One of God’s Names, Shakai, normally translated as Almighty, is profoundly linked with π.
The gematria of Shakai ( שַׁדַי ) is 314, easily suggesting π’s first 3 digits:
3.14.
It is essential to note that the Torah uses base 10 numbering but does not use the decimal fraction.Therefore, a number such as 3.14 can only be suggested by its integer counterpart, 314. The two numbers are of course equivalent.
The sages explain that Shakai is an acronym for “He told His world, ‘Enough!’” ( שֶׁאָמָר לְעֹולָמֹו דַי ). As expounded in the midrash, the world was created by expansion, but the expansion was at first without limit,prompting God to limit it so that reality should have boundaries. This limiting process is the equivalent of the mathematical finite limit, whereby, through a series of infinite operations, we can come progressively closer to the value of π, but of course can never define it exactly.
The implication that π can be known through a limiting process is found in the first verse of parshat Va’eira,I have appeared to Abraham, to Isaac, and to Jacob by the Name of Kel Shakai, but by my Name, Havayah, I was not known to them.
In this verse, first, the 3 patriarchs are mentioned.
Then the Name Kel ( אֵ־ל ) appears, whose numerical value is 31.
Then, the Name Shakai, whose numerical value is 314.
Hence, this verse reveals the first three numbers in the infinite progression that is π : 3, 3.1, 3.14, …
The identification of π with 3 made by the sages is itself an allusion to the 3 patriarchs, since the sages themselves denote the intrinsic connection between
the patriarchs and the number 3,
“Only three are called patriarchs, Abraham. Isaac, and Jacob.”
The ratio of 3:1 alludes to the clear-cut faith that the patriarchs bestowed unto the Jewish nation, the faith in one God.
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The numerical value of the three patriarchs ( אַבְרָהָם יִצְחָק יַעֲקֹב ) is 638, or the product of 22 and 29. If 638 is the circumference of a circle then the closest integer to its diameter would be 203, or 7 · 29. Indeed, this illustrates the closest ratio to π for integers under 100,which is 22/7. 203 is also the value of just the first three letters of “Abraham” ( אבר ) before the hei that God added to his name upon commanding him to circumcise himself. The literal meaning of these three
letters is “limb” or “wing” (i.e., “arm”), suggesting the diameter of a circle. In addition, these three letters permute to spell bara ( ברא ), “create,” the first verb of the Torah, and indeed the first three letters of the Torah.
If we then add the circumference to the diameter we get 841, or 292 or 21 (which denotes the inspirational number of 21, or 212 ┴ 202).
Amazingly, the value of the words “to Abraham, to Isaac, and to Jacob” אֶל אַבְרָהָם אֶל יִצְחָק וְאֶל יַעֲקֹב) ) is 737, the exact value of “He told His world, ‘Enough!’” ( .(שֶׁאָמָר לְעֹולָמֹו דַי 737 is also the value of the words, “With all of your heart, with all of your soul, and with all of your all”2 בְּכָל לְבָבְךָ וּבְכָל נַפְשְׁךָ וּבְכָל מְאֹדֶךָ) ), whose three
aspects, three levels of love of God, correspond to the three patriarchs.
The most important example of two words in the Torah that exhibit a circumference to diameter ratio (within integers) are the first two words of the Ten
Commandments, “I am Havayah [your God]…”: The word for “I am” ( אָנֹכִי ) equals 81, while Havayah (י־הוה) equals 26.
If we follow the same rationale for “Shabbat”,(שָׁבָּת702 which we saw above corresponds to the suppression of the remainder of π, we find that its diameter would be 223, or “There is none but Him” ( אֵין עֹוד מִלְבַדֹו ), or “This is the thing,” ( זֶה הַדָבָר ), both
indicating a revelation of the Radla, as above. The latter,in particular, is the idiom that differentiates between the clarity of Moshe’s prophecy and the riddled nature of other prophets’ prophecies. Indeed, Moshe Rabbeinu received prophecy through his faculties of thought (particularly his wisdom), whereas all other prophets received it through their faculties of imagination.
3
Returning to the finite limit of π, the next decimal fraction is 3.142 (this is the closest decimal approximation to 3.14159…) and its integer equivalent is 3142. Does 3142 also appear in this verse? If it does, we ב"ה would expect it to appear in the words following the Name Shakai (314), “but by my Name, Havayah, I was not known to them.” Rashi writes that God is saying that by not revealing the Name Havayah, “I did not reveal to them my measure of truth.” In our context,God’s measure of truth refers to the secret of the true measure of π. In other words, what the verse is saying is
that God’s Name Havayah ( וּשְׁמִי י־הוה ) contains the secret of the remainder of π. Let us see how.
