Investigation on the effect of golden ratio in two monuments (Taj Mahal, Notre Dame) using vector calculus

From my childhood, I always had this keen interest in learning computer. My younger brother is pursuing Bachelors in Computer science so I used to have this discussion about the computer technologies, the chipset, the concepts behind the working of the computers, how and why it is faster than human brains, etc.

 

The day I learned about binary code, the language understood by the computer systems, I even got more attracted to the world of computers. My brother introduced me to computer programming and from there I picked up the word “Fibonacci Series” and decided to research on this topic. On further investigating the topic, I came to know about the “Golden Ratio” and its physical significance of how every beautiful thing follow the Golden Ratio. The Golden Ration finds its application in almost every aspect of life. I came to knew that monuments like the Taj Mahal, Norte Dame as well as the Pyramids also follow the Golden Ratio which makes them even more beautiful. Therefore, I have decided to take up this IA as the relationship between the Golden Ratio and the real-life objects, if proven will even make it more beautiful. Thus, I thought about exploring more about the relationship of Golden ratio with the architectural masterpieces like the Taj mahal in India and the Norte Dame in France in this IA.

To determine the value of Golden Ratio from the architectural monuments of the Taj mahal in India and the Norte Dame in France.

The Taj Mahal, the name itself is of Persian origin where “Taj” means Crown and “Mahal” means a palace thus it symbolizes “the Crown of Palace”. It is situated on the southern bank of the River Yamuna in Agra city of India. The Mughal Emperor of the 17^th century, Shah Jahan was the main orchestrator of the Taj Mahal which was built to house the tomb of his beloved wife, Mumtaz Mahal. The construction of this architectural masterpiece was completed by the end of 1643. The Taj mahal cites one of the finest examples of Muslim architecture due to which UNESCO designated it as the UNESCO Heritage Site thus indicating the achievements Indian art and architecture during that time. The amount of precision and knowledge of mathematics that was used while the construction of the white marble ivory structure is unthinkable in the modern generation. The concepts of geometry that was used to construct the Taj mahal could also be used to construct other real-life monuments and architectural structures.

 

The weight of the minarets along with their size and angles were considered while constructing the masterpiece to calculate the after effects of earthquake. The minarets surrounding this architectural structure is of slightly outward lean to withstand the after effects of a severe earthquake. The distance between the doors and windows are all of equal distance to indicate the astounding symmetry of the Taj Mahal. It is also believed that the reflection of the Taj Mahal in the water has a line symmetry with respect of the original structure.

 

The use of precious and semi-precious stones was encouraged to form the decorations of the place. The tiling pattern follows the arrangement of regular hexagons with a six-pointed star. For the decoration purpose of the nearby garden, the stones were arranged or laid in such a way that they combine squares and hexagons finally giving the structure of octagons. The main building of the tomb stans on the top of a platform of height 50 meters. The height of the tomb is approximately about 58 meters while the height of the corner minarets are about 137 feet. The concept of Golden ratio was used while the construction of the Taj mahal due tow which it appears as an architectural masterpiece by the Muslims. The Rectangle serving as the basic outline were all in Golden Proportion thus making it accurate.

One of the finest examples of the French Gothic architectural structure, the Notre-Dame de Paris means “Our Lady in Paris”. Situated on the Eastern site of Ile De La Cite, the cathedral church was constructed over the ruins of other two churches. Pope Alexander III conceived the idea of constructing one single large scaled structure in the year 1163 and the initial construction of the French masterpiece was completed by the end of 1250. Though the structure was not fully developed by 1250, the choir and the construction of the western façade along with the nave were all completed back then.  The Gothic carvings completes the decoration of the internal Cathedral which is of 427 by 157 feet. The roof height is about 35 meters high. The Gothic towers surrounding the Western façade are of height 120 feet approximately. The structure of Notre dam also follows the golden Ratio.

