Why do some particular human faces look beautiful?
Ask the same question with reference to objects. Artists and researchers alike have been striving to solve this problem for a long time.
How many out there who hate Maths? Well, I bet Soham Chowdhury will tell you a thing or two!
Designers, architects and engineers all over the world have very recently grasped the use of these two mathematical concepts in making their work more aesthetically pleasing.
All human faces have an underlying pattern that follows the aesthetical algorithms developed using the Fibonacci Series and 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 1:0.618, or 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. It’s a 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 a 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).
In layman’s terms, that basically means that 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. Not always naturally, but if led to do so.
Another aspect is that the main subject, when 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 understand why, 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 a proof that the Rule of Third arranges subjects in such a way that it is aesthetically pleasing to the human eye.
Why is it to be so was unjustified, until recently; when neuroaesthetics researchers established the link between the golden ratio, and the golden spiral to aesthetics. However, the reason why it should happen is still a mystery. Its answer neuroaesthetics and neurology researchers alike are striving to find.
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, is also designs houses using the golden ratio.
You see, 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.
Else why 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 maths, 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
See that value! It’s 1.619, only 0.001 more than the golden ratio! That means, 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 solve is the purpose 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! Despite the golden ratio being a very public fact in today’s arts society. The evidence is slowly building up.
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 can be defined as the following:
Fibonacci Number = Sum of Last Two Fibonacci Numbers
To demonstrate with an example, the Fibonacci Number after 13 and 21 would be 13 + 21 = 34.
Now, let’s get started with the Golden Ratio. Now, 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 also 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 explain the connection between aesthetics and mathematics, along with 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 just meant to be an introduction to the study of aesthetics, and its fundamental connection with mathematics.
That raises several questions.
How come the golden ratio and the Fibonacci Series of Numbers, along with other concepts of mathematics, evolved in a way that suits our sense of aesthetics?
But then, there is a larger question lurking much deeper.
How is that the human mind is always attracted to objects which follow the Golden Ratio, which is a rigid value; whereas the mind is known to be flexible?