The gematria of the words in the verse up to this final phrase
( (וָאֵרָא אֶל אַבְרָהָם אֶל יִצְחָק וְאֶל יַעֲקֹב בְּאֵ־ל שַׁדָּי is 1292.4 The remaining two words in the verse, “But, by my Name, Havayah” ( וּשְׁמִי י־הוה ) both have 4 letters. The secret here is that instead of just the regular gematria of these two words, we need to calculate the equivalent of their dot product (this is called “particular multiplication,“ הַכָּאַָה פְּרָטִית , in Kabbalah).
י ∙ ו ┴ ה ∙ ש ┴ ו ∙ מ ┴ 10 = ה ∙ י ∙ 6 ┴ 300 ∙ 5 ┴ 6 ∙ 10 ┴ 10 ∙ 5 = 1850
And indeed, 1292 ┴ 1850 = 3142!
Continuing the above logic, the next whole number to look for is 31416. To discover this number in our verse we need to divide it into two parts, 31 and 416.5
First note that the two-letter Name of God Kel ( ,(אֵ־ל 31, appears 3 times before in the verse as “to” ( אֶל ), “to Abraham, to Isaac, and to Jacob.”6 In the final phrase of the verse these same two letters invert to spell lo ( ,(לא “no” or “not,” “I was not known to them,” alluding to the unknowable remainder of π.
Now note that the first word of the verse (for which
the Torah portion is named) is “I have appeared” ( (וָאֵרָא
shares the same number, 208, as one of the three
patriarchs mentioned immediately in the continuation:
Isaac. This phenomenon is understood to mean that the
Divine revelation to all of the three patriarchs was
through the conduit of the soul-root of Isaac (208 = 8 ·
26, or 8 times Havayah). The sages teach us7 that Isaac is
the patriarch of the future. The light inherent in his
soul and name will be the Divine revelation of the
future.
The word after “I have appeared” is “to” ( אֶל ), whose
value is 31. The word before “Isaac” is also “to” ( .(אֶל
The word between the two “to” is Abraham, whose
name equals 248, or 8 times “to” ( אֶל ), meaning that the
three words between “I have appeared“ and “Isaac“ are
equivalent to 10 times “to” ( אֶל ), or 310, which
according to the logic of our present analysis is the
same as 31! And the gematria of these two words, “I have
appeared” ( וָאֵרָא ) and “Isaac” ( יִצְחָק ) both equal 208 so
their combined value is 416. So here we have discovered
the two parts of 31416 in our verse! This is the
beginning of the revelation of the future.
In the future the exact value of the transcendental
number π will be known to all those who will connect
to God in awe, as it is said, “the secret of Havayah is to
those who fear Him, and His covenant He will make
known to them.”8 The secret of π’s remainder relates to
name Havayah, as we saw, and this secret will be made
known to those who fear Him. In Hebrew, the word
used in reference to the patriarchs, “I did not make
known [to them]” ( נֹודַעְתִּי ) is the same as the word in
this verse, “I will make known to them” ( .(לְהֹודִיעַם
Furthermore, both verses refer to God’s covenant,
implicitly implying that the secret of π’s remainder is
the secret of God’s covenant with the Jewish people.
The finite limit approach to π’s remainder reflects the
stage of separation. The very concept of a limit, a
dividing line, is an expression of separation. In
addition, the particular verse in which this approach to
π’s remainder refers to God’s covenant, which was
given to Abraham as the commandment of circumcision
(itself separating between the Jewish people and all
other peoples of the earth). We noted above that
separation corresponds to the first mal, as in
circumcision. God’s covenant refers specifically to the
sefirah of foundation (in the human form, the
procreative organs corresponds to foundation). God’s
Name that corresponds with foundation is Shakai and
as learned from this verse, it is the Name associated
with His covenant.
Notes: 1. 3 is the 3rd prime number (counting 1 as a prime). Most
interesting is the fact that the 31st prime is 113, whose relationship
with π will be explained in the next chapter. The 314th prime is
2081. If we add these three together, we get 2197 or 133. 2.
Deuteronomy 6:5 3. Rambam Hilchot De’ot 7:6. 4. The gematria of “I
was not known to them” ( לֹא נֹודַעְתִּי לָהֶם ) is 646, which is exactly
half of 1292, an example of a whole and a half, as explained
elsewhere. 5. 31 is the value of God's Name, Kel ( 416 .(אֵ־ל is 16 times
the value of God's essential Name, Havayah. 6. Note that the
average value of “to… to… and to…” ( אֶל אֶל וְאֶל ) is 33 or “by [the
Name of] Kel” ( 7 (בְּאֵ־ל . Shabbat 89b. 8. Psalms 25:14
Dedicated in honor of the wedding of
Yosef Yitzhak Eliayhu and Naomi Tilles
Kof Kislev, 5770
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