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Can we mathematically define it? You’d be surprised to know: yes, we can.
This is a blog on the mathematics of beauty.
The Answer Does Not Lie in Aesthetics
The study of beauty is known as aesthetics. Aesthetics is a form of arts. It involves determining how to make people and objects look beautiful.
Contrary to popular conception, aesthetics does not strive to find out the reason for such beauty: in other words, it is unconcerned with why beautiful things are so, and if there are any patterns to it.
If we are looking for an answer as to why certain things are beautiful, I’m sorry, we’re gonna have to wait. But beyond the realm of the arts, mathematics can help us predict when certain things will look beautiful.
Finding Mathematical Patterns in Beauty
Beauty is a very hard concept to define. Perhaps the best explanation is the old cliche, “Beauty lies in the eyes of the beholder.” It is very hard to predict what will appear beautiful to whom.
However, there are certain patterns we have begun to notice.
We have noticed that there are certain configurations which always appear beautiful across cultures, to anybody and everybody who is a human.
And the most interesting part? All of it is connected by some mathematics!
The Mathematics in Such Beauty
Two things in mathematics form the fundamentals of aesthetics: the Fibonacci Series of Numbers and the Golden Ratio.
By applying these two concepts of mathematics, you can determine whether a precise configuration would be aesthetically pleasing to the human eye.
Mind you, if the math thus gives you the result that the configuration will not be aesthetically pleasing, to some people, it still might be. Beauty is a crooked mistress, after all.
But if the math thus gives you the result that the configuration will be aesthetically pleasing, it will be, to every human eye.
Surprised? It’s not rocket science, the math is quite simple. And it involves something called the Golden Ratio.
The Golden Ratio
The golden ratio is precisely 1.618. It has hundreds of applications in both Mathematics and the Arts. In arts, aesthetics is one of them. We shall use the diagram given below to illustrate the Golden Ratio.
This line segment, consisted of two smaller parts a and b, obeys the golden ratio. Not because it is a particular line segment, but because of the relative lengths of the two parts a and b.
The rules of golden ratio states that for a and b to be in a golden ratio, the ratio of the length of a to that of b should be 1.618.
Two quantities are said to be in the golden ratio if the ratio of their sum, a+b, to that of the larger quantity a, is equal to the golden ratio, 1.618.
To understand what I mean in layman’s terms, consider the example illustrated below.
(a+b) = 9.999 cm,
a = 6.180 cm,
(a+b) : a = 9.999 cm / 6.180 cm = 1/0.618 = 1.618
The sum is coming as 9.999 cm instead of 10 cm, since the values of a and b were unrounded approximations. That proves that our the quantities a and b in our line segment (a+b), are in the golden ratio 1.618. This kind of a line segment is known as a golden line segment.
Value of the Golden Ratio
The precise value of the golden ratio, also known as the golden mean, goes into infinite decimal places. It is an irrational number. Normally, this formula is used to calculate the golden ratio, or the golden mean.
The same principle can be applied to create a golden rectangle, which has been illustrated below. Mind you, it’s a very special shape when it comes to aesthetics!
The Golden Rectangle
This golden rectangle consist of a square of side a, and a rectangle of width b, as illustrated in the figure given below.
A golden rectangle is a rectangle with length a+b and breadth a.
An easier representation would be thinking of it as a square with side a, and a rectangle with breadth b and length a standing side by side, as illustrated in the figure above.
I mentioned that the golden ratio (a:b) should be 1.618. In this rectangle too, the ratio of a:b should be the golden ratio 1.618.
Any rectangle cannot be a golden rectangle. A golden rectangle has to meet two conditions:
- Made up of a square and a rectangle: A golden rectangle has to have a square of side a and a standing rectangle of breadth b and length a, as shown above. The rectangle may be positioned in any side: left, right, top or bottom. But a separate square and a rectangle are necessary to form a golden rectangle.
- Golden Ratio: The ratio of a to that of b has to be 1.618, which is the golden ratio.
The Golden Spiral
Suppose, you construct a golden rectangle, as I had illustrated above. That would contain a square and a rectangle. We now concentrate on the rectangle of the golden rectangle, leaving out the square. Can you make that rectangle a golden rectangle?
Yes, you can.
- If you divide the whole area of the rectangle, into a square and a rectangle, you get a golden rectangle. More or less.
- We can divide any rectangle into a square and a rectangle. Then how do you know it is a golden rectangle? You divide the rectangle in such a way that the ratio of side a of the square to that of breadth b of the rectangle is 1.618, which is the golden ratio. Once done, that rectangle is a golden rectangle.
- Realise that the rectangle you just made into a golden rectangle was actually a rectangle inside a golden rectangle. So, you have constructed a golden rectangle inside a golden rectangle!
