We all share a somewhat similar taste when dealing with aesthetics and proportion, and tend to associate these attributes to beauty. There’s a mathematical proportion that often appears when we measure what we consider beautiful.
This proportion has been studied since ancient times, from the Greeks Phidias and Plato, to some outstanding science figures as Da Vinci and Johannes Kepler and lately by the Oxford physicist Roger Penrose, such is the fascination provoked by the the so-called “golden ratio”.
Golden ratio
But what is this golden ratio and why is so interesting? well for starts it appears repeatedly in nature, and is also found on different geometric shapes. Its value is derived from a simple mathematical proportion; consider the ratio between two segments, where the length of the whole segment (“a+b”), divided by the larger section (“a”), is the same as the division between the larger section (“a”) by the smaller section (“b”); this gives an approximate value of 1.6180339887498.
GoldenRatio (gofiguremath.org)
Such proportion is called “the golden number” or “golden ratio”, and can be expressed by the formula (1+√5)/2, and is identified by the Greek letter φ (Phi).
This value φ, has very interesting properties; for starts, it is an irrational number; so, similar to the value for Pi (П), it has no predetermined or repeated pattern in its decimal values, and cannot be expressed as a simple fraction. One case where this ratio appears is in the pentagon; If you trace lines between all vertex of a pentagon, the resultant segments cut each other in a position that divides them precisely in a proportion equal to the golden ratio.
Pentagon and Phi (matematicasvisuales.com)
Fibonacci
Leonardo Bonacci, better know in the mathematical community as Fibonacci (diminutive of “filius” or “Son” of Bonacci), was one of the greatest mathematician of the middle ages; he was the one who introduced Arabic numbers to XIII century Europe. Most of his known work in mathematics comes from his book “Liber Abaci” or “book of calculations” used for merchant’s finance calculation, but the most famous section is based on how math can be used to express the ratio in which rabbits breed, and this result became known as the Fibonacci sequence, which is expressed in the formula Fn = Fn-1 + Fn-2. This means that every number in the sequence is the result of the sum of the two previous numbers of it. Resulting in the following sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
From this series you can deduce that:
1+1 = 2
1+2 = 3
2+3 = 5
3+5 = 8 … And so on; The value one is repeated at the beginning since the first value doesn’t have to be added (or added with zero).
But a far more interesting relation comes up from the Fibonacci series. If you divide the values of the contiguous Fibonacci numbers, their values successively start to get closer to the golden ratio value:
| Fibonacci Numbers | Ratio |
| 3/2 | 1.5 |
| 5/3 | 1.6~ |
| 8/5 | 1.6 |
| 13/8 | 1.625 |
| 21/13 | 1.615384615384 |
| 34/21 | 1.619047619047 |
| 55/34 | 1.617647058823 |
| 89/55 | 1.618~ |
Is notorious that the golden ratio and Fibonacci numbers are closely related, and this relation becomes more and more interesting.
If you take a “Golden rectangle” with a long side that’s 1+ φ (1.618), times longer than the short side, and trace a square with dimensions equal to the shorter side, the remaining section is itself another golden rectangle; and you can repeat this process over an over in a increasingly smaller scale.
GoldenSpiral (wikipedia.com)
Another curiosity is that tracing a rectangle with lengths matching Fibonacci numbers (34 x 21 in the image above), the resultant smaller golden rectangles also follow the Fibonacci sequence, in this example, 1,1,2,3,5,8,13 and 21. And if you trace a curve through the resultant squares it forms what is known as a “golden spiral”.
The golden ratio and Fibonacci numbers in nature
The golden spiral appears frequently in nature; one of the the most recognized examples is the shape of the nautilus shells, but it also appears in the shape of different plants, spiral galaxies and in the shape of hurricanes, and can even be distinguished in the human face’s proportions.
NautilusShell (p-mtimes.blogspot.com)
Galaxy-GoldenRatio (io9.gizmodo.com)
Cases like pine cones, leaves in some plants and petals in flowers also show a spiral distribution that often matches with Fibonacci numbers; the seeds in sunflowers also have a spiral patter that if counted both, clockwise and counterclockwise, the number of lines in each direction are itself contiguous Fibonacci numbers, like 13 and 21 in the image below. Such examples of the golden ratio, golden spiral and Fibonacci numbers appearances are frequent in nature.
Semillas en espiral (mathematica.stackexchange.com)
Golden ratio everywhere
Of course, this proportion, so prevalent in nature, is used in arts architecture and engineering, with several examples ranging from the ancient pyramids of Egypt and the Parthenon in Greece, where the phi (φ), proportion appears repeatedly; it also appears in the Cathedral of Notre Dame in Paris, the Taj Mahal in India and also in modern buildings, such as United Nations Secretariat building In New York, USA; which was built as a rectangle that follows the dimensions of the golden ratio, or the CN tower in Toronto Canada, the tallest free standing structure in the world, which has a height between its base and the observatory of 342 meters, and a total height of 533 meters, having a ratio that’s close to the 1.618, or phi (φ).
There are different and very interest numbers in math, but phi (φ) and the Fibonacci sequence are cases in which their properties are simply beautiful, and we can witness it not only in mathematical formulations, but also in multiple examples in nature. This is a great example of how mathematical symmetries and proportions can be captivating and enthralling.
Regards, Alex – ScienceKindle.
