How the irrational number Pi can be derived from the Torah?

Question

Some traditions say, based on a Gemmorah in Taanis 9a (also מדרש רבה נשא י' ו', זוהר פנחס רכ"א א, and more):

"ליכא מידי דלא רמיזא באורייתא"

"There is nothing that is not alluded to in the Torah"

That is, everything can be derived from the written Torah (the Pentateuch of 304,805 letters, and Gr"a adds from Parashat Bereshis alone etc). Many tried (and succeeded) to find certain integers, such as numbers and dates.

In what way can the irrational and infinite number Pi be derived from the written Torah?

(to clarify, the question is not about approximation of Pi, it is about the possibility to derive an irrational number from a list of letters)

Answer 1

Pi can be derived from the Torah by rolling up a Torah scroll and measuring the end's diameter and circumference. The ratio between them will be pi.

Answer 2

The Vilna Gaon saw that the word “line” in the parallel Kings and Chronicles texts of the passage are spelled קוה and קו respectively. The extra ה is the key.

Take the ratio of the sums of the standard numeric values of the Hebrew letters (kuf = 100, vav = 6,Hey = 5) for each of these words, obtaining 111/106; multiply by 3 – the apparent value of π and we obtain π is equal to 333/106 ≈ 3.141509, agreeing with π to four decimal digits.

• Solomon's Sea and π

This article also finds pi using gematria Genesis and pi

Answer 3

Pi is computable, i.e. it can be defined using finite rules, and those rules explain a set of infinite steps that can be followed, and the limit of those steps is pi.

This proves that there is no inherent issue with the Torah being "composed of letters" as a barrier to it containing the definition of pi* (and what is wrong with the geometrical definition of pi?), and maybe isn't any better than asking how is a googol or Graham's number alluded to in the Torah?

One could imagine such a set of rules exists in the Torah, one day to be discovered or revealed, but it is clear we don't have that information at the moment. Not that that matters much.

^{[What might be a better question is how are the non-computable numbers found in the Torah?]}

^{* Mādhava of Sangamagrāma, who wrote the most famous series expansion for pi, wrote the steps in verse}

Answer 4

There are claims that already in the old kingdom of Egypt everybody used approximation of pi as 22/7 = 3.142857. For sure, it was known at the time of Archimedes. Hebrew alphabet has 22 letters and 7 is the number of the original double letters בגד כפרת according to Sefer Yetzira. In the commentary to the translation by Arieh Kaplan, it states the sound of resh with dagesh was lost or deliberately concealed (1). So pi is alluded by a reciprocal of a fraction of double letters, which themselves hint to rolling/repetition of the alphabet to invoke a second sound.

Alternatively, 7 is the number of days in a week. Incidentally, these are the first two complete quantities defined by the Torah, which brings the question, what can be learned from it?

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(1) Quote from Aryeh Kaplan "Sefer Yetzira", Revised edition, Weiser, 1997, p. 160 (the book provides a big list of original sources):

Highly significant is the fact that the Resh (ר) is considered one of the doubles. Most post-Talmudic grammarians take presently the opposite view, and state that the Resh never take a Dagesh. Not only is there no verbal distinction between the hard and soft Resh, but modern Hebrew grammar does not even recognize such a difference in the written form.

There are, however ten different words, appearing in fourteen places in the Bible, which are written with a Resh containing a Dagesh.

See Table 28, containing: 1. haRimah Shmuel A 1:6, 2. haR'item Shmuel A 10:24, 17:25, Melachim B 6:32; 3. Ra Yirmiyahu 39:12, Mishlei 11:21, 20:22; 4. Karat Yechezkel 16:4; 5. Sharekh Yechezkel 16:4; 6. Rosh Habakuk 3:13; 7. LeSharekha Mishlei 3:8; 8. Marat Mishlei 14:10; 9. Rakh Mishlei 15:1; 10. sheRoshi Shir haShirim 5:2.

It is obvious, however that the usual rules applying to letters בגד כפת, do not apply to Resh.

The present sound of Resh is a fricative, and and is therefore most probably the soft sound. The hard Resh was either lost or deliberately concealed after destruction of the Temple. In earlier times, its use was standard, and there is evidence from their transliteration of names, that its pronunciation was known to the authors of the Septuagint (which used double for Sarrah). By the 10th century, however the double Resh was only used by the members of the small Mazya community in Tiberias. Tiberias had been the last city in which the Sanhedrin, the great court which preserved the tradition, had flourished. This was one of the mysteries that the Sanhedrin had entrusted to the community of Tiberias.

According to the Sefer Yetsira 2:3, Resh is in the group of Dentals (זסשרצ). Alon with letters Zayin, Samekh, SHin, Tzadi, it is pronounced with teeth. According to Long Version (of Sefer Yetsira) 2:1, it is sounded "between teeth with the tongue lying down, spread out." We cannot say that it is a rolled /r/ sound, since this involves the tip of the tongue. It would then be closest to the /l/ sound, and should be included among the Linguals (דטלנת). Furthermore, the hard Resh should be a plosive, like all other hard doubles.

There is no /r/ sound in use today that meets these criteria. Furthermore, there is no plosive sound pronounced with the teeth that could be a candidate for the hard Resh. The original pronunciation of this letter therefore remains a mystery.

Some people believe that pronunciation of /ř/ in Czech language as well as /rz/ in Polish language is the original pronunciation of Resh with dagesh.