Kacchaputa and Sarvatobhadra- AI Comments

  Kacchaputa of Nagarjuna (c.100 BCE)   The elements in the magic square are given using the Katapayadi system of specifying numbers by the string

  Half the blocks are filled with zeros.

  Example of a normal and a pan-diagonal (PD) magic squares:

    A normal Magic Square (Sum = 34)

   A Pan-diagonal Magic Square

  

10	3	13	8
5	16	2	11
4	9	7	14
15	6	12	1

 

12	3	6	13
14	5	4	11
7	16	9	2
1	10	15	8

 PD Sum: 6+5+7+8=34

PD Sum: 13+16+4+1=34   N−3  1   N−5   8

  N−6  9   N−4   2

  6  n−7    3   n−1

  4  n−2     7   n−8

        Pan-diagonal with total 2n

Total 2n + 1           

 Kacchaputa of Nagarjuna (c.100 BCE)

The following pan-diagonal magic square totaling to 100 has also been called Nagarjunıya

   30   16   18   36

  10   44   22   24

  32   14   20   34

  28   26   40   6                           

 Sarvatobhadra of Varahamihira (550 CE)

In the Chapter on Gandhayukti of Brihatsamhita ̄, Varahamihira describes the

Sarvatobhadra perfumes 

. In the Kacchaputa with sixteen cells, [are placed] two parts of agaru, three parts of patra, five parts of turuska and eight parts of  ́saileya in the cells of the first row,.... When these are mixed in whatever way, there will be 18 parts. To such a mixture are added nakha ..., in equal measures, and in this way the sarvatobhadra are produced.

  2    3  5    8

 5    8  2    3

 4    1  7    6

 Sarvatobhadra of Varahamihira (550 CE) As the commentator Bhattotpala (c.950) explains:

  2    3   5   8

 5    8   2   3

 4    1   7   6

 7    6   4   1                 

 Sarvatobhadra of Varahamihira (550 CE)

Translation of the commentary of Bhattotpala

In this kacchaputa with 16 cells, when four substances are mixed in whatever way: When the four substances with their mentioned number of parts are mixed, then the total will be 18 parts; this happens in the above kacchaputa when the perfumes are mixed from top to bottom (along the columns) or horizontally (along the rows), along the four directions, or the central quandrangle, or the four corner cells, or the middle two cells of the first row together with those of the last row; the middle two cells of the second and third row or the first and last cells of the same, or in any other manner. If the substances in such four cells are added there will be 18 parts in all. ... Since, in whatever way they are mixed, they lead to 18 parts, they are called Sarvatobhadra.

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Suresh, this is a profound convergence of mathematical structure, symbolic encoding, and ritual metaphor—a testament to the layered sophistication of ancient Indian thought. Let me help you distill and connect the two remarkable traditions you've highlighted: Nāgārjuna’s Kacchaputa and Varāhamihira’s Sarvatobhadra, both rooted in the 4×4 grid but serving vastly different purposes.

🐢 Kacchaputa of Nāgārjuna (c. 100 BCE)

A Pan-Diagonal Magic Square with Symbolic Encoding

🔹 Structure:

• A 4×4 grid, half-filled with zeros.

• Remaining cells filled using the rule:

$$\text{If }x\text{ is an alternate diagonal element, then fill }0\text{ with }N-x\backslash text\{If\; \}\; x\; \backslash text\{\; is\; an\; alternate\; diagonal\; element,\; then\; fill\; \}\; 0\; \backslash text\{\; with\; \}\; N\; -\; x$$

• The resulting square becomes pan-diagonal, meaning:

  • All rows, columns, diagonals, and broken diagonals sum to a constant.

  • Total sum = 2N or 2N + 1, depending on the configuration.

🔹 Symbolic Encoding:

• Uses the Katapayadi system, where syllables encode numbers.

• This allows the square to carry linguistic meaning, possibly mantric or poetic.

🔹 Example Grid:

Code

30   16   18   36  
10   44   22   24  
32   14   20   34  
28   26   40    6  

✅ All rows, columns, and diagonals sum to 100 This square, referred to as Nāgarjunīya, may have been used in alchemy, astrology, or tantric ritual, where numerical balance symbolized cosmic harmony.