- Similarly, you can continue doing this for infinite times. Considering you go for a finite number of times, you get something as illustrated below.
However, do you notice that there is a spiral running right through the whole spiral? That is something known as a golden spiral.
The initial starting point of this spiral is the smallest golden rectangle in the series of golden rectangles.
This is a special kind of spiral known as a logarithmic spiral, whose growth factor is φ, which stands for the golden ratio (1.618).
Loosely speaking, it is kind of a “curved diagonal” of all the squares and rectangles inside the golden rectangle, joined together.
Neuroaesthetics studies have proved that in a series of golden rectangles, as illustrated above, the path that the eyes would follow is the golden spiral. Perhaps not always naturally, but definitely if led to do so.
Another aspect is that the main subject, when placed at the starting point of the golden spiral overlaid onto an object, looks aesthetically pleasing to the human eye.
That’s the catch. If led to do so. I’ll come back to this a bit later. Before that, I want you to pay attention to one more detail.
The Rule of Thirds
Those who have studied aesthetics in photography would be aware of this.
This rule of thirds is actually a variation of the golden spiral.
I publicly lay down before you the fact that most photographers do unconsciously apply it while photographing.
Why the rule? There’s a reason.
First, let’s understand what exactly this is. First, you need a grid overlay over the image, to understand the rule.
This rule states that a aesthetically-pleasing would have its main subject at one of the six intersections of grid lines, as illustrated in the image above.
To answer that, we must go back to the golden spiral. Take the above photo as an example of a photo that follows the Rule of Thirds perfectly.
Now, you can overlay a golden spiral onto the photo, by tracing the diagonals of a continued golden rectangle series. That’s precisely what has been done here.
If you overlay the Rule of Thirds grid over the same picture, you would get the answer. The starting point of the Golden Spiral would be coincident with the lower right grid intersection!
If you change the orientation of the golden rectangle by 180 degrees left or right, then the starting point of the Golden Spiral would be coincident with another grid intersection. In this way, you can flip and rotate the golden rectangle to make the starting point of the Golden Spiral coincident with all the grid intersections.
The rule has been formulated based on aesthetics studies.
Coincidentally, it turns out that most photographs actually follow the Rule of Thirds, although the photographer never consciously used it during his photography session!
This is kind of an empirical proof that the Rule of Thirds arranges subjects in such a way that it is aesthetically pleasing to the human eye.
Why should it be so was unjustified, until recently; when neuroaesthetics researchers established the link between the golden ratio and the golden spiral to aesthetics.
Application in Aesthetics
The golden rectangle has allegedly been used in several historical structures, the most alleged among them being the Pantheon in Rome.
Many such allegations have come up for the use of the golden ratio in ancient architecture.
Although there is no conclusive evidence for the application of the golden ratio in constructing the Pantheon, there are for others.
In 2004, studies regarding this issue were conducted at the Great Mosque of Kairouan. It was evident that the golden ratio was applied consistently throughout the structure.
However, critics have brushed this off as a vague example since the structures which had that property were determined to be structures that were added later. Suspicions are unconfirmed, and the debates still continue.
The Swiss architect, Le Corbusier, based his design philosophy, known as Modulor, explicitly on the golden ratio. Another Swiss architect, Mario Botta, also designs houses using the golden ratio.
Even the human face, to a certain extent, follows the golden ratio, since it can be divided into several golden rectangles, all making up a single large golden rectangle.
No wonder it is it said that the eyes make the first expression of beauty!
It is so because if you construct a golden spiral on the human face, based on the already constructed series of golden rectangles, the starting point would be at the eyes.
This pyramid has been found to follow the golden ratio! You might be wondering how a pyramid can do so, by the virtue of its shape. Evidence of the Golden Ratio has been found in pyramids by egyptologists ever since.
Based on the overall study of all pyramids, researchers found some surprising conclusions:
- Pyramid Base Edge: Based on the overall study of pyramids, it has been found that all the edges of the base of all pyramids is within the range 755-756 feet. Surprisingly accurate with all the pyramids.
- Height of the structure: The height of the structure is the more striking feature. The height is actually the length of a perpendicular dropped from the top of the pyramid to the base. Surprisingly, all the measured structures have an exact height of 481.4 feet!
If we work out the math, it turns out the perpendicular bisector of each side of the pyramid, represented as a in the diagram, to be 612 feet. We can take that as the slant height of the pyramid.
Dividing the slant height by half the base length of the pyramid, would give us base b of the triangle. That comes to, after calculations, at 378 feet (756 / 2 feet).