The Fibonacci series shares a very close relationship with the Golden Ration or the golden proportion. Considering a Fibonacci series, a particular term is the summation of the previous two terms. Each term in the Fibonacci sequence is expressed as - 

 

T_n = T_{n - 1} + T_{n - 2}…………………(equation - 1)

 

Example of a Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …

Vector and matrix are both mathematically related to each other. Now if a vector is having only 1 row or is only containing 1 column then it can be termed as row vector and column vector respectively.

 

The Vector \(\vec{v}\) of a Matrix A has been depicted in the equation below:

 

Here, k symbolizes the Vector Value,

 

\(A\vec{v}=k\vec{v}\)

 

\(=>A\vec{v}=\alpha I\vec{v}……………………(equation - 2)\)

 

Here the identity matrix is I having the order of the matrix A.

 

Vector Value -

 

Vector Value of a particular can be identified based on the scaling factor:

 

From Equation (2), we are deriving Vector Value formula:

 

\(A\vec{v}=\alpha I\vec{v} \)

 

\(=>A\vec{v}=\alpha I\vec{v}=0\)

 

\(=>[A-\alpha I]\vec{v}=0………………………(equation - 3)\)

 

According to Vector definition, \(\vec{v}\)is an example of a non-zero vector. So, we can write:

 

det det (A - αI) = 0…………………………(equation - 4)

 

This equation stated above is referred to as the Characteristic Polynomial of Matrix or Characteristic Equation of Matrix.

In this exploration I will be finding the value of Golden ratio from the two monuments: The Taj mahal in India and Notre Dame in Paris. So, we all know that these monuments follow the Golden ratio, I would say the architecture of these monument follow the golden Ratio. Now first of all, I will be considering two cases, one for the Taj Mahal and other for the Notre Dame in Paris. So, first of all, I will present a sectional breakup of the Taj mahal and my motive will be to represent the length of each section (lengthwise) as a Fibonacci sequence where the next term will be the summation of the present and previous term as discussed in the above section. Value of the next term of the Fibonacci sequence can be derived by using the concepts of matrices and vectors and calculus. The length or here the height of the architectural structures will be represented using the Fibonacci sequence, then I will find the value of the generalized term of the Fibonacci sequence, because Fibonacci sequence also follows the golden ratio. Here the Golden ratio factor will be considered in order to obtain the next term from the present and previous term. The factor will be the golden ratio and the value will be verified as well in this investigation

The Taj Mahal in India and Norte Dame in Paris both follow the Golden Ratio. In order to understand the architectural structure, the entire height has been considered and divided into several sections to draw correlation with the Fibonacci series. The images provided below are taken from a survey cnnducted by Achaeological Survey of India and Preventive Archeaology in France respectrively:

The total height is taken in each monument and divided into several sections, the values of which are provided in the table mentioned below:

The sections are divided in such a way that they fall under the Fibonacci series i.e., the next term is the summation of the present and previous term.

In this exploration section, the equation of Fibonacci Sequence has been derived from the given equation of characteristic matrix equation,

 

\(\Bigg(\frac{T_n}{T_{n-1}}\Bigg)=(1\ 1\ 1\ 0)\Bigg(\frac{T_n}{T_{n-1}}\Bigg)…..equation(5)\)

 

For simplicity, let us consider: (1 1 1 0) = Y

 

\(∴\Bigg(\frac{T_n}{T_{n-1}}\Bigg)=Y\Bigg(\frac{T_{n-1}}{T_{n-2}}\Bigg) \)

 

\(=>\Bigg(\frac{T_n}{T_{n-1}}\Bigg)=Y\ \bigg\{Y\Bigg(\frac{T_{n-2}}{T_{n-3}}\Bigg)\bigg\}=Y^2\Bigg(\frac{T_{n-2}}{T_{n-3}}\Bigg)\)

 

\(=>\Bigg(\frac{T_n}{T_{n-1}}\Bigg)=Y^{n-1}\bigg(\frac{T_{1}}{T_0}\bigg)=Y^{n-1}\bigg(\frac{1}{0}\bigg)\)

 

So, substituting the values of T and considering different values of n with the equation (5), a relationship of nth term with respect to the first two terms can be derived.

