Existence of the golden ratio in Tanjavur Brihadeeshwarar temple

In this study, we discussed the existence of golden ratio in Brihadeeshwarar temple, Tanjavur, Tamil Nadu, India, built in 1010 AD. It is listed on the UNESCO’s world heritage site of the Chola temples in southern India. This temple represents an outstanding creative achievement in the architectural idea of the pure form of the Dravida temples. Golden ratio has great influence in architecture, mathematics and art. We analyzed existence of the Golden ratio in structural design of Tanjavur Brihadeeshwarar temple prakaram. We used the Phi Grid and Phi Spiral software to measure the golden ratio and verified our result.


Introduction
T he Golden ratio, mathematics and aesthetics are intricately associated between each other. The Golden ratio is an irrational number that is approximately equal to 1.618, which is represented by the Greek symbol known as phi (φ). This ratio is considered as the human optical sense and not limited to aesthetic beauty but also found its existence in natural world through the body proportions of living beings, the growth patterns of many plants, insects and also in the model of enigmatic universe. The properties of Golden section can be instituted in the pattern of mathematical series and geometrical patterns. Because of its unique and mystifying properties, many researchers and mathematicians have studied the Golden ratio. Akhtaruzzaman and Shafie [1] discussed the geometrical substantiation of φ, the Golden ratio and the Baroque of nature, architecture, design and engineering. Golden ratio is key to the secret of beauty Saraf et al., [2]. Thapa et al., [3] studied the relation of Golden ratio, mathematics and aesthetics. Craciunet et al., [4] analyzed the generalized Golden ratio as a fixed point of an operator defined by an arbitrary mean satisfying certain conditions. Iosa et al., [5] discussed Phi in physiology, psychology and biomechanics: The golden ratio between myth and science. Sen and Agarwal [6] investigated Golden ratio in science, as random sequence source. Proportions in architecture are discussed by Singh [7]. Fett [8] studied an in-depth investigation of the Divine ratio. Recently, many authors discussed golden ratio, see [9][10][11][12].
Keeping this in mind, in present work we investigated existence of the Golden ratio in structural design of Tanjavur Brihadeeshwarar temple prakaram. We used the Phi Grid and Phi Spiral software to measure the golden ratio.

Golden ratio analysis: Tanjavur Brihadeeshwarar temple
The Brihadeeshwarar temple is a Hindu temple dedicated to god Shiva located in Tanjavur. This temple plan and improvement make practical and effective use of the axial and symmetrical geometry rules. The temple wall consists of an entrance with a double row of pillars. The perimeter wall forms a rectangular monastery which could be divided into 2 squares. The centre of the first square contains the Nandhi pavilion and second contains the god Shiva statue. In this temple main tower (Gopuram) height is 60m. The tower of the main gateways is 30 m high. In this space there are five main sections: the sanctum with the towering superstructure (srivimana), the Nandhi pavilion in front (Nandi-mandapam) and in between these the main community hall (mukhamandapam), the great gathering hall (mahamandapam) and the pavilion that connects the great hall with the sanctum (ardhamandapam) (Geva and Mukherji [13] and Croker [14]). The Figure 1  points out the ideal places are represented by numerically. In Figure 1, A is the sanctuary, B is the Artha mandapam, C is the mahamandapam, D is the Nandi-mandapam, E is the Gopuram and F is the Shrine of subrahmanya.

Working rules
The following rules are helpful to measure the Golden ratio in the Brihadeeshwarar temple courtyards (see Figure 1 and 2).  Now, we get four points S, T, U and V from that construction. Using that points draw square S 1 T 1 U 1 V 1 where S, T, U, V are midpoints of sides of that square. The square S 1 T 1 U 1 V 1 identify the golden section of BDGI (see Figure 3).
In that figure we consider the portion between middle of karpagraham to middle of nandhi pavilion and evaluate the Golden ratio between them from middle of temple prakaram.
From line segment LM: From line segment CH: From line segment BG: From line segment ID: Similarly repeat the same procedure for other two portion.

Golden ratio in entrance of temple to middle of temple[EFHC]
In Figure 4, we consider the portion between entrance of the temple to middle of the temple and evaluate the Golden ratio between them from nandhi pavilion.
From line segment C 0 N: From line segment DG: From line segment CF: From line segment HE:

Golden ratio in middle of temple to back end of temple [ACHJ]
From Figure 5, we consider the portion between back end of the temple to middle of the temple and evaluate the golden ratio between them from karpagraham.
From line segment KC 0 : From line segment BI: From line segment AH: From line segment CJ:

Golden ratio between outer and inner region of temple Prakaram
From Figures 6 and 7, we measure the ratio between the sides of each outer region (i.e. [EFHC], [BDGI] and [ACHJ]) of temple Prakaram to outer region of Karpagraham, Nadhi pavilion and middle of temple respectively.
Let the portion between entrance of the temple to middle of the temple.ie.
[EFHC](see Figures 6 and 7), if we take CE = EF = FH = HC = φ (say) then Hence we get the ratio And middle of temple to back end of temple, i.e., [ACHJ] (see Figures6 and 7), we get

Measuring Golden ratio using software
Software,s are used to reduce the complexity of calculation and measurements. We used some available software to ensure our results. The following two software's are used to measure the Golden ratio and we get the following outputs.
(i) Phi Grid: using the Phi grid software, we get the same result (see Figure 8).
(ii) Phi Spiral: using the Phi Spiral software, we get the same output (see Figure 9).