Temples of India are living testimonies of a past where spirituality found stunning expressions through art and architecture. Every facet of a temple structure is suffused with aesthetics meant to evoke an emotional response that is beyond oneself. The sculptural and architectural forms of temples are guided by a vast body of design rules codified in the shilpa shastra and vasthu shastra texts handed down, and adapted to their present forms. These texts exemplify the heights of creative achievements of the past in the fields of art, architecture, science, and engineering; feats that remain extraordinary measured against even the best of present standards.
The Sri Padmanabhaswamy temple in Thiruvananthapuram is a classic example of this. The temple stands in seven acres of land with four main entrances facing the prime directions. The principle deity of the temple, Sree Padmanabha Swamy, is in meditative Anantha-Shayana posture, reclining on Adi Sesha. The 18 feet long idol’s interior is filled with 1208 salagramas (sacred stones) transported from the Gandaki River in Nepal. The idol has an orientation precisely perpendicular to the East – West. Some of the additional deities of the temple are also positioned this way. The orientation along the east – west adheres to the agama design principles in the classical texts such as the samarangana sutradhara and tantrasamuchaya which recommend Vishnu temples fronting east such that the main deity of the temple is facing the rising Sun [1, 2]. Texts like manasara also maintain that the temples of Vishnu should have their main entrance facing the town. In olden days, the central part of Thiruvananthapuram city was primarily to the east of the Padmanabhaswamy temple.
A visual hallmark of the temple is the gopuram, a tower made of granite and brick that rises 100 feet above the ground, with a foundation that is nearly 40 feet deep. It has a pyramidal structure and rests above the main east entrance to the temple. Ornate sculptures adorn the walls of this lofty edifice. As per old temple records, the foundation of the gopuram was laid during the 16th century CE with the construction largely getting completed in the 18th century during the reign of the Travancore king Anizham Thirunal Marthanda Varma . The temple however steeps far further back in antiquity.
There are seven floors to this gopuram, with a pair of window-like openings along the center on opposite sides of six of these floors. The opening on the lowest floor is also the grand doorway to enter the temple. The temple door and the window openings on the five floors are aligned vertically from the bottom to the top of the gopuram. At the top of the edifice are a set of seven kalasha evenly spaced.
The precise orientation of the gopuram becomes evident through a rare visual spectacle that unfolds twice every year. On the days of equinox, one gets to see the setting Sun passing exactly through all the window openings in the gopuram in succession, in roughly five minute intervals (see Figure). On days past the equinox, the setting Sun gets blocked by the tower structure.
These photographs were taken during September 23, 2017 equinox. Credits: Vijayan Madhavan
Equinoxes correspond to the two days of a year when the day and night are of equal duration. On these days, the Sun rises exactly east and sets exactly due west. The equinoxes now fall roughly on March 21st and September 22nd give or take a day from year to year. On the other days, the sunrise and sunset happens along directions north (or south) of east and north (or south) of west (see appendix 1). To accomplish this visual spectacle on the days of equinox, the imposing gopuram structure had to be constructed at right angles to the exact east – west line, a feat requiring precision engineering.
The layout of the Padmanabhaswamy temple with the gopuram at the East entrance, and the E-W, and N-S directions marked.
Defining the precise east – west direction is not an easy task. What techniques did the sthapathis employ to determine this? The answer to this can be found in the mathematics and astronomy texts of ancient India. Though magnetic compasses were in use for navigational purposes as early as the 11th century CE, it was known that the magnetic directions of north and south do not coincide with the geographical north and south, the difference though minimal for regions close to the equator. Much earlier to the development of compasses, there were simple geometrical techniques widely used to establish the cardinal directions. The earliest records on this can be found in the collection of texts called sulbasutras.
The sulbasutras are some of the oldest written records of mathematics (see appendix 2). They primarily contain geometrical techniques for the construction of structures for different rituals. One of the topics dealt with in sulbasutras is the determination of prime directions by observing the Sun on any given day of the year. Mahadhira (16th century CE) in his commentary to Katyayana sulbasutra explicitly states that it would be incorrect to assume East based on the direction of dawn on any arbitrary day [3, 4].
तस्य उदयस्थानानां बहुत्वात् प्रतिदिनं भिन्नत्वात् अनियमेन प्रची न्जातुं न शक्या । तस्मात् शङ्कुस्थापनेन प्राचीसाधनमुक्तम् । दक्षिणायने चित्रापर्यन्तमर्कोअभ्युदेति । मेशतुलासङक्रात्यहे प्राच्यां शुध्दायामुदेति । ततो अर्कात् प्राची न्जानं दुर्घटम् ।
As the rising points (of the Sun) are many, differing from day to day, the east-west direction cannot be known just by observing the sunrise or sunset (on any arbitrary day). It is suggested that the east-west be determined by fixing a shanku (pointed stick) etc.