 

Diagonalizing the value of Y and ignoring Y^n-1,

 

Considering two matrices M and N which are non-zero, such that -

 

Y . M = M . N

 

=> Y = M . N . M ^-1

 

=> N = M^-1 . Y . M ……………………(equation - 6)

 

Now we are using characteristic vector equation as discussed above to find the value of K- Considering equation (4), we can write -

 

det det (A - αI) = 0

 

Here,  A = Y = (1 1 1 0)  and I = (1 0 0 1)

 

∴ det det {(1 1 1 0) -α(1 0 0 1)} = 0

 

=> det det (1 - α 1 1 - α) = 0

 

=> (1 - α) (- α) - (1)(1) = 0

 

=> α²- α - 1 = 0………………………(equation - 7)

 

Using Sreedhar Acharya’s Formula, to find the roots of the quadratic equation -

 

 \(α_1 =\frac{1+\sqrt{5}}{2}\)

 

 \(α_2 =\frac{1-\sqrt{5}}{2}\)

 

From equation (3), we can write:

 

Considering Case 1 - Let,α = α₁

 

\([Y - αI]\vec{v}_1= 0\)

 

\(=>\big(1- \alpha1\ 1 -\alpha_1 \big)\big(\frac{x}{y}\big)=0\)

 

\(=>\bigg(\frac{x-x\alpha_1+y}{x-y\alpha_1}\bigg)=0\)

 

From the equation stated above, we can further simplify,

 

x - yα₁ = 0

 

=> x = yα₁

 

When y = 1, x = α

 

\(\vec v _1=\bigg(\frac{\alpha_1}{1}\bigg)\)

 

Case 2: Let, α = α₂

 

\([Y-\alpha_2I]\vec{v}=0\)

 

\(=>(1-\alpha_21 \ 1-\alpha_2)\big(\frac{x}{y}\big)=0\)

 

\(=>\bigg(\frac{x-x\alpha_2+y}{x-y\alpha_2}\bigg)=0\)

 

From the equation stated above, we can further conclude,

 

x - yα₂ = 0

 

=> x = yα₂

 

Let y = 1 then the value of x = α₂.

 

\(\vec{v}_2=\big(\frac{\alpha_2}{1}\big)\)

 

Therefore, concluding -

 

M = (v₁,v₂) = (α₁ α₂ 1 1)

 

 

\(M^{-1}=\frac{1}{det\ det(\alpha_1\ \alpha_2\ 1\ 1)}adj(\alpha_1\ \alpha_2 \ 1 \ 1)\)

 

\(=>M^{-1}=\frac{1}{α_1-α_2}(1-α_2-1α_1)\)

 

\(=>M^{-1}=\frac{1}{\frac{1+\sqrt{5}}{2}-{\frac{1-\sqrt{5}}{2}}}(1-α_2-1α_1)\)

 

\(=>M^{-1}=\frac{1}{\sqrt{5}}(1-α_2-1α_1)\)

 

From equation (9), we can write:

 

N = M ^-1 . Y . M

 

\(=>N=\frac{1}{\sqrt{5}}(1-α_2-1α_1)\cdot(1\ 1\ 1\ 0)\cdot(α_1\ α_2 \ 1\ 1)\)

 

\(=>N=\frac{1}{\sqrt{5}}(1-α_2-1α_1)\cdot(α_1+1\lambda_2+ 1α_1\ α_2)\)

 

\(=>N=\frac{1}{\sqrt{5}}(α_1+1-α_1α_2-α^2_2+α_2+1α^2_1-α_2-1-α_2-1+α_1α_2)\)

 

 

\(=>N=\frac{1}{\sqrt{5}}\bigg(​α_1+1-\frac{1+\sqrt{5}}{2}\cdot\frac{1-\sqrt{5}}{2}0\ 0-α_2-1+\frac{1+\sqrt{5}}{2}\cdot\frac{1+\sqrt{5}}{2}\bigg)\)

 

 

Since, α² - α - 1 = 0

 

\(=>N=\frac{1}{\sqrt{5}}(α_1+1+1\ 0\ 0-α_2-1-1)\)

 

