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The Remarkable Golden Ratio - Part 1

Updated on June 19, 2013

Throughout human history, an integral part of science and design has been the observation of the world around us on the quest to discern and interpret the patterns and relationships that, theoretically, would define and govern the inner workings of the universe. One attempt at such an understanding has been accomplished through the use of a mathematical model discovered by a Greek mathematician over two thousand years ago, a mysterious model which is now known to the general public simply as the Golden Ratio. Known to Renaissance artists as the divine proportion, the Golden Ratio exhibits itself nearly everywhere in the natural world, and it is so much a part of life that if objects do not exhibit this ratio, they are not as attractive to the human eye. This ratio, since its discovery – and even before its discovery - has been used to provide beauty and balance in the design in many aspects of human design and art including architecture, product design, and art.

To relate the golden ratio to its effects on design, art, and architecture it is necessary to begin with the very basic elements defining the ratio – it’s discovery, what exactly the ratio is numerically and how it was derived, and why a ratio derived from simple numbers could have such a vital importance in all aspects of design and human life, even taking an integral part in how people perceive and subconsciously judge architecture and designs.

This is the first article of a five part series of hubs through which I will explain many aspects of the Golden Ratio's uses and relationships to the nature of Design.

The Golden Ratio Defined

So, what is the renowned and mysterious Golden Ratio ? Simply put, it is the ratio of the line segments that result when a line is divided according to a very unique method. It is a ratio of 1 to phi, where Through close observation this ratio can be noticed as present in nearly all aspects of biological life, and even elsewhere in the universe. The ratio is also closely related to the Fibonacci sequence numbers, which are, starting at 0, a sequence of numbers which are simply the sum of the two numbers before it in the sequence – i.e. 0, 1, 1, 2, 3, 5, 8, 13, 21, etc… Surprisingly, after about the 13th Fibonacci number, the value of any Fibonacci number divided by the number directly before it in the sequence produces a ratio which is always very close to the value of phi. As the Fibonacci sequence continues the ratio between each set gets even closer to the golden mean, continuing to approach 1.618 as the sequence continues out infinitely with exponentially higher numbers. The ratio cannot be exactly met by the Fibonacci sequence though, because the Golden Ratio is an irrational number.

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History

Greek philosopher Plato first theorized (along with his contemporaries) about the existence of a Golden Ratio or mean, proposing that if a line were divided unequally into two parts so that the larger was related proportionally to the smaller one there would be some sort of relationship. Circa 300 B.C., the ratio was first defined mathematically by a Greek philosopher by the name of Euclid in his work Euclid’s Elements, where he called it the “extreme and mean ratio” nearly 1,500 years later, Italian Mathematician Leonardo Fibonacci mentioned a sequence of numbers originally from Indian mathematics now called Fibonacci numbers, and around 1600 A.D. Johannes Kepler proved a relationship between Fibonacci’s numbers and the Golden Ratio. Many other scientists and mathematicians studied the golden ratio and Fibonacci numbers as well, and eventually Mark Barr suggested the Greek Letter Phi as a symbol to denote the golden ratio.

A large part of the study of the Golden Ratio, as evidenced by its denotation in the renaissance of being the “Divine Proportion”, is that it is seen as a means to look into the divine, and attain a deeper understanding of beauty and spirituality in life, a closer connection with God.

The Vitruvian man
The Vitruvian man

Phi in the body

As it turns out, much of our environment follows rules seemingly set by the Golden Ratio. For example, the ideal human body is proportioned according to the golden ratio, at least according to Vitruvius, a Roman architect, writer, and engineer, who wrote about human proportions in his only surviving workDe Architectura. He proposed the idea of a perfect system of proportions based upon the ideal human body. The “Vitruvian Man” was later popularized by an illustration of the human body following these proportions drawn inside a circle by Leonardo Da Vinci.

One example of this Vitruvian man concept would be the length measured from foot to naval when compared to the overall height of a person turns out to be 1 to 1.618. Also, if the pinky finger is compared to the middle finger, they will measure 1 to 1.618. The first two joints of any of finger (starting from the tip) compared to the third at the knuckle is, yet again, 1 to 1.618. In human lungs, the bronchi in the left and right lungs, when compared, have a ratio in length of 1 to 1.618. DNA, the foundations of life itself, is found to have ratio of 1 to 1.618 in width and cycle length, respectively. This proportion is evident again and again and can be used to measure the ratios of nearly any features of the body. This practice has become commonplace to people curious about the ratio, and many have turned to creating scales based upon the ratio.

Dr. Steven Marquardt, creator of the “beauty mask” which utilizes the golden ratio, said "All life is biology. All biology is physiology. All physiology is chemistry. All chemistry is physics. All physics is math." Keeping true to this ideal,Dr. Marquardt has patented a “beauty mask” which he says is mathematically modeled to conform to the golden ratio, and is, according to Dr. Marquardt’s studies, essentially a grid of the ideal human facial proportions. In his study, Marquardt placed his mask over the images of models from many cultures and with very few exceptions, the mask fit remarkably well, regardless of race. Even more interestingly, the mask was tested on sculptures and paintings, and it seems that our ancestors unknowingly made their artwork to match the ratio. The mask can be placed over a photo of a person or drawn on their face in order to gauge their attractiveness according to the golden ratio, no matter their gender or race. Marquardt hopes for his “beauty mask” to become a vital part of aesthetic surgery and dentistry.

An image of a Fibonacci spiral: note how the pattern can be continued indefinitely.
An image of a Fibonacci spiral: note how the pattern can be continued indefinitely.

Phi in biology

As the golden proportion can be continued to measure nearly all of the organic elements in the human body, it can also in fact be found that this ratio is prevalent in nearly all organisms. A widely used example of the golden ratio and Fibonacci numbers in nature can be found in the shape of the Fibonacci spiral, which occurs as you connect the opposing corners of the squares in theFibonacci Rectangle. The Fibonacci rectangle is simply a rectangle made by squares of sides following the Fibonacci sequence placed in such a way that they create a rectangle. The Spiral is then created by connecting the diagonally opposite corners of the Fibonacci rectangle. Notice that if the sequence is continued, and the next Fibonacci number is added as a square, the pattern is continued. The resulting pattern can be seen in biology and the shapes we see in everyday life. The spiral growth of seashells is one simple example that is made very obvious. The outline of the human ear has the exact same spiral. Another amazing example of the spiral occurring naturally can be found in many plants. The spiral appears in pine cones, pineapples, and most famously in the natural growth of seeds on sunflowers.

In other animals and plants, the ratio is prevalent as well: penguins’ eyes, beak, and wings all fall at golden markings proportional to its height. Ants multiple body sections are golden sections, as well as the sections of its legs. (Similar to how human fingers are divided) In dolphins, the eye, fins, and tail are placed at golden sections according to the total length of the dolphin’s body as well. Interestingly enough, in a branching plant, many times the branches they produce are produced in Fibonacci quantities, i.e. one branch branches into two, two into three, three into five, five into eight. This often continues as the plant grows larger.


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    • Trish_M profile image

      Tricia Mason 5 years ago from The English Midlands

      Hi :)

      I have found this subject fascinating, ever since I first read about the 'Golden Ratio'.

      I look forward to reading more :)

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