Last year I noticed an intriguing correlation between numerical sums within Pascal's triangle and the sums of the letters of the English alphabet numerated under the three most commonly-employed systems of gematria. I looked at it after discussions with Leo Tavares, who claimed that the Hebrew alphabet was self validating, based on the sums and mirror sums of its letter numerals.

The three substitution systems I normally apply to the English alphabet are given below, along with their sums and the sums of the mirrored values (eg 21 becomes 12).

The standard value system (based on the Hebrew

A 1, B 2, C 3, D 4, E 5, F 6, G7, H 8, I 9, J 10, K 20, L 30, M 40, N 50, O 60, P 70, Q 80, R 90, S 100, T 200, U 300, V 400, W 500, X 600, Y 700, Z 800

Sum =

Mirror sum =

The ordinal value system (same as the Hebrew ordinal method)

A 1, B 2, C 3, D 4, E 5, F 6, G7, H 8, I 9, J 10, K 11, L 12, M 13, N 14, O 15, P 16, Q 17, R 18, S 19, T 20, U 21, V 22, W 23, X 24, Y 25, Z 26

Sum =

Mirror sum =

The reduced value system (based on the Hebrew

A 1, B 2, C 3, D 4, E 5, F 6, G7, H 8, I 9, J 1, K 2, L 3, M 4, N 5, O 6, P 7, Q 8, R 9, S 1, T 2, U 3, V 4, W 5, X 6, Y 7, Z 8

Sum =

Mirror sum =

All of these numbers are found within rows 0 to 11 of Pascal's triangle, shown below. Note that the blue outline triangle encloses the number 126 twice at its base and as the digits of 126 down the spine of Pascal's triangle, giving a '126' associated with each vertex of the blue triangle.

Note: 729 is 9 cubed and this formula for calculating cubes within Pascal's triangle was discovered by Tony Foster. https://www.cut-the-knot.org/arithme...InPascal.shtml. It can be used to calculate any cube by centering the hexagon over any member of the natural number sequence running down the triangle. The formula calculates the cube of that central number.

I thought it interesting that the central 9 is found on the same row as the base of the blue outline triangle I drew, where two of the 126s are found.

It works without adding the mirror sums. Then you would have 4095 (sum of rows 0 to 11), 351 (sum within the blue triangle) and 126 (the number at each vertex of the blue triangle), for the standard, ordinal and reduced sums. But the mirrors are all there too and Pascal's triangle itself is mirrored either side of its central spine, so the idea of mirroring is found within the triangle itself.

I think it's intriguing evidence that the structures of the English alphabet and of those three systems of English gematria were designed. The numbers are what they are, of course, and couldn't be any different, but there needn't have been 26 letters in the alphabet, and the denary counting system wasn't inevitable either. There are many possible systems of gematria, but those three are arguably the simplest and the most natural, as simple and natural as Pascal's triangle itself. 26 is the ordinal value of 'God' and the standard and ordinal values of