A Piece of the Torah's Pi

Here’s a semi-obscure controversy from the Hebrew bible that you may be aware of: The Torah (in two different locations,  I Kings 7:23 and II Chronicles 4:2) gives the incorrect value for π. For those who have forgotten since math class, Pi or π is the ratio of a circle’s circumference to its diameter. It has the same value for every circle, and though it is impossible to calculate its precise value it approximately equals 3.14159.

Here is one of the offending verses, from I Kings: “And he made the molten sea, ten cubits from brim to brim; it (was) round all about, and the height thereof (was) five cubits; and a line of thirty cubits did compass it round about.” And even if you have indeed repressed every memory of math class you probably see that thirty divided by ten is exactly three, and that three does not equal 3.14159.

The Torah using 3 as π’s value is sub-par whether or not you are religious. A secular observer would (and several have) note that other civilizations at the time of the events of I Kings had already approximated Pi much more closely, had at least realized that this fundamental constant of the universe exceeds the round number of three. And a religious observer obviously has trouble squaring how the eternal truths of mathematics can contradict a verse that clearly says the Molten Sea was thirty, ten, and round.

One answer to this quandary is that Jewish law obviously is willing to approximate when demanding of human beings to construct “perfect” circles. That is, the verse is not defining Pi, but explaining that the craftsmen of the Temple vessels measured a circumference of exactly 30 cubits and approximated the diameter, rounding 9.549 upward to ten. The Talmud notes that one is permitted to make approximations in measurement in the direction of stringency. In this case, using the absolute value of Pi would decrease the diameter of the Molten Sea, perhaps making it too small. The approximation is thus called for and perhaps even legally necessary so as not to err on the too-small side of the precise 9.549. Alternatively, the Lubavitcher Rebbe argued that the workmen used the exact 9.549 value, and the Torah was simply describing the circle with approximation, with the maximum precision necessary for practical purposes.

Either way, the sages of the Talmud were clearly aware that Pi does not equal three, as is stated clearly in the ancient Mishanat Ha’Middot, as Maimonides argues in his commentary on the Mishna, and is implied in the Talmud’s complex discussion of the required size of a round Sukkah in Tractate Sukkah 7b-8a.

However, the question remains — it is good and fine that the later sages were aware of more accurate values of Pi, and that the Biblical verses are mere approximations, but ultimately, the Book that is supposed to speak the truth presents a ratio that is misleading, and not misleading merely in practical matters but in a similar eternal truth of our universe, the hard-to-calculate, definitely-not-a-round-number phenomenon that is π.

An elegant and astonishing discovery of Rabbi Max Munk provides an answer of sorts, and to understand it we must first take a moment to speak of traditional Jewish biblical exegesis. The advent of computers has seen the rise of the controversial (and possibly downright-debunked) Bible Codes, an attempt to apply massive computing power to one ancient method of deriving truth from the Tanakh, that of counting spaces between the letters. However, most Orthodox Rabbis would tell you that it was the switch to computers that indicated the project was doomed to fail; like the story of the biblical Pi itself, a demand for ever-more “precision” tends to overlook the fact that the Torah was given to humans to grasp on the scale of human understanding. Besides, the Bible Codes apply only one method of exegesis, and far from the most common or important one.

Rabbi Munk, on the other hand, applies two better-established methods, and applies them not with a broad brush entering anything he can think of into a search box but with surgical precision, to the extent that it seems almost inconceivable that the verse should give rise to the meaning he discovered by accident. But you be the judge. Here is the verse that was translated earlier into English in the original Holy Tongue:

Rabbi Munk’s first method is to observe the parenthetical statement on the last line. It is a note on the three-letter word preceding it (remember, Hebrew read right-to-left). That word, וקו, is quite important to the verse; it is the “and a line” that refers to the 30-cubit circumference of the Molten Sea. The parenthetical statement tells the reader of the verse that even though the word is to be pronounced וקו (“V’kav”) it is written וקוה, (“V’kavah”). Now, the difference in the meaning of these two words is quite slight, the difference between “and a line” and “and its line.” However, the difference between the written form and the pronounced form has exegetical significance; the tradition of the book of Kings says that there are two superimposed realities in the verse, its written form and its pronounced form. One of the ways of dealing with this bifurcation is to view the pronounced version as the “revealed truth” whereas the written version is a deeper or “inner truth” of the verse.

Rabbi Munk’s second method is to apply Gematriah, the classic Jewish numerology in which each letter of the Holy Tongue is assigned a numerical value, to this verse. In the reckoning of the Gematriah, קו, the word that means “line” in the pronunciation of the verse, has a value of 106. The word that means “line” in the verse’s written form, however, is קוה, with a value of 111.

So, to sum up the two methods, the pronounced, revealed truth of the verse for the world has a value of 106. The written, secret, deeper truth under the surface, however, has a value of 111.

Rabbi Munk reasons that if we’re looking at a verse whose revealed meaning is problematic (because its approximation of a circle’s dimensions are so far off), maybe we can fix it by applying the verse’s hidden meaning. That is, we can perform an operation a little like dimensional analysis with the verse’s numbers. One can find the number of inches in three feet by multiplying (3 feet) x (12 inches / 1 foot), with the foot units cancelling out and leaving us with 36 inches. Similarly, our verse has a possible conversion: The revealed value of Pi into the verse’s deeper, truer meaning. The formula for this is:

(3, the revealed value of pi from the verse) x (the deeper truth of the verse, 111 / the “revealed” value of the verse, 106).

This, using only numbers put into the verse when the book of Kings was written, yields 3 x (111/106) = 3 x 1.04716981132 = 3.14150943396. And that is Pi to four decimal places.

QED.

1. Someone says:

Myeh. What about the 5th, 6th and 17th decimal place? Why only 4? Why decimal places – the system used in the Torah for describing fractions of a whole is fractions of real positive integer numbers (e.g. “vchamishiso yosef olov”)- not necessarily fractions of 10.

My own suggestion is that (a) of course Tanach was written before the discovery of irrational numbers, and that (b) the sea of Shlomo, like any real object, was not a perfect circle – it had width for starters, which gives you a choice of different places to measure from. In fact (though I don’t believe this to be true) you could derive a width of something like (10 Amos – 30 Amos / Pi) / 2 which is approximately 0.22 Amos.