\(=>N=\frac{1}{\sqrt{5}}\bigg(\frac{1+\sqrt5}{2}+2\ 0\ 0-\frac{1-\sqrt5}{2}-2\bigg)\)

 

\(=>N=\frac{\sqrt{5}}{5}\bigg(\frac{5+\sqrt5}{2}0\ 0\frac{-5+\sqrt5}{2}\bigg)\)

 

\(=>N=\bigg(\frac{5\sqrt5+5}{10}0\ 0\frac{-5\sqrt5+5}{10}\bigg)\)

 

\(=>N=\bigg(\frac{\sqrt5+1}{2}0\ 0\frac{-\sqrt5+1}{2}\bigg)\)

 

=> N = (α₁ 0 0 α₂)

 

Now, we know that

 

Y = M . N . M^-1

 

=> Y^n-1 = (M . N . M ^-1)^n-1

 

=> Y^n-1 = M . N . M ^-1M . N . M ^-1M . N . M ^-1M . N . M ^-1…………(upto n -1 times) M . N . M ^-1

 

=>Y ^n-1 = M . N . N . N . N…………(upto n - 1 times) K.M^-1

 

=> Y ^{n -1} = M . N ^{n -1} . M ^-1

 

\(∴N^{n-1}=\big(α^{n-1}_{1}0\ 0α^{n-1}_{2}\big)\)

 

Initially the nth term in terms of first two Fibonacci terms has been represented as:

 

\(\bigg(\frac{T_n}{T_{n-1}}\bigg)=Y^{n-1}\bigg(\frac{T_1}{T_{0}}\bigg)=Y^{n-1}\big(\frac{1}{0}\big)\)

 

\(\bigg(\frac{T_n}{T_{n-1}}\bigg)=M.N^{n-1}.M^{n-1}\big(\frac{1}{0}\big)\)

 

\(=>\bigg(\frac{T_n}{T_{n-1}}\bigg)=(α_1α_2\ 1\ 1)\cdot\big(α^{n-1}_{1}\ 0\ 0\ α^{n-1}_{2}\big)\cdot\frac{1}{\sqrt{5}}\big(1-α_2-1α_1\big)\big(\frac{1}{0}\big)\)

 

 

\(=>\bigg(\frac{T_n}{T_{n-1}}\bigg)=\frac{1}{\sqrt{5}}\big(α_1\ α_2\ 1\ 1\big)\cdot\bigg(\frac{α^{n-1}_{1}}{-α^{n-1}_{2}}\bigg)\)

 

\(=>\bigg(\frac{T_n}{T_{n-1}}\bigg)=\frac{1}{\sqrt{5}}\bigg(\frac{α^{n}_{1}-α^n_2}{α^{n-1}_{1}-α^{n-1}_{2}}\bigg)\)

 

\(∴T_n=\frac{1}{\sqrt{5}}\big(α^n_1-α^n_2\big)\)

 

\(=>T_n=\frac{1}{\sqrt{5}}\big(\frac{1+\sqrt{5}}{2}\big)^n-\big(\frac{1-\sqrt{5}}{2}\big)^n\)

 

\(∴T_{n-1}=\frac{1}{\sqrt{5}}\big(α^{n-1}_{1}-α^{n-2}_{2}\big)\)

 

\(=>T_{n-1}=\frac{1}{\sqrt{5}}\big(\frac{1+\sqrt{5}}{2}\big)^{n-1}-\big(\frac{1-\sqrt{5}}{2}\big)^{n-1}\)

 

Here, the two consecutive term and limiting ratio is given below:

 

\(\frac{T_n}{T_{n-1}}\)

 

\(\frac{\frac1{\sqrt5}(α^n_1\ -\ α^n_2)}{\frac1{\sqrt5}(α^{n-1}_1 \ -\ α_2^{n-1})}\)

 

\(\frac{α^n_1\big(1 - \frac{α^n_2}{α^n_1}\big)}{α^{n-1}_1\big(1 - \frac{α^{n-1}_2}{α^{n-1}_1}\big)}\)

 

