from the ground. For the first few months, it only grows from the ground. Branches sprout from meristem points, which are locations from which new branches can grow. Those branches continue to rise for a little more, and finally form two distinct points of their own. Using the Fibonacci sequence, the total number of sprouting points grows.
The spirals that the Fibonacci Series produces are the most well-known. When you look at the petals of a cauliflower or fruitlets on a pineapple, you’ll notice that they all spiral outward. The number of seeds, florets, bumps, leaves, or tubercles in each of those spirals is exactly equal to the number of Fibonacci numbers. While it’s possible some people choose to disregard certain species of spiraling plants because they don’t spin out their seeds according to the Fibonacci number sequence, we now know why: it’s the best method to pack the seeds.
In order for the seeds of a sunflower to thrive, they must be given the same amount of room. They need to be packed as evenly as possible and in the most efficient manner. Why do plants use spirals to cram seeds into a small area? Because they don’t construct a structure and then stock it with seeds, as we do when we build a storehouse. During the bulb’s growth, it produces seeds.
These spiraling seedpods are the result of plants growing seeds at the centre and then expanding the space they occupy. If the seeds were placed one on top of the other in the flower pod, the stem would be strained because the seeds would be squashed together on the top and bottom. Seeds mature in a circular pattern on the bloom because the flower positions the growing seeds at an angle to one another and allows them to spread outward. However, how may this be done in the most effective manner? The fourth seed would be deposited immediately underneath the first seed if the flower made four seeds for every circular ‘turn.’ This would result in four rows of seeds pushing outwards. Even while it’s better than a straight line, it’s still a waste of valuable real estate. The optimal number of seeds to deposit per turn is roughly 1.618, according to the results. There are no seeds buried directly under their predecessors since flowers can only produce whole numbers of seeds. The seeds, on the other hand, spiral outwards from the core.
A Fibonacci number can be found by counting the seeds in any spiral because of the Divine Proportion of 1.618 seeds per turn. Plants that produce their leaves from a single stalk also display the Fibonacci Sequence. The number of finished leaves per ’round’ around the stem will be a Fibonacci number as the plant develops higher.
Fibonacci For Fun:
When mathematicians get their hands on a problem, they hold it tight. Fibonacci’s series features a number of oddities, but they’re also fascinating. The total of any ten successive Fibonacci numbers, for instance, is divisible by eleven. Two terms prior, the squares of the two preceding Fibonacci numbers will provide the next Fibonacci number. Eight white, five black, and thirteen white and black keys make up an octave on a piano.
Over the course of history, the Fibonacci sequence has enthralled mathematicians and artists alike. Nature’s remarkable use of the Golden Ratio, often known as the ‘Golden Spiral,’ shows that it is an essential property of the cosmos.
The Fibonacci sequence begins with zero, one, one, two, three, five, eight, thirteen, twenty-four, and so on indefinitely. The sum of the preceding two numbers is the number itself. To me, it looks like the universe has an in-built numbering system. 15 amazing natural examples of phi.
In flowers, the number of petals follows the Fibonacci sequence in a predictable pattern. The flower, for example, has three petals; buttercups have five; the chicory has 21; the daisy has 34; and so forth. It’s because of Darwinian processes that each petal is located at 0.618034 of a 360° circle, allowing for the best potential exposure to sunlight and other circumstances, that Phi emerges in the petals.
The Fibonacci technique can also be used to the flower’s head. Once a seed is formed, it moves outwards to fill the rest of its environment. Spiraling patterns can be seen in abundance in sunflowers.
There have been reports of seed heads containing as many as 144 or more seeds in a single plant in rare situations. These spirals, when counted, tend to have a Fibonacci number as their total. A highly irrational number, it turns out, is necessary to optimize filling rates (namely one that will not be well represented by a fraction). Phi is a good fit for this description.
Pineapples and cauliflower also have similar swirling patterns.
One way to see this is by looking at the way tree branches grow or divide. Eventually, a tree’s main stem will sprout a branch, establishing two new growth points. While one of the new stems continues to grow, the other remains inactive. Each new stem follows the same pattern of branching. The sneezewort is a nice illustration of this. This pattern can be seen in everything from root systems to algae.