Katyayana sulbasutra explains a simple set of observations that can be done on any given day to establish the E-W line [3, 4]:
समे शङ्कु निखाय शङ्कुसम्मितया रज्ज्वा मण्डलं परिलिख्य यत्र लेखयोः शङ्कुग्रच्छाया निपतति तत्र शङ्कु निहन्ति सा प्राचि । (Katyana Sulbasutra 1 2)
Fix a stick on a level ground; draw a circle with a string measured by the stick; fix nails at points on the line (of the circumference of the circle) where the shadow of the tip of the stick falls (at two different times of the day). The line joining the two points forms the east-west line.
Here the word shanku refers to gnomon, or most simply a stick with a pointed tip, rajju refers to rope, and prachi is east (whereas udichi is north)
(An illustration of the technique suggested in sulbasutras for determining exact E-W. The shadow of the stick changes its size and orientation with the passing of time from sunrise to sunset. The exact E-W would correspond to the two points where the edge of the shadow touches the periphery of the circle drawn with the stick as the center.)
In projective geometry, there is the concept of vanishing point. Lines that are drawn parallel to each other will appear to meet at infinity. Thus it would not matter from where we do the above experiment. The line drawn by every person doing the same experiment will be pointing East – West. This method is now famously known globally as the “gnomon-circle method” or the “Indian circle method”.
The technique is staggeringly simple, wanting barely any paraphernalia except for a stick and a rope. One finds more elaborate mentions of the same approach in some consequent texts of mathematics. For example, in the highly celebrated work tantrasangraha written by the brilliant mathematician Nilkanta Somayaji who lived during the 15-th century CE in Kerala, there is a complete description of this method of finding the East-West .
Tantrasangraha goes further by pointing out a minor correction that needs to be applied to get the direction precisely right. Since the entire set of observations lasts for several hours, the apparent position of the Sun relative to the distant set of background stars would change within that duration. The Sun would have moved a little bit north or south in its journey depending on the time of the year. This truly happens due to the incremental changes in the position of the Earth in its orbit around the Sun. Tantrasangraha recognizes this, and suggests purva-para-bindhu-shodanam as the procedure for correcting this offset (see appendix 3). The method is also mentioned in the famous Siddhanthasiromani of Bhaskaracharya, who lived during the 12th century CE.
The stick-circle method, first described in the sulbasutras, and which finds recurrent mention in many subsequent texts, is what the stapathis (architects) of the Padmanabhaswamy temple are likely to have used to get the orientation of the temple right. It takes the setting Sun only about half an hour to move down through the windows of the 100 foot long gopuram of the temple. The time frame is too short for the additional fine correction to be of consequence. But having access to this knowledge, and an impulse to achieve perfection, it is likely that the correction was also taken into consideration while fixing the orientation of the gopuram orthogonal to the east-west.
(The top view layout of the temple. The gopuram and the idol of Padmanabhaswamy are positioned exactly perpendicular to the east west line.)
The tower and the temple of the Sri Padamanabhaswamy stand as a testimony to the architectural brilliance and scientific knowledge of the sculptors who built this beautiful edifice three centuries ago. Many local residents are aware of the visual spectacle that unfolds during the equinox.
But beyond that, these facts are not widely known. No science or history curriculum of the state mentions in detail the mathematics of sulbasutras or the tantrasangraha, despite the latter being a monumental treatise written in Kerala five centuries ago. The temple is a living tangible example of how mathematics and astronomy can be unified into complex architectural design, but it is seldom taught to children in those terms as a locally accessible example. At a time when educationalists and policy makers are thinking long and hard for ways to blend STEM education with the arts, and allied fields, the absence of such examples in our formal curriculum could only mean one of two things. Either we have not looked deep into regional and micro-history to pick out these brilliant nuggets from the past, or that we have chosen to consciously ignore them. One can hope that it is not the latter.
ANAND NARAYANAN / ILLUSTRATIONS BY HARSHA SIMHA
Anand Narayanan and Harsha Simha are faculty at the Indian Institute of Space Science and Technology, where they teach astrophysics and control systems respectively.
