Mathematical and Scriptural aspect of Mekhala in the Construction of Yagya Kund

. Yagya kund has five parts - Khat, Nabhi, Kant, Mekhla and Yoni. The construction of Yagya kund shall be done with proper consideration of all of its parts. The final shape and appearance of the Yagya kund came after the construction of Mekhala. Mekhala is an object which surrounds another object by keeping that object in the middle. For Yagya kund, Mekhala surrounds the fire pit which provides the shape to the Yagya kund. The present paper attempts to describe the mathematical and scriptural aspects of Mekhala i.e. description of Mekhala and Khat, shape, number of Mekhala, and significance. The number of Mekhala varies by the size of Yagya Kund, the number of offerings, and the purpose of Yagya. The study also described the dimension of all types the Mekhala construction for 1 hast Kund indicating the important role of Mekhala in the Yagya Kund construction.


Introduction
Yagya is one of the main foundations of Indian Vedic culture. Yagya is not just a mere ritual for local welfare but it has wider applications. Veda describes that Yagya brings harmony and stability in the ecological cycles (1). In fact, the purview of Yagya includes the whole universe and all of its activity and functions. The proper flow of all the activities and functions of the universe is only possible because of Yagya (harmony & ecological balances), and hence, Atharveda states that अयं यज्ञो विश्वस्य भु िनस्य नावभिः । (Ayam Yagyo vishwasya buvanasya Nabhihi) -it means -Yagya is the nucleus of the universe (2). Besides universal philosophy, the practical portion of the ritual containing Tools, Kund, Mandap, etc has also been very well defined in terms of purpose, construction and performance proper consideration of all of its parts. The final shape and appearance of the Yagya kund comes after the construction of Mekhala. In scripture, Rishis recommend worshipping Yagya kund (shape and appearance) as the physical body of Deity (5) (6, shloka 31). The present paper discussed the shape, construction, numbers and significance of Mekhala.

Types of Khat (depth of Kund) with reference to Mekhala
Khat is a depth of Kund. There are two kinds of considerations for deciding the depth (Volume) of Yagya Kund i.e. 1) Kund Volume containing Mekhala with Khat, 2) Kund volume containing only Khat without Mekhala (8, Page 72). 1) Kund Volume containing Mekhala with Khat -In this type of Yagya kund the total volume of yagya kund includes depth of khat and height of Mekhala. For example, if the kund is 1 hast (24 angul), then the total volume of Kund will consist of the depth of khat (15 angul)  It implies that make 2 angul Kanth for 2 hast, 3 angul kanth for 3 hast kund, 10 angul kanth for 10 hast kund, and likewise (3, chapter अथ कं ठ: (atha kanthah)).
As above described, Sharada Tilak Tamtram recommends an increment of ½ angul of Kanth per increment for every 1 hast of Kund, while Yagya-Madhusoodan book recommends an increment of 1 angul of Kanth per increment for every 1 hast of Kund.

Construction of Mekhala
The size-dimension of Mekhala depends on the dimensions of Kund (1 hast, 2 hast, etc) and the number of Mekhala aimed for performing Yagya. The following description implies the consideration of the size variation of Mekhala to be considered for constructing Yagya kund with different numbers of Mekhala.
Sharda Tilak Tantram mentioned the same formula as described above and stated that it has been also mentioned by Indian scripture Siddhant Shekhar, Somshambhu, Mahakapilpanchratre and Pratham (7, page 208).
Other opinions in addition to above stated opinions for the dimension of Mekhala for 1 Mekhala Yagya Kund also mentioned in the scripture. Yagya-

For Yagya Kund with only Two Mekhala
According to 'र्ं िान्िोकक्' (Trantantrokte) in Sharda Tilak Tantram, for construction of 2 Mekhala yagya kund it has been mentioned that 'षष्टाम् शे नाष्टमां शे न मे खलाविर्यं मर्म् ।' (Shashtam shenashtmanshen mekhladvitayam matam) (7). It means for 2 Mekhala construction the first Mekhala dimension (outer height and width) will be ⅙ of the hast of the Kund and the outer height and width of the second Mekhala will be ⅛ of the hast of the Kund. For example for 1 hast kund (24 angul), the dimension (outer height and width) of the first Mekhala is 4 angul (⅙ of 24) and the dimension (outer height and width) of the second Mekhala is 3 angul (⅛ of 24) (7, page 208). The same reference of 'र्ं िान्िोकक्'

