Mathematically, the golden ratio is an irrational number, represented as phi (Φ). One way to find this amount is through the equation x2 – x – 1 = 0. Once solved, we find that:
The Golden Ratio is equal to 1.6180339887498948420…
An easier way of comprehending the golden ratio is through geometry, since people are more commonly exposed to the concept visually. In the simplest sense, consider a line segment that has been divided into segments a and b in such a way that their ratio is the same as the ratio of their sum to the larger of the two line segments.
This can be understood as a/b = (a+b)/a = 1.618 (phi).
One of the greatest applications of the golden ratio in geometry is the golden rectangle. This quadrilateral figure contains sides that are in proportion to the golden ratio (their ratio and the ratio of the sum of two nonparallel sides to the larger of the parallel sides is equal to 1.618). From its basis in this structure, the golden rectangle is certainly visually appealing, and it is recognized as one of the most perfect shapes that can be formed.
The golden rectangle has been very prevalent in art, and some would argue that it initially was utilized in the construction of the Parthenon in 447 to 488 BCE Ancient Greece. However, while much of the Parthenon does adhere to the 1.618 ratio very closely, there is no evidence indicating that it was intentional during construction. Despite this, these measurements do make the ancient structure visually alluring, even after the damaging exposure it has succumbed to throughout time.
In truth, the first mentioning of the golden ratio in recorded history (without actually using its name) wasn’t made until 300 BCE, by Euclid in “Elements”. In this work, he states:
“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”
After its official recognition, the golden rectangle served as a major point of guidance in countless works of art, gaining massive popularity during the Renaissance. Even today, outside of the arts, many formed rectangles are based in the golden ratio.
However, the presence of the golden ratio isn’t simply limited to the creativity of human minds, but it acts as an overarching structural blueprint in nature. This includes many naturally occurring structures, even anatomical ones. For example, while standing, if you measure the distance from your navel to the floor, along with the distance between your navel and the top of your head, you will discover a ratio of 1.618.
In addition, the golden ratio is a part of many natural things that colloquially are considered beautiful, such as flowers. The presence of the golden ratio in flowers derives from a concept closely tied to the irrational value, known as the Fibonacci sequence. Named after its founder while he was attempting to determine mathematically the annual birth count of rabbits based off particular breeding habits, the Fibonacci sequence is a series of numbers where the next number in the series is found by adding up the two numbers that preceded it. Starting with 0 and 1, this sequence goes:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
Written as a rule, the expression is xn = xn-1 + xn-2.
As the numbers increase in this sequence, the closer the ratio of the current number to the preceding number becomes to the golden ratio (34/21 = 1.619, while 21/13 = 1.615). As for flowers, the number of petals on a flower species generally follows the values present in the Fibonacci sequence. For example, lilies have 3 petals, roses have 5, delphiniums have 8, marigolds have 13, etc. Speaking from a Darwinian perspective, each petal is placed the way it is to allow for the best possible exposure to sunlight.
The Fibonacci sequence and golden rectangles are also responsible for the formation of what is known as a golden spiral. This spiral is given shape by the successive points that divide a golden rectangle into squares. The logarithmic spiral appears as such:
are also golden spirals.
In fact, the presence of golden spirals extends beyond Earth’s evolutionary patterns, and the perfect swirls are present on a cosmological scale. Particularly in the structure of spiral galaxies, the category of galaxy that is known for its winding spiral shape where the present young stars and solar systems exist in the arms of a flat disk. Like hurricanes, the shape of spiral galaxies is a perfect golden spiral. Spiral galaxies account for 77 percent of all galaxies discovered so far, including our very own Milky Way.
Furthermore, some scientists have theorized that the golden ratio exists on an even grander, all-encompassing scale. These individuals have been proposing that, since the feature of a golden spiral has appeared in so many instances in our universe, the golden ratio could be a property of space-time. Some have even gone as far as saying that a golden spiral is present in the very topology of space-time. As to why the universe follows this rule, however, it is not known.
It is also important to mention that some are critical of the golden ratio being seen as this idea of perfection, noting the many misconceptions that have been held with it. The biggest of these is probably the commonly mentioned claim that Leonardo Da Vinci made use of the ratio in many of his works, as he found it to be behind the measurements of the perfect face. It has been said to have played a part in the painting of the face of the Mona Lisa, the perspectives of The Last Supper, and even the structure of the Vitruvian Man. However, there is no evidence in his records indicating that Da Vinci made any use of the golden ratio. And, while it does appear that the Vitruvian Man is a representation of the golden ratio, its measurements actually don’t correlate with that value.
Despite this, the golden ratio is still present in many works of art and natural structures. Being found in people, shells, hurricanes, and galaxies, the golden ratio might literally be a universal standard.
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