The phrase we will be looking at has 4 words and 14 letters in the original Hebrew, “it has turned some white hair” (ןבר לעה שהכפ האיהו). The value of the entire phrase is 784, or 28 squared. Apart from being an independent phrase grammatically, when the value of a phrase is a square number, that too indicates its inner coherence.
Since 28 squared is the same as 2 squared times 14 squared, and 2 squared is 4, the average value of each word in this phrase is 14 squared, or 196.
Since the total value of the phrase is 28 times 28, which is the same as 28 times 2 times 14, and we have 14 letters, it follows that the average value of each letter is 2 times 28, or 56. Let us take a closer look at this product, 2 times 28. 28 is a figurate number, meaning that we can take 28 dots (or letters) and arrange them graphically in the form of a triangle. In this case, 28 is the triangle of 7, which looks like this:
Now, 2 times 28 can be arranged as two triangles back-to-back, or a diamond shape. Thus, 56, the average value of each letter in the phrase is the diamond of 7.
Furthermore, we now have seen that the value of the entire phrase is 28 times 28, whereas the average value of each letter in the phrase is 28 plus 28!
The final letters in the phrase, “it has turned some white hair” (ןבר לעה שהכפ האיהו) spell “Aaron” (ןרהא), in sequence, whose value is 256 or 16 squared. The remaining letters then equal 784 minus 246, or 528, which is also a triangular number, specifically the triangle of 32, which we write using the following notation: r32. But you will note that 32 is 2 times 16. So, we have found a very interesting, not entirely trivial, mathematical relationship:
= 16 ┴ r(16 ┴ 16)
The value of a triangular number is easily calculated using the definition: rn = n(n ┴ 1)/2 Thus, we have that = 16 ∙ 16 ┴ 32/2 ∙ 33, or = 16 ∙ 16 ┴ 16 ∙ 33 = 16(16 ┴ 33) = 16 ∙ 49 = 4 ∙ 7
So we have found that the square of 28 is equal to the product of two other squares, 4 and 7! This is especially beautiful because 28 is the product of 4 and 7. A challenge for the reader: is there another triplet of numbers for which this is true?
Alternating Words and Letters
16, the root of the value of “Aaron,” figures even more prominently in this phrase when we consider alternating words. The sum of the first and third words—רעשיהו—is 592, or 16 times 37. The sum of the second and fourth words—ןבל הכפה—is 192, or 16 times 12! Of course, we already know that the sum of 37 and 12 must be 49, or 7 squared, as above.
Let us do the same for the letters in the phrase. The sum of the first set of alternating letters—ויהכשרב—is 543, which incidentally is the value of God’s connotation, “I will be that which I will be” (היהר אש היהא). The sum of the second set of alternating letters—האפהעלנ—is 241, the secret of the acronym רמא, which stands for the three stages of formation: light, water, and firmament (יעיקמרוא).
Mathematically, 241 is the covenant number of 16, which means that 241 dots can be arranged in the form of two triangles of 16, placed apex-to-apex, with another dot called the “sign of the covenant” in between the two apexes.
But what is even more interesting is that just as the sum of the two alternating sets is 784, a square number, so their difference: 592 minus 192 is also a square number, 400, the square of 20.
We can divide the second set of alternating letters—האפהעלנ—further by looking at the two sets of alternating letters within. The first set will be the letters הפענ, whose sum is 205, and the second set will be אהל, whose sum is 36. Now we have divided the phrase into 3 parts whose values are 543, 205, and 36.
We can use Newton’s method of “finite differences” to create a discrete quadratic series out of these 3 numbers:
The number in the last row, 169, is known as the series base. 169 is the square of 13.
We can use the base to extend the series forward:
The next number is thus 1050, and the sum of all four numbers in the series (36, 205, 543, and 1050) is 1834—a number that has two interesting properties:
First, 1834 is the sum of the first 14 interface numbers (also known as “centered square numbers” in mathematics). Interface numbers are defined as the sum of two consecutive square numbers, such that f[n] = n ┴ (n - 1). The first few interface numbers are thus 1, 5, 13, 25, 41, etc.
Second, 1834 is a special 3-dimensional figurate number known as an octahedral number. Here is what the first few Octahedral numbers look like:
It should be clear that an octahedral number is composed of two square-pyramids, where each pyramid is a sum of squares. Graphically, it should not be very hard to convince ourselves that every Octahedral number is actually a sum of interface numbers. And so we have that:
If we would like a simpler function for the Octahedral numbers, we can use the following: Oct[n] = n(2n ┴ 1)/3
Incidentally, whenever referring to figurate numbers, we usually reference the Online Encyclopedia of Integer Sequences (oeis.org), where these series, or sequences of numbers are cataloged. The series we have mentioned in this article are: triangular numbers are sequence A000217, square numbers are sequence A000290, interface or centered-square numbers are sequence A001844, diamond numbers are also known as pronic numbers and they are sequence A002378, covenant numbers do not have an exact representation in the oeis.org, octahedral numbers are sequence number A000590.