Appendix 1: The orbital plane of the Earth around the Sun, and the Earth’s equatorial plane projected on to the sky keep an angle of 23.5 degree with respect to each other. As a result, the rising (and setting) of the Sun does not always happen exactly due East (and West). The diagram below explains how the rising point of the Sun drifts along the Eastern horizon over a full year.
(The drawing shows the rising positions of the Sun drifting over the course of a year from North to South and back.)
A year is thus divided into two halves. The 6 months from December to June when the rising point of the Sun slowly drifts from South to North is uttarayana, and the 6 months from June to December when the Sunrise point drifts from North to South is called dakshinayana (where ayana means journey). The setting Sun follows the same pattern. If the sunrise happens north of east, the sunset will be north of west. On March 21 and September 22 (give or take a day), the Sun would rise exactly from East and would set exactly into West.
Appendix 2: Sulbasutras are part of kalpasutras, belonging to vedangas, which are sometimes categorized as auxiliary limbs of the main corpus of Vedas. These texts deal with the practical aspects of vedic rituals, primarily the construction of enclosures and altars for the rituals. The word sulba is derived from the root sulb which means “to measure”. Presently, seven Sulbasutras have been identified. These are Baudhayana sulbasutra, Manava sulbasutra, Apastamba sulbasutra, Katyayana sulbasutra, Maitrayana sulbasutra, Varaha sulbasutra, and Vadhula sulbasutra. According to scholars, the oldest among these is Baudhayana, dating back to approximately 800 BCE, with all the others written somewhere between 800 BCE and 300 BCE. These texts thus form one of the oldest written material on mathematics in India .
Appendix 3: Tantrasangraha is a positional astronomy treatise written by Nilakanta Somayaji, from the illustrious lineage of guru-sishya parampara starting from Sangamagrama Madhava. All of them lived in the Northern parts of Kerala from 14th to early part of 19th century CE. Historians collectively refer to it as the Kerala School of Mathematics and Astronomy . The chapter 3 of trantrasangraha titled छायाप्रकरणम (chaaya-prakaranam) deals with sticks and shadows, with changing positions of the Sun . The first two verses of this chapter bears instructions on how to how to position a stick on a flat surface, and how the shadow would move over the course of a day. The third verse mentions the correction to the east point because of the changing declination position of the Sun during the course of the observation of the shadows from morning to evening.
In the figure (adapted from ) the hemisphere is the sky, and AB represents the stick. The smaller circle is what one would draw with the stick as the center. The larger circle is the horizon. Here W and E1 are the points that we get on the circle corresponding to the points of intersection of the tip of the shadow with the circumference in the forenoon and afternoon respectively. Without the fine correction mentioned in tantrasangraha, E1-W would be identified as the east – west line. The correction is the distance OE2. The corrected east-west direction will be E2 – W. The offset distance is given by
where Φ is the latitude of our location, and the two δ angles are the declination positions of the Sun in the forenoon and afternoon when the shadow of the stick touches the periphery of the circle. The same is also given in the siddhanthasiromani text written by the 12 century CE astronomer and mathematician Bhaskaracharya (Bhaskara – II).
 Sree Padmanabhaswamy Temple, Aswathi Thirunal Gouri Lakshmi Bayi, Bharatiya Vidya Bhavasn/Mumbaii/India; 2013. ISBN-13: 978-8172764791. For a metaphysical reasoning on why the east orientation is important, the readers are encouraged to refer chapter 6 of the book.
 The Hindu Temple (Volume – 1), Stella Kramrisch, 2015, Motilal Banarsidass Publishers; Ninth Reprint edition, ISBN-13: 978-8120802223
 Katyayana Sulbasutra – with Sanskrit commentaries of Karka & Mahidhara, English Translation of Hindi commentary, Explanation and Geometrical Figures, D. P. Kularia, Devesh Publication; 2009, ISBN-13 : 978-8189580810
 NPTEL lectures of IIT-Madras on Mathematics in India – From Vedic Period to Modern Times, by Dr. K. Ramasubramanian, Dr. M. D. Srinivasan, and Dr. M. S. Sriram – https://nptel.ac.in/courses/111/101/111101080/
 Tantrasangraha of Nilakanta Somayaji – K. Ramasubramanian, and M. S. Sriram, Springer London Dordrecht Heidelberg New York, co-published with Hindustan Book Agency, 2011, ISBN 978-0-85729-035-9. Chapter 3 is particularly relevant
 Sriram M.S. (2008) Kerala School of Astronomy and Mathematics. In: Selin H. (eds) Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-4425-0_8683.
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