Numerical Nataraja

 108 - Mathematical significance.

An attendant of Lord Shiva, Tandu teaches and instructs Bharta muni about 108 Karanas and angharas. Later, Bharta muni passed the knowledge of Karanas to his successors. Karnas are intricate and complicated dance movements of lord Shiva the creator of the cosmic dance. I always wondered why the number 108 is significant in the Hindu tradition and why the numbers of Karanas are restricted only to the number 108.

One can observe 108 Karanas on the wall of temples in south India. Few temples have all of 108 and some have less than 108. The sculptural illustrations are evident in the four temples of south India. The Rajarajeshwara temple in Tanjore, Nataraja temple in Chidambaram, Arunachaleshwara temple in Triuvanallmalai, and Vriddagirishwara temple in Vriddachalam

The numbers of sculptural illustrations vary in each temple, few temples represent Karanas by the carvings of male dancers and few by female dancers. The following is the list of a number of Karanas available in the five ancient temples dedicated to Lord Shiva.

Temple	Number of Karana illustrations
Rajarajeshwara temple	81
Natrajai temple in Chidambaram	108
Arunachaleshwara temple	108
Vriddagirishwara temple	101

 

Geometrical significance of 108.

Lord Shiva exists in nature in five different forms, Earth, Water, Fire, Wind and Space or sky. There are five dedicated temples to the Lord, where each represents one form of the lord. These temples are known as Pancha bhoota Sthalams. Among the five temples, only one exists in Andra Pradesh and the rest are situated in Tamil Nadu. 

 

	Forms of Lord	Temples	Location
1	Earth	Ekambreshwara temple	Kanchipuram
2	Water	Jambukeshwara temple	Thiruvanikaval
3	Fire	Arunachaleshwara temple	Thiruvannamalai
4	Air	Kalahasti temple	Sri kalahasti
5	Space	Thulllai Natarajar temple	Chidambaram

 

   The five forms of Shiva are five dots placed at equidistance on the circle of time.

   The line joining the dots creates a pentagon in the circle. 

   Observe the pentagon.

TThe Sum of interior angles of the pentagon is 540°. To find the angular value of each angle, I divided the sum 540° by 5. Therefore

540°/5 = 108°.

Connecting the five dots on the circle will result in a pentagon and drawing five diagonals to the pentagon will result in a start like structure called a pentagram. The pentagram is made of five triangles and one pentagon at the centre. The regular pentagon has an angle equal to 108°. 

 

108 is the product of the first three numbers raised to themselves: 

108 = 1¹ * 2² * 3³ 

So, it is the hyper factorial of 3 since it is of form 1¹ * 2² *3³. The value of each number is 1, 4, and 27 respectively.

 

According to the India calendar system, there are 27 Nakstras (stars). The 27 Nakshatras cover 13°20’ of the ecliptic each. Each Nakshatra is also divided into quarters or padas. Hence 27*4 = 108. Shiva is represented as the start in four quarters. When you observe the Nataraja, you can find the idol is enclosed within the star.

An interesting fact to observe here is the number 108 – 27= 81. The total number of Karanas in the Rajarajeshwara temple. Also, the sum of the prime factors of 108 is 2 + 2 + 3 + 3 + 3 = 13. the partition {1, 3, 3, 3, 3} has a product of 81.

This gave me an idea to factorise each number using the prime factorisation technique. Surprisingly I found, 108 is the smallest number which can be partitioned into six distinct primes such that the sum of any five numbers is also a prime: 

108 = 5 + 7 + 11 + 19 + 29 + 37. IT can be noted that the sum of any five out of 5, 7, 11, 19, 29 and 37 is always prime. 5 + 11 + 19 + 29 + 37 = 101. This is the total number of Karanas in the Vriddagirishwara temple.