For Yagya Kund with only Three Mekhala
The description for three Mekhala Kund is widely available in the scriptures. The dimension of Mekhala for Yagya kund with three Mekhala is given in figure 3. In addition, the present study described 3 different opinions for Mekhala construction for Yagya kund with 3 Mekhala, which are depicted in Figure 4 and described as following.    It means for the Yagya Kund with three Mekhala, the inner-height of the first Mekhala that is from Kanth (from the ground) is 9 angul, followed by the inner-height of second Mekhala which is of 5 angul and the third Mekhala which is of 2 angul. The width of the same would be 4 angul, 3 angul and 2 angul respectively (Figure 3). In addition, the same dimensions for 3 Mekhala represented in Kriyasaren book has been mentioned by Maharshi Abhay katyayan, Mandap kund siddhi chapter 3, (8, page 81). The same dimensions also mentioned by Sharda Tilak Tantram book (7, page 206).

3) Three Mekhala calculations as per Vashist Sanhita
Besides two opinions of Mekhala calculations as described above, another opinion is also given in  (3, page 52). It implies that for Yagya kund with five Mekhala, the width of the first Mekhala will be 6 angul, followed by 5, 4, 3 and 2 angul. This is also said by Sharada Tilak as मे खला: पञ्च िा कायात िः षट् पञ्चावधधविपपक्षकै :' (Mekhalah panch va karya shatpanchabdhitripakshkaeh) (7, page 208)). The total width (from Kath to outside) would be 20 angul. The inner-height of the first Mekhala (total inner-height of Mekhala) is 9 angul and innerheight of the second, third, fourth and fifth Mekhala can be calculated by subtracting 1/5 of the total inner-height from the inner-height of previous  The inner-height of 1 hast Yagya kund with five Mekhala and three Mekhala has been the same i.e. 9 angul. For the five Mekhala kund, the total-innerheight (9 angul for 1 hast) is divided by five to find each of the Mekhala's outer-height i.e. 9 angul /5= 1/6/3 angul/yav/yuka for 1 hast Yagya kund.

Consideration of Number of Mekhala based on number of offerings (Yagya kund capacity) and purpose of Yagya
The number of Mekhala vary with capacity of the Yagya Kund and number of offerings of Yagya kund (6). Scriptures describe Kunds with the number of Mekhala varying from 1 to 7 as per the purpose as well as per the capacity of the Kund and purpose of Yagya (3, 6) (7, page 206) (8, page 80). Kund which is also not mentioned in Sharada Tilak book.

Preference of three Mekhala and Five Mekhala
In addition, the inner-height of the Mekhala is considered from the surface (Kanth or ground). To understand it better, it can be given in the following formula as 'Inner-height of the given Mekhala = outer-height of the given Mekhala plus sum of all the lower Mekhalas' outer-height'.

Consideration of total height of Mekhala for five or higher Mekhala Kund
For 1 hast Kund with Khat, it is defined that a total 9 angul of height shall be used for Mekhala construction. Total Mekhala height for 3 Mekhala kund is 9 angul which can be calculated based on the formulas given for each of Mekhala. However, for deciding the total height for the five Mekhala kund, there is no formula given as per the books referred to by the study. Though it covered the calculation of height for all five Mekhala by subtracting one-fifth of total height (first Mekhala's inner-height) from previous Mekhala, there is no indication how to choose the total height taken 9 angul as for 1 hast kund. Hence, the study recommends a total height of three Mekhala kund for different hast kund calculation. This consideration can also be applicable for seven, twelve or other higher numbers of Mekhala.
In the five Mekhala kund, total-height (of all Mekhal) is divided into five equal parts to find each individual Mekhala's height (outer-height). This consideration can be applicable for seven, twelve or any higher number of Mekhala kund construction i.e. 7 Mekhala kund and 12 Mekhala kund by dividing the total-height into 7 and 12 parts respectively. Thus, the height calculation of Mekhala for the five or higher number of Mekhala kund is defined.
The width and height of each Mekhala for any hast of Kund can be calculated using ratio of width to the hast used for 1 hast kund. The five Mehala kund width for the 1 hast kund is found in the scripture and is defined in the study. However, the present paper could not find the width of Mekhala for the seven and twelve number of Mekhala kund having a scope of further research.