The Most Beautiful Number in Mathematics
1.618…
The Golden Ratio. The Divine Proportion. The number 1.618… goes by many different names. However, mathematicians mostly call it phi. Everywhere around us, we see this beautiful number. It isn’t the number itself that is special, but rather all the other beautiful objects that revolve around it. Mathematically, this number can be expressed in the form:
The symbol on the left is the Greek letter for phi. If you notice, this is also the positive root for the equation x²-x-1. One very interesting property of this number can be found in the Fibonacci sequence. The Fibonacci sequence basically starts with 2 ones and then the next numbers can be expressed as the sum of the previous 2 numbers. Recursively, it can be expressed as:
Numerically, it can be expressed as:
In this sequence, if you divide each term by the term that precedes it, you will notice that the quotient slowly begins to approach phi. We can construct a graph to visualize this.
Rather than manually calculating the quotients, I made a simple python code to calculate the quotients for me:
Code:
num_1 = 1
num_2 = 1
fib_no = [1, 1]
update = 0
flag = 0
while update < 14:
if flag == 0:
num_1 += num_2
fib_no.append(num_1)
flag = 1
elif flag ==1:
num_2 += num_1
fib_no.append(num_2)
flag = 0
update += 1
print(fib_no)
quo_list = []
for ind in range(0, len(fib_no)-1):
quo = fib_no[ind+1]/fib_no[ind]
quo_list.append(quo)
print(quo_list)Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987] [1.0, 2.0, 1.5, 1.6666666666666667, 1.6, 1.625, 1.6153846153846154, 1.619047619047619, 1.6176470588235294, 1.6181818181818182, 1.6179775280898876, 1.6180555555555556, 1.6180257510729614, 1.6180371352785146, 1.618032786885246]Using this data, we will construct a graph with the index of the dividend on the x-axis and the quotient, up to 3 decimal places, on the y-axis:
As you can clearly see on this graph, the quotients all approach y = 1.618. This is the traditional mathematical origin of phi. Despite this brilliant derivation of phi, the number is more commonly seen in nature. Due to it being in almost every beautiful or elegant thing on Earth, phi got the name ‘Divine Proportion’ as it was believed that only God could have put such a number in every single beautiful thing. Phi is also argued to be one of the fundamental building blocks of nature. Let us have a look at a few examples where phi can be notably seen.
One of the most common examples that people cite when they talk about phi is in the center of a sunflower.
As you can see in this image, the seeds of the sunflower grow in opposing spirals. Now, if you calculate the diameters of successive spirals and divide them, you will be left with phi! Quite brilliant isn’t it? Since we’re talking about spirals at the moment, let’s have a look at this marine animal called a nautilus. The nautilus has a very interesting shell, and if you look at the next two photos, you will understand what I mean:
The left photo is how the nautilus looks on the outside. The nautilus doesn’t look very special when viewed from the outside. But when seen on the inside, you see a magnificent spiral pattern within its shell. What’s more, if you divide the diameters of consecutive spirals, like what we did with the sunflower, you will again obtain phi as the the quotient! Interestingly, the Golden Ratio is often showed in the following form:
As you can clearly see in this image, if you divide the length of the bigger squares by the length of the preceding smaller square, you will be left with phi again. Very often, you will see this curve superimposed on images like that of the sunflower and nautilus we saw above.
The Milky Way galaxy is really quite beautiful. The radiating arms of of the galaxy never cease to catch one’s eye. However, since it’s a spiral, like the other examples, do you think that dividing the diameters would yield phi? If you thought that, then you are are right! Indeed, the quotient that you obtain when you divide the consecutive diameters of the spirals in the galaxy is the Golden Ratio. No doubt, our ancestors would have been awestruck seeing this number everywhere and believed it to have been put there by God himself.
Hurricanes are one of the most deadly natural disasters in the world. But, have you ever seen a satellite image of them from above? If you have, you would know that the hurricane looks like a spiral. And no doubt about it this time, if you divide the diameters of each spiral, you will be left with phi again. Seeing this number in nature so many times truly shows that the number is one of extreme beauty and elegance.
In a beehive, it is a known fact that the female bees always outnumber the male bees. However, what’s truly special is that if you divide the number of female bees by the number of male bees, you will always be left with phi! Isn’t that amazing? The Divine Proportion can also be seen outside of nature, inside our own bodies! “Where?”, you may ask. Well, let’s have a look!
If you divide the length of your shoulder to your fingers by the length of your elbow to your fingers you obtain phi again! Similarly, dividing the distance of your hip to toes by your knee to toes, not surprisingly, you get phi again. There are multiple more examples of the Golden Ratio that can be seen within the human body. In the words of Robert Langdon in Dan Brown’s international bestseller The Da Vinci Code:
My friends, each of you is a walking tribute to the Divine Proportion.
Not surprisingly, after seeing the beauty of this number, a lot of different artists, architects and musicians began featuring this number in their works. From Da Vinci to Michelangelo, from Beethoven to Mozart, all of these people followed a strict adherence to the Divine Proportion. The number appears in many monuments like the Great Pyramids in Egypt, the Parthenon in Greece and even modern buildings like the headquarters of the United Nations in New York. Legend states that the famous violin making family, the Stradivari, used phi as the basis for marking the f-holes on their violins! With such legends using the Golden Ratio in their works, there is no doubt that phi is indeed the most beautiful number in mathematics.
All of these examples show how the Golden Ratio is seen everywhere we go. With its elegant roots in mathematics, the number phi transcends all borders and extends into many different fields in our life. It shows us how just by strict adherence to the Golden Ratio, you can truly create objects of beauty.
