15 Uncanny Examples of the Golden Ratio in Nature

This is divided into a square, labelled 21, and another, smaller, horizontal rectangle. The square labelled 21 is overlaid with another quarter circle, from the top left, to the bottom right corner. It is overlaid with a curved blue line from the top right to the bottom left. The vertical rectangle is further divided into a square labelled 8, and a horizontal rectangle that is divided again. Next to it, another vertical rectangle contains a square labelled 3 and a smaller horizontal rectangle, which, in turn, contains a square labelled 2. The blue line continues to curl smaller across these shapes.

Shown is a colour diagram of a rectangle divided into a pale purple square and a smaller pink rectangle. The top and left edges of the square are each labelled with a blue, lower case, italic a. The top edge of the smaller rectangle is labelled with a red, lower case, italic b. The entire bottom edge of the larger rectangle is labelled with a + b, in green italics. Look at the array of seeds in the center of a sunflower and you’ll notice they look like a golden spiral pattern. Amazingly, if you count these spirals, your total will be a Fibonacci number.

(This equation has two solutions, but only the positive solution is referred to as the Golden Ratio \(\varphi\)). Therefore, the golden ratio may be the fundamental constant of nature. Mozart made use of the Golden Ratio when writing a number of his piano sonatascloseSonataA piece of instrumental music, usually for a solo instrument, or a small group.. In Mozart’s sonatas, the number of bars of music in the latter section divided by the former is approximately 1.618, the Golden Ratio. For instance, phi enthusiasts often mention that certain measurements of the Great Pyramid of Giza, such as the length of its base and/or its height, are in the golden ratio. Others claim that the Greeks used phi in designing the Parthenon or in their beautiful statuary.

We celebrate Fibonacci Day Nov. 23rd not just to honor the forgotten mathematical genius Leonardo Fibonacci, but also because when the date is written as 11/23, the four numbers form a Fibonacci sequence. Leonardo Fibonacci is also commonly credited with contributing to the shift from Roman numerals to https://1investing.in/ the Arabic numerals we use today. The famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientists for centuries. Also known as the Golden Ratio, its ubiquity and astounding functionality in nature suggests its importance as a fundamental characteristic of the Universe.

1. Since both the terms are not equal, we will repeat this process again using the assumed value equal to term 2.
2. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone.
3. This spiral gets wider by a factor of 1.618 every time it makes a quarter turn (90°).
4. Mathematician George Markowsky pointed out that both the Parthenon and the Great Pyramid have parts that don’t conform to the golden ratio, something left out by people determined to prove that Fibonacci numbers exist in everything.
5. The Parthenon in Athens and Leonardo da Vinci’s Mona Lisa are regularly listed as examples of the golden ratio.

And that is why Fibonacci Numbers are very common in plants.1, 2, 3, 5, 8, 13, 21, … So that new leaves don’t block the sun from older leaves, or so that the maximum amount of rain or dew gets directed down to the roots. The spiral happens naturally because each new cell is formed after a turn. Indian poets and musicians had already been aware of the Fibonacci sequence for centuries though, having spotted its implications for rhythm and different combinations of long and short beats.

There are many examples of the golden ratio in nature — yet many people have no idea what it is or how to appreciate the planet’s stunning geometry. This might be because the US as a nation, does not appear to excel in the subject of mathematics. As Hart explains, examples of approximate golden spirals can be found throughout nature, most prominently in seashells, ocean waves, spider webs and even chameleon tails! Continue below to see just a few of the ways these spirals manifest in nature.

But a Golden Spiral is made by nesting smaller and smaller Golden Rectangles within a large Golden Rectangle. We can take the Golden Rectangle one step further by adding a line that forms a quarter circle in each square. The rectangle has a long side of a + b and a short side of a. In the second month, one pair of rabbits move in, but they don’t have any babies for the first two months. By the sixth month, both the first and second pairs are having a pair of babies every month. Additionally, if you count the number of petals on a flower, you’ll often find the total to be one of the numbers in the Fibonacci sequence.

The second fascinating thing about Fibonacci numbers is, like the golden ratio in nature, that we see them everywhere. This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure.

Practice Questions on Golden Ratio

DNA molecules follow this sequence, measuring 34 angstroms long and 21 angstroms wide for each full cycle of the double helix. Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers.

Flowers and Branches

Note that we are not considering the negative value, as \(\phi\) is the ratio of lengths and it cannot be negative. Since both the terms are not equal, we will repeat this process again using the assumed value equal to term 2. The value of the golden ratio can be calculated using different methods. Why don’t you go into the garden or park right now, and start counting leaves and petals, and measuring rotations to see what you find.

Why Does the Fibonacci Sequence Appear So Often in Nature?

The second primordium grows as far as possible from the first, and the third grows at a distance farthest from both the first and the second primordia (Seewald). In the 1830s, scientist brothers found that the rotation tends to be an angle made with a fraction of two successive Fibonacci Numbers, such as 1/2, 1/3, 2/5, 3/8 (Akhtaruzzaman and Shafie). “As the number of primordia increases, the divergence angle eventually converges to a constant value” of 137.5° thereby creating Golden Angle Fibonacci spirals (Seewald). New growth may simply form spirals so that the new leaves, petals, and branches will not block older leaves, etc. from sunlight or air, or so that the maximum amount of rain or dew will get directed down to the roots (Akhtaruzzaman and Shafie).

Shown is a black and white illustration of a rectangle divided into smaller squares and rectangles that get smaller as they move around the page, towards a spot in the bottom right quadrant. The smallest square is not labelled, but it looks like this pattern could continue, becoming smaller and smaller with each iteration. In 1868, Wilhelm Hofmeister suggested that new cells destined to develop into leaves, petals, etc. (primordia) “always form in the least crowded spot” on the meristem (growing tip of a plant). Each successive primordium of a continuously growing plant “forms at one point along the meristem and then moves radially outward at a rate proportional to the stem’s growth” (Seewald).

But it’s always useful to step outside a particular perspective and ask whether the world truly conforms to our limited understanding of it. The dimensions of architectural masterpieces are often said to be close to phi, but as Markowsky discussed, sometimes this means that people simply look for a ratio that yields 1.6 and call that phi. Finding two segments whose ratio is 1.6 is not particularly difficult. Where one chooses to measure from can be arbitrary and adjusted if necessary to get the values closer to phi.

Objects designed to reflect the golden ratio in their structure and design are more pleasing and give an aesthetic feel to the eyes. It can be noticed in the spiral arrangement of flowers and leaves. Emergence theory is based deeply on the notion of language. A powerful language is one which has the ability to express the golden ratio in nature maximum amount of meaning with the least number of choices, since each choice requires resources. A resource in this sense can be a unit of electricity spent for a logic gate to be opened to activate a binary choice in a computer language, or a calorie or two of energy needed to make a mental choice of what shirt to wear.

The Fibonacci sequence works in nature, too, as a corresponding ratio that reflects various patterns in nature — think the nearly perfect spiral of a nautilus shell and the intimidating swirl of a hurricane. Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral). For example, the ratio between two pine needles is 0.618, as well as the ratio of leaf venation.