**Q: What is the Jewish source for the curvature of natural numbers?**

A: There are several sources for the curvature of natural numbers to become either irrational or transcendental numbers like pi and e, or the inverse of the fine structure constant, 137. One of them is the *Baal Mishnat Chasidim* who explains that the number pi, which is the circumference of a circle whose diameter is 1, begins with the parameter of a hexagon. The hexagon is like the inside of a *Magen David* (Star of David). The hexagon has six sides, each of them equal to the radius of the circle. If the radius is 1/2, the diameter is 1, and the parameter of the hexagon is 6×1/2, which is 3. That is the approximate value that the sages give for pi, 3. Then if the polygon becomes more sided, with 7, 8, etc. sides, to infinity, that infinitely approaches the exact number pi. Pi is the secret of progression from a six-sided polygon to an infinite polygon.

This is a very beautiful phenomenon that is obviously connected to the secret of creation. The world was created in 6 days, which creates a hexagon. The secret of the 7th day, Shabbat, is the transcendental addition to the hexagon, to make it approach and become an infinite limit, pi.

Another example of the importance of approximation of irrational or transcendental numbers to natural numbers is in the Torah. The first two words of the Ten Commandments read "*Anochi Havayah*," (I am God). *"Anochi"* equals 81, which is 9 squared. "*Havayah*" equals 26. The secret of the relation between the two words and numbers is that if 81 is considered to be the circumference of a circle, to discover the diameter, the circumference must be divided by pi. This gives us 25.79 or approximately 26. There is a beautiful mathematical phenomenon here. "*Anochi*" is a transcendent level, *sovev kol Almin* (God’s transcendent light), like the circumference of a circle, the secret of the *igul hagadol (*great circle*).* The *kav* (line) or, as in "*Az yibakah kashachar orech*a (Then shall your light break forth as the morning)[1] the secret of *tiferet*, the inner dimension of which is compassion, corresponding to Yaakov (Jacob), a permutation of "*yibakah*," the Name Havayah. That is why Havayah is the name of *tiferet*, the name of Yaakov. This is the diameter that connects the two semi-circles of the circle. This is a beautiful secret of "*Anochi Hashem*." It is based upon an approximation.

To make it even more clear that this approximation is intended according to the Torah, we can abstract this concept. We know that throughout all generations, mathematicians tried to square the circle, unsuccessfully. Just over 100 years ago, it was proven that it is impossible to square the circle because pi is a transcendental number. Nonetheless, if we take any natural square number, and consider it to be the circumference of a circle, and then divide it by pi in order to learn the diameter, we can abstract the expression as n squared/ pi = d. If we start with a circumference of 1, 1squared is 1, divided by pi will give us a diameter of approximately .32. If we take 2 as the circumference, the diameter is approximately 1.27. For 3, the diameter is 2.87. The diameter for 4 is 5.1 . For 5 it is 7.96 . For 6, 11. 47. For 7, 15.61. For 8, 20.38. These are all approximate numbers. Now we have the diameter for all the numbers leading up to "*Anochi*," 9 squared. The sum of the first 8 diameters is approximately 65. 65 is the way God's Name of Adnut is read. Just as God’s Name Havayah, 26, is 2×13, 65 is 5×13. The sum of all the numbers from 1, *alef*, which is the first letter of "*Anochi*," to "*Anochi*," which is 9 squared, is 65 + 26, which is 91, the triangle of 13, all numbers from 1 until 13, 7×13= "Amen." This is the inclusion of Adnut in Havaya. The first eight equal Adnut, which, when included within Havaya is the Yichuda Ila’ah (higher unification). This can also be interpreted as the inclusion of Havaya in Adnut, which is the *Yichuda Tata'ah* (lower unification), as explained in *Sha'ar Hayichud v'ha'amanah*, in Tanya.

