The Golden Ratio & Its Impact On The Design Of The Universe
Abstract:
The Golden Ratio is one of the most commonly found blueprints in the design of the universe. Derived from the infamous Fibonacci Sequence, the mathematical ratio is notorious for producing aesthetically pleasing visuals and compositions (Caryle, 2012). The most common design elements produced by The Golden Ratio are its infamous nautical spirals, appearing in various iterations in the natural world from the spiral of seashells to the strategic placement of succulent leaves, to the swirl-like patterns of our planet’s closest galaxies (Edison, 2020). However, The Golden Ratio’s sphere of influence extends far beyond spirals but has made its mark on the artistic realm, capturing the minds of many Renaissance figures, as well as has been vital to the proportion and function of our bodies. These recurring appearances of The Golden Ratio have consequently aided in the structure of the world around us for the better.
Origins in Fibonacci & Mathematical Properties:
The Fibonacci Sequence is a series of numbers in which the next number is found by adding up the previous two consecutive numbers before it. Starting with zero, the application of this rule results in 0, 1,1, 2,3,5,8,13,21, 34, and so on extending into infinity (Bourne 2019). This sequence was first discovered by Indian mathematicians 1300 years ago before later being introduced to Europe in 1202 by Leonardo of Pisa, otherwise known as Fibonacci in which the sequence obtained its name. Fibonacci first described this sequence in his famous mathematics book Liber Abaci, in which the mathematician remarked on the breeding patterns of rabbits (Mann 2019). Dividing any Fibonacci number by the number before it will bring you close to the irrational number approximated at 1.61803, a number in which Greek mathematician Euclid named Phi, or more commonly dubbed as The Golden Ratio for the ratio’s ability to conjure proportions or visuals aesthetically pleasing to the eye when put to use. (Caryle, 2012) Given a length broken into two portions, Phi can be described as the ratio between the two portions equal to the sum of the overall length and the larger portion, otherwise expressed as the equation a/b = a+b/a. As a quadratic equation, The Golden Ratio can be written as x2-x-1=0 with its solution being x = (1 + √5)/2 Through Both The Golden Ratio and its link to The Fibonacci Sequence, much of the design of the world around us, both natural and manmade have been organized around these two unique mathematical concepts to visual perfection (Carlson 2019).
The image below displays the equation a/b = a+b/a by showing that if the Golden Rectangle was cut into a square, the ratio of the remaining rectangle is equal to the ratio of the original rectangle. The areas of consecutive squares correspond with the numbers from the Fibonacci Sequence (Wolfram Research, 2020)
Evidence of The Golden Ratio in Nature:
One of the easiest ways to spot the Golden Ratio in nature is by investigating the natural structures of plants and flowers. Phyllotaxis relates to the arrangement of leaves around a plant stem, and many plants display Fibonacci Phyllotaxis through utilizing the angle 137.5, otherwise known as The Golden Angle. Derived from dividing the circumference of a circle in adherence to the qualitatives rules of phi, this angle equivalency to the Golden Ratio is used in the natural world to optimize the development of plants. (Atela and Gole, 2003; Go Figure, 2015) For example, the development of seeds in sunflowers are made in such a way so that they create intricate nautical spirals. The resulting design is thanks in part to the Golden Angle being used to complete each turn after a new seed is made so that there are no gaps in between.
The reason behind the Golden Angle being used to turn the seeds in each spiral rather than other possible angles is due to Phi being an irrational number. Using any number that could be converted into a simple fraction in the turns of sunflower seeds would result in a series of lines stacking up, creating gaps in the process (Math Is Fun, 2020). Because Phi is irrational, the resulting effect is the creation of well-distributed seeds in intricate spiral patterns with no gaps in between. Often the number of these spirals ends in Fibonacci numbers, hence the term Fibonacci Phyllotaxis (Atela and Gole, 2003) However, usage of the Golden Angle in the natural world extends far beyond proper seed distribution. Many plants utilize the spiral pattern growth that the Golden Angle provides so that each branch, leaf, and petal can soak up the greatest amount of sunlight and/or rain without any overlapping or gaps, enabling them to grow and photosynthesize properly (Go Figure, 2015).