\(\frac{α_1\big(1 - \frac{α^n_2}{α^n_1}\big)}{\big(1 - \frac{α^{n-1}_2}{α^{n-1}_1}\big)}\)

 

\(\frac{\big(1 - \frac{α^n_2}{α^n_1}\big)}{\big(1 - \frac{α^{n-1}_2}{α^{n-1}_1}\big)}\)

 

 \(= λ1 =\frac{1 + \sqrt5}{2} = 1.61803399\) 

 

Since,

 

\(\frac{α_2}{α_1}\)

 

\(=\frac{\frac{1-\sqrt5}{2}}{\frac{1+\sqrt5}{2}}\)

 

\(=\frac{(1-\sqrt5)^2}{(1+\sqrt5)(1-\sqrt5)}\)

 

\(=\frac{1-2\sqrt{5}+5}{1-5}\)

 

\(=\frac{6-2\sqrt{5}}{-4}\)

 

\(=\frac{3-\sqrt{5}}{2}\)

 

≈ - 0.3819

 

So,

 

\(-1<\frac{α_2}{α_1}<0\)

 

\(=>0<\bigg|\frac{α_2}{α_1}\bigg|<1\)

 

\(=>(\frac{α_2}{α_1})^n=0\)

 

Thus, it can be concluded from the above deductions, the ratio between two Fibonacci terms is 1.618. The Fibonacci series has been expressed in the characteristic vector equation, so this value remains a constant between any two terms of the Fibonacci series. The dependency of the architectural dimensions of the Taj Mahal and Notre Dame has been verified in the next section -

 

Verification - Verification of Golden ratio = 1.618 for the Taj Mahal architectural structure

Sample Calculation -

 

Value observed of 4^th Term = Expected Value of 3^rd Term × 1.618

 

= 16 × 1.618 = 25.888 ≈ 25

 

Error while calculation of 4^th Term = 25.888 - 25 = 0.888

 

Verification - Verification of Golden ratio = 1.618 for the Norte Dame architectural structure

The graph plotted above is a scattered plot. Here the relationship between absolute error and the nth term of the Fibonacci sequence by the application of the Golden ratio concept with respect to the number of terms considered for the calculation has been represented.

 

The absolute error can be observed to be declining which is exponential in nature with respect to the number of terms considered, the value of which has shown a decline from 2.326 to 0.888 when the number of terms shows an increment from 2 to 4. The intervals between the consecutive points are regular which is 1 term. The equation of trendline:

 

y = 6.0938×e^-0.481x

 

Here the absolute error while calculating a term has been represented by the y-axis and x represents the total number of terms considered for the expansion. Further we are analyzing the trendline:

 

0 = 6.0938 × e^-0.481x

 

e^-0.481 x = 0

 

ln ln e^-0.481x = ln ln 0

 

-0.481 x × ln ln e = -∞

 

x = ∞

 

Thus, we are concluding that the absolute error while calculating the nth term should be 0 where, n = ∞.

 

No significant outliers were observed in the graph. The value of R² is maximum indicating the accuracy of the graph.

The graph plotted above is a scattered plot. Here the relationship between absolute error and the nth term of the Fibonacci sequence by the application of the Golden ratio concept with respect to the number of terms considered for the calculation has been represented.

 

The absolute error can be observed to be declining in an exponential manner with respect to the number of terms, the value of which has shown a decline from 2.326 to 0.888 when the number of terms has been increased from 2 to 4. The intervals between the consecutive points are regular which is 1 term. The equation of trendline:

 

y = 6.9473 × e^-0.483x

 

Here the absolute error while calculating a term has been represented by the y-axis and x represents the total number of terms considered for the expansion. Further we are analyzing the trendline:

 

0 = 6.9473 × e^-0.483x

 

e^-0.483x = 0

 

ln ln e^-0.483x = ln ln 0

 

- 0.483x × ln ln e = - ∞

 

x = ∞

 

Hence, it can be concluded that the absolute error remains zero for nth term where n = ∞. No significant outliers were observed in the graph. The value of R² is maximum indicating the accuracy of the graph.