Before proceeding any further, we need to know about another geometric shape: the golden triangle. For a triangle to be a golden triangle, any two sides a and b need to be in the golden ratio, i.e. 1.618.
Now, back to the math. Let’s get back to the slant height of the pyramid a and the half-length of the base of the pyramid b.
a = 612 feet,
b = 378 feet,
a:b = 612 feet / 378 feet = 1.619
I bet that value appears eerily familiar. It’s 1.619, only 0.001 more than the golden ratio.
What that implies is that that triangle is a golden triangle. And that implies that somehow, all Egyptian pyramids contain the golden ratio in their construction!
Whether ancient Egyptians ever knew about the golden ratio is a questionable act, and it could be based on other geometric shapes, such as 3-4-5 Pythagorean Triangle, Kepler’s Triangle, etc.
But it does, in a way, answer why the Egyptian pyramids are aesthetically pleasing to look at, at least structurally.
The Mona Lisa
Arguably the most famous portrait of the world, the Mona Lisa too has golden ratio used in the proportions of the subject.
Even the subject of Mona Lisa is shown to have the proportions of the Golden Ratio! Now, what do we determine of this? Leonardo da Vinci used the golden ratio while painting the Mona Lisa? That would be an absurd idea indeed.
Even artists of today do not sit with a ruler, pen, pencil, and protractor to mark the golden ratio proportions for their paintings!
But somehow, it mysteriously appears in things of the most beauty.
The Affair between the Golden Ratio and the Fibonacci Series of Numbers
1 1 2 3 5 8 13 21 34 55 89 144 233 377 ………….
This is the Fibonacci Series of Numbers, which is, as usual, infinite. It is named after Leonardo Fibonacci, who introduced the concept in Western mathematics through his book Liber Abaci. Despite the name, the Fibonacci Series has its beginning in Indian Mathematics.
In layman’s terms, the Fibonacci Series of Numbers is defined as:
This Fibonacci Number = Sum of Last Two Fibonacci Numbers
For instance, the Fibonacci Number after 13 and 21 would be 13 + 21 = 34.
Now, let’s get started with the Golden Ratio. Let’s compare the ratio of any two numbers of the Fibonacci Series. Comparing 3 and 5, we get 1.66, which is far from the Golden Ratio.
Compare 13 and 21, we get 1.615, which is close to the Golden Ratio. We are getting close.
Compare 21 and 34, you get 1.619. That is only 0.001 more than the golden ratio!
Compare 144 and 233, we get 1.618, which is the Golden Ratio!
From 144:233 onwards, the ratio between any two numbers remains equal to the Golden Ratio, 1.618.
Therefore, the ratio between the Fibonacci Series of Numbers from 144 onwards, is coincident with the Golden Ratio. And the numbers before 144 have ratios quite close to the golden ratio.
We have already studied the application of the Golden Ratio in aesthetics. You might have guessed what I mean to say.
We can use the Fibonacci Series of Numbers in aesthetics! How? Remember that series of Golden Rectangles which I showed to you a little while ago? Kind of a similar pattern can be created by applying the Fibonacci Series of Numbers, as illustrated below.
The ratio of the areas of the squares would be in the proportion of the Fibonacci Series of Numbers. The largest rectangle formed would also be a golden rectangle!
So, in order to make a golden rectangle look more aesthetically pleasing, you can divide it into further sections based on the Fibonacci Series of Numbers.
So, there you go. The Fibonacci Series of Numbers too has an application in aesthetics!
These are a special kind of tile designs, which consist of dissimilar rhombuses of two types; as illustrated below.
As the name implies, these tilings have been designed by the famous mathematician Roger Penrose, who is, by the way, one of my favourite mathematicians!
Notice that there are two kinds of rhombuses: thick and thin rhombuses. The thin ones are coloured green, whereas the thick ones are coloured blue. This is where the Golden Ratio comes into play.
Regardless of the size of the Penrose Tiles, the ratio of the thick to thin rhombuses is equal to the golden ratio, 1.618!
Study it more carefully, and you would find that there are certain patterns which have been repeated. It has also been found that the distance between similar patterns grows in proportion to the Fibonacci Series of Numbers, as the number of similar patterns increases. The Fibonacci Series of Numbers has a long-standing relationship with the Golden Ratio.
In this blog, my aim was to very briefly delve into the connection between aesthetics and mathematics, along with empirical proof. This blog itself is too short to go into the complex mathematical procedures, which I have avoided as far as possible at all places.
This was meant to be merely an introduction to the mathematical study of aesthetics, and its fundamental connection with mathematics.
However, that raises a more fundamental question. Why are objects that follow the Golden Ratio always aesthetically pleasing to the human eye?
Ahh, that’s a great enigma! At least for today.