This also suggests another beautiful intention when we say *Havayah Echad* in the *Shema Yisrael* prayer. We know that in the Torah portion of *Va'etchanan,* the Ten Commandments are repeated by Moshe. The conclusion of the retelling of the account of the Giving of the Torah is to arrive at the epitome of Jewish consciousness, which is the verse "*Shema Yisrael, Hashem Elokeinu Hashem Echad*." This is not the place to explain the connection between "*Anochi Hashem Elokecha*" and *"Shema Yisrael Hashem Elokeinu Hashem Echad*." However, the first two words of the Ten Commandments are A*nochi Hashem* (“I am God). The two concluding words of *Shema Yisrael* are "*Hashem Echad* (God is One). According to the Rambam these are the first two basic mitzvot of the foundation of the Torah, the first two of the thirteen principles of faith. We are commanded to believe in God's existence and Providence and to believe in His absolute unity. "*Hashem Echad*" numerically is the simple relationship of *shalem vachetz*i, whole to half. Hashem = 26, and Echad = 13. "*Anochi*" is the transcendent circumference of the *igul hagadol* and Hashem is the *or ein sof *of *malei kol Almin* (God’s immanent light). It goes together very beautifully. Hashem is the diameter of the circle whose circumference is 81. The radius is half the diameter. The phrase "Hashem Echad" is the full diameter (This is like the concept in Kabbalah and Chassidut that in the future, *Bayom hahu yihyeh Hashem echad ush'mo echad*, there will be two *echad*s, the radius of the line, of the *kav or ein sof*, will penetrate and permeate the entire *chalal hapanui* and will reach and connect to the other side. What in our world is considered a radius will become a full diameter.) The relationship between the two words, Hashem Echad, now becomes the simple relation between the diameter and the radius. When we calculated the perimeter of the hexagon, it was based on the radius, 6 radii. Here also, Hashem's Name is basically 26. That is a full diameter, which possesses a symmetric radius, which repeats itself twice. That radius is 13, Echad.

The series that we mentioned before also fits in very beautifully with this explanation. If we meditate upon the word Echad, alef, chet daled, Kabbalah explains that the alef chet is a concept in its own right, the nine sefirot of Z'a. The daled is the addition of malchut, the spreading of the revelation of the Unity of Hashem throughout reality. These are two different *partsufim* (personas), the partsuf *Z'a *and the partsuf *Nukbah d'zah*. This is sometimes explained as *Partsuf Leah*. In our series we had n squared/ pi from 1 to 8 = approximately 65. That is the *alef chet*, which is a code which says from 1 to 8. When the 1 to 8 are combined, they become 9. The 9 in Kabbalah is divided into 1 and 8. There are 9 *sefirot* of *zeir anpin* from *keter* (crown) to *yesod* (foundation). The keter, is in a category by itself. That is why it is 1. Then there are 8 specific *sefirot* from *chochmah* (wisdom) to *yesod* of *zeir anpin*. As a total, it is 9, which is the root of "Anochi," (I) 81.

This also explains in detail the relation between "*Anochi*" and "*Echad*" (One). If we take all the three letters of the word "Echad" (*alef, chet, dalet*) and square each one individually, this is called in Kabbalah *ribuah prati*. The *alef* squared and *chet* squared gives us 65. According to the Arizal, the *dalet* is a *dalet rabati*, written as a big dalet. It should be 4 times dalet, which explicitly alludes to 4 squared , 16. When this is added to 65 it becomes 81, or 9 squared. 9 squared is the *square* of the first two letters alone. This is the way to understand that essentially the dalet is included within the *alef* and *chet*.

To summarize, we have the allusion in the word "*Echad*" to the first eight squares from 1 to 8, each one divided by pi, altogether equals approximately 65, which is itself the exact value of 1 squared plus 8 squared. Then the 1+8 is nine, 9 squared, whose diameter is 26, God’s Name, Havayah. When all the first 9 are added together, it is 91, the triangle of the whole word, 13. Now we see why the *dalet* has to manifest by itself. Then all together it equals 13. The triangle of 13, from 1 to 13 equals 65 plus 26. This is a very beautiful example of approximate natural numbers. Clearly this concept appears and is of great significance in the Torah.

[1] Isaiah 58:8.