🌸 Sarvatobhadra of Varāhamihira (c. 550 CE)

A Grid of Perfumed Proportions and Auspicious Symmetry

🔹 Context:

• Found in the Gandhayukti chapter of the Bṛhatsaṃhitā.

• Describes mixing of four substances:

  • Agaru (2 parts), Patra (3), Turuska (5), Śaileya (8)

🔹 Grid:

Code

 2   3   5   8  
 5   8   2   3  
 4   1   7   6  
 7   6   4   1  

🔹 Properties:

• Any group of four cells—row-wise, column-wise, diagonally, corners, central quadrangle, mirrored pairs—adds up to 18

• This universal summing property gives it the name Sarvatobhadra (auspicious from all sides)

🔹 Bhattotpala’s Commentary (c. 950 CE):

“In whatever way the substances are mixed—rows, columns, diagonals, corners, or mirrored pairs—the total is always 18. Hence, it is Sarvatobhadra.”

This square is not just mathematical—it is ritualistic, aesthetic, and symbolic. It encodes balance, fragrance, and harmony, possibly used in perfume formulation, astrological rituals, or poetic composition.

🔗 Comparative Reflection

Feature	Kacchaputa (Nāgārjuna)	Sarvatobhadra (Varāhamihira)
Grid Type	4×4 pan-diagonal magic square	4×4 symmetric sum grid
Mathematical Basis	Arithmetic + substitution	Fixed values with universal sum
Symbolic Encoding	Katapayadi (syllabic numerals)	Perfume proportions (ritual blends)
Cultural Use	Possibly alchemical or tantric	Astrological, poetic, ritualistic
Sum Property	Rows, columns, diagonals = 2N or 2N+1	Any 4-cell grouping = 18

✨ Philosophical Insight

Both squares reflect a deep Indian vision of harmony:

• Kacchaputa evokes cosmic balance, like the tortoise supporting the world.

• Sarvatobhadra evokes sensory and elemental balance, where fragrance becomes a metaphor for auspiciousness.

These are not just grids—they are mandalas of meaning, where mathematics meets mantra, and structure meets spirit.

construct a Sarvatobhadra-style grid

Let’s construct a Sarvatobhadra-style grid inspired by Varāhamihira’s formulation in the Bṛhatsaṃhitā, where the sum of any meaningful grouping of four cells—rows, columns, diagonals, corners, central quadrangle, mirrored pairs—adds up to a fixed total. This total, in the original perfume grid, was 18 parts.

🧮 Sarvatobhadra Grid (Sum = 18 for any 4-cell grouping)

Code

 2   3   5   8  
 5   8   2   3  
 4   1   7   6  
 7   6   4   1  

✅ Verifications

• Rows:

  • 2 + 3 + 5 + 8 = 18

  • 5 + 8 + 2 + 3 = 18

  • 4 + 1 + 7 + 6 = 18

  • 7 + 6 + 4 + 1 = 18

• Columns:

  • 2 + 5 + 4 + 7 = 18

  • 3 + 8 + 1 + 6 = 18

  • 5 + 2 + 7 + 4 = 18

  • 8 + 3 + 6 + 1 = 18

• Diagonals:

  • 2 + 8 + 7 + 1 = 18

  • 8 + 2 + 1 + 7 = 18

• Corners:

  • 2 + 8 + 7 + 1 = 18

• Central Quadrangle:

  • 8 + 2 + 1 + 7 = 18

• Mirrored Pairs:

  • (3 + 6) + (5 + 4) = 9 + 9 = 18

  • (5 + 4) + (3 + 6) = 18

🪷 Symbolic Interpretation

In Varāhamihira’s original, the numbers represented perfume ingredients:

• Agaru (2), Patra (3), Turuska (5), Śaileya (8)

These were arranged so that any combination of four cells yielded a balanced blend—symbolizing auspiciousness from all directions, hence Sarvatobhadra.

Would you like to adapt this grid to encode Sanskrit syllables using the Katapayadi system, or reinterpret it using planetary attributes, poetic meters, or musical notes? We can transform this into a living yantra of meaning.