The below diagram illustrates that the underlying principles from The Golden Ratio still applies with the Golden Angle, as the ratio of the red arc to the blue arc equals 137.5 degrees, or a+b/a = a/b (Go Figure, 2015)
The Golden Ratio in Art Composition:
Seeing how the Golden Ratio contributes to the natural world’s beauty, it’s understandable that artists would want to replicate this astounding perfection tool in their own works. In art, the Golden Ratio can be implemented via a ‘phi grid’, in which the ratios of its columns and lines is 1: 0.618, creating a series of nine unequal sections. Visually, the phi grid is similar to the infamous Rules of Thirds in photography, yet the uniformity of the phi grid proves to be more advantageous by providing a pleasing perspective while also seeming realistic at the same time (Scott, 2020). For example, placing the phi grid on Raphael’s School of Athens and its evident that the grid provides a holistic perspective of the image’s surroundings. Placing the same grid on various sections of Leonardo Da Vinci’s The Last Supper and there are clear instances of the golden ratio being used, from the dimensions of the table to room, to how Jesus is seated next to disciples (Meisner, 2016; The Artists 2018). Of all famous Renaissance artists, Da Vinci by far has the most compelling evidence of being influenced by the mathematical concept, as he was the creator of the illustrations of famous Italian mathematician Luca Pacioli’s book De Divina Porportione (The Divine Proportion). Published in 1509, Pacioli’s book detailing the relationships between mathematics and artistic proportions, specifically art and The Golden Ratio (Mann, 2019). Da Vinci’s handiwork in the manuscript consists of hundreds of brilliantly illustrated three-dimensional shapes such as dodecahedrons and icosahedrons, which have natural golden proportions within their structures. Many contemporary artists would later take inspiration from these geometric illustrations and incorporate them into their own work. 20th-century artist Salvador Dali’s rendition of The Last Supper implements dodecagons as the windows of the biblical scene, setting the scene’s hallowed atmosphere (Meisner, 2016).
The Golden Ratio and the Development of the Human Body:
Not only is the Golden Ratio tangible in both natural and manmade settings, but it can also be identified in different aspects of our own bodies. The discussion between proportion and beauty has been carried out as far back as 25 B.C when Roman Architect Marcus Virtrtruvius Pollio compared the human body to a perfect building. Pollio’s probings on human proportion would later be picked up and illustrated in Da Vinci’s Vitruvian Man depicting the idealized human body placed inside a circle and square (Bourne, 2020). Examining an adult human body, the distance between the navel and foot to the human body’s height comes out to 1: 1.601. Investigating human fingers. which comprise three sections, the ratio of the first two sections to the entire length of the finger comes out to The Golden Ratio (Edison, 2020; Bourne, 2020).
When it comes to the rhythmic patterns of the human heart, a study by the British Medical Journal suggests that people who’s blood pressure counts aligned with The Golden Ratio were less prone to fatal cardiac arrests than those whose blood pressure came out higher (Edison, 2020).
The proportions between the eyes, lips, chin, and mouth in the Marquardt Beauty Mask all conform to The Golden Ratio.
The proportionality and subsequent symmetry in the human body are
what has kept our species intact after thousands of years of evolution
due to human preference for individuals who have symmetrical bodies,
as we perceive them to be more healthy. The same concept is also
applicable to our attraction to faces in particular. The more symmetrical
a person’s face is, the more we find it attractive, and, like with all other
aspects of the design of the human body, this symmetry is rooted in Phi
(Bourne, 2020). In modern times, renowned facial expert Dr. Stephen
R. Marquardt’s work in human perception of beauty has led to the
creation of the ideal human face: the Marquardt Beauty Mask.
Structured in the likes of Phi from the pentagon and hexagon who have inherent golden ratios in their dimensions, the Marquardt Beauty Mask reveals the universality in our perception of beauty, as it can be successfully fitted on faces of individuals across all races and time periods (Meisner, 2016; Marquardt Beauty Analysis, 2015). Though the beauty mask illustrates the ideal human face, Marquet notes from his own research that no one is a perfect fit for the mask as he explains, “We are all a variation of the mask. Some of us vary only a little and some vary a lot, but most of us are somewhere in between”(Meisner, 2016).
Conclusion:
The ubiquity of The Golden Ratio and its universal methodology across various fields has contributed to the efficiency of our society. Its usage in the natural world has been crucial to the survival and development of both plants and people, while its use in the realm of art has captured the imaginations of many famous artists and designers, further linking the beauty between mathematics and art. Though it is not the only mathematical concept found in the natural world, it stands apart from the rest as being the most efficient and perfect in use
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