0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. is the Fibonacci sequence. Interestingly enough, ratios between consecutive terms of this sequence seem to be approaching the same number: 34/21 is ~1.619, 55/34 is ~1.618, 89/55 is ~1.618. Mathematicians have discovered the equation describing this relationship: a/b = (a + b)/a = 1.618…

Great, so what? Well, this irrational, or never-ending, ratio starting with 1.618 has been given many names over the years including  the “golden ratio” and the “divine proportion”. But what’s so divine about this random number? This ratio has been discovered in and applied to a multitude of scientific, mathematical, and even aesthetic design disciplines. From the time of Pythagoras and Euclid to present day investigations of our intertwined natural and numerical worlds, people have been intrigued by the universality of this ratio.  

Furthermore, the golden ratio has been used extensively in the arts for its aesthetic qualities.

The golden ratio can be easily studied using flower anatomy. It has been noted that flower seeds generally appear in varying Fibonacci sequence numbers. This seed model of minimizing gaps between petals while maintaining a round shape for the flower has been mathematically explained using the golden ratio. Apparently, starting a new petal at a 1.618 or .618 rotational ratio from the prior petal is how many flowers have evolved in petal design. Similarly, this ratio can be found in the curves on a shell, the arrangement of seeds, the arms of a spiral galaxy, the spatial organization of hurricane winds, length ratios in DNA, and animal body structure proportions. Why is this ratio so prevalent in nature? Going back to the flower seed example, starting a new seed column at another rotational ratio such as .5 would create a linear arrangement of seeds. Similarly, any rotational seed ratio that is a definite fraction or rational number leads to seed overlap and gaps. This is why other irrational numbers like pi don’t work as well in this model – they are too close to fractions. Essentially, the divine proportion is an irrational number furthest from a rational fraction.  

  Some examples of what flowers would look like with varying rotational ratios

Some examples of what flowers would look like with varying rotational ratios

Furthermore, the golden ratio has been used extensively in the arts for  its aesthetic qualities. Artists like Salvador Dali and Le Corbusier explicitly used  the divine proportion in many  of their works. It has also been theorized that even DaVinci’s “Mona Lisa” follows the law of the divine proportion. The appearance of this ratio in the natural world has been understood through mathematical models like the one examined above, but the golden ratio in art, a subjective field, implies that there exist psychological and perceptual implications to this ratio. This has been a controversial field of study. Some researchers claim that applying the golden ratio to all disciplines overextends its natural importance. Others, like the pioneering physicist and psychologist Gustav Fechner, disagree. Through empirical studies, Fechner found  that rectangles in the golden ratio 1.618:1 were  the most pleasing to the eye.  While the ultimate significance of the golden ratio is still under investigation, it is undoubtedly still a unique and interesting scientific phenomenon,

Teja Ravivarapu is a freshman from Sid Richardson college at Rice University.

Resources

  1. Weisstein, Eric W. "Golden Ratio." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GoldenRatio.html

  2. “Nature, the golden ratio, and Fibonacci” Math is Fun, https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

  3. Livio, Mario. “The Golden Ratio and Aesthetics” https://plus.maths.org/content/golden-ratio-and-aesthetics


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