Another illustration is provided by the Golden Rectangle’s special properties. If the ratio of the sides (a/b) is equal to the golden mean (phi), this shape can be repeated infinitely and take on the shape of a spiral. The logarithmic spiral can be found all over the place in nature.
Snail and nautilus shells, as well as the cochlea of the inner ear, follow the logarithmic spiral. Other examples of this phenomenon include some goats’ horns and certain spider web shapes.
It’s no surprise that spiral galaxies follow the Fibonacci sequence. The diameter of at least one spiral arm of the Milky Way is approximately 12 degrees. Spiral galaxies, on the other hand, appear to defy Newtonian physics in this regard. Galaxies’ rotational angular speed varied with distance from the centre, which meant that their radial arms would bend inward as they rotated. Spiral arms are expected to form around the galaxies in the following rotations. Since they don’t, there’s a problem with the “winding” process. A unique feature of the universe is that the stars beyond our galaxy appear to be moving faster than expected, which helps to preserve its shape.
It is common to see the Golden Ratio in both human and nonhuman faces. The lips and nose of a person are perfectly in line with the eyes and jaw at each of these golden spots. When viewed from the side, it appears that the dimensions of the eye and ear are identical (which follows along a spiral).
Despite the fact that every person’s body is unique, the population as a whole tends to be phi. According to some theories on human attractiveness, we become more “beautiful” as our proportions get closer to phi. This grin is considered to be the “loveliest” when it is 1.618% wider than the other incisors, which are themselves 1.618% wider than the canines, and so on. Our predisposition to prefer physical forms that follow the golden ratio may be an indicator of reproductive health and fitness, as suggested by evolutionary psychology.
Using the ratio of phi, which is the length of our fingers, we can see that each finger area is larger than the previous one.
The Fibonacci sequence even governs the proportions of our bodies. Your navel-to-head distance is determined by a mathematical formula known as the golden ratio, which you can see by looking at it. The Golden Sections include the eyes, fins, and tails of dolphins, starfish, sand dollars, sea urchins, ants, and honey bees.
There are numerous ways in which honey bees follow the Fibonacci sequence. A colony’s female population can be divided by the number of males, as an extreme example (females always outnumber males). The answer is approximately 1.618. It’s no surprise that honey bee ancestry follows the well-established pattern. For males, there is only one biological parent, in contrast to females (a female and male). When it comes to great-great-grandparents and great-great grandparents, men have three times as many as women. It is the same for both sexes: 2, 3, 5, 8, 13, etc. Additionally, as previously stated, the physiology of bees follows the Golden Curve quite well, as well.
It’s at an angle to their flight path that a hawk gets the best view of its prey, which is also the spiral’s pitch.
Physician Jasper Veguts of the University Hospital Leuven in Belgium says that doctors can tell if the relative proportions of the uterus are normal and healthy by conducting a thorough examination of the uterus. According to the Guardian, this is the quote:
In the last six months, ultrasound measurements were used to calculate the length-to-width ratio of 5,000 women. A table displays the findings. During her early twenties, this ratio is roughly 2 and gradually decreases until it reaches 1.46 in her late forties, according to the data gathered.
When Dr. Verguts discovered that during the most fertile period of women’s reproductive lives, between ages 16 and 20, the length-to-width ratio of a uterus is approximately the golden ratio, he was delighted.
He was pleased with the outcome because it was the first time anyone had seen it.
Fibonacci can be found even in the tiniest of places. This is the ratio of the Fibonacci numbers of 34 and 21, which roughly corresponds to the ratio of Phi: 1.6180339. 6. Shellfish
Another illustration is provided by the Golden Rectangle’s special properties. If the ratio of the sides (a/b) is equal to the golden mean (phi), this shape can be repeated infinitely and take on the shape of a spiral. The logarithmic spiral can be found all over the place in nature.
Snail and nautilus shells, as well as the cochlea of the inner ear, follow the logarithmic spiral. Other examples of this phenomenon include some goats’ horns and certain spider web shapes.
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