Pi & Its Torah Source

For a more details/sources, please see my source sheet:

https://www.sefaria.org/sheets/60352?lang=bi

&

Search: ‘Game Theory’ on this site for a related post

“V’es dasei hamelech einum osim” –the laws of the king they do not observe.

(Esther 3:8)

The above quoted verse was among the complaints Haman lodged against bnei yisroel so as to get Achashveirosh to agree to their annihilation.

Chazal (Megillah 13b) elaborate on and reveal the true meaning of this particular complaint. “All year the Jews succeed in getting out of work, claiming, ‘Today is Shabbos’, ‘Today is Shavous’, ‘Today is ‘Pesach’, etc.”

Haman was, in effect, introducing the world to an old anti-Semitic standard of ‘too many Jewish holidays’, Rachmana l’tzlan.

Ironically, today, our calendar simply can’t compete with the amount of national holidays that exist in the secular one!

There is National Donut Day (June 5^th), International Beer Day (first Friday in August), Love Your Lawyer Day (first Friday in November), the list goes on and on. Based on a loose count, I have learnt of over sixty such days!

Some of these holidays are often referred to cynically as ‘Hallmark Holidays’. The suggestion being that the Hallmark greeting card company invented these holidays only to benefit their bottom line.

In fact, several years ago, in an effort to quell these rumors, Hallmark themselves issued the following press-release:

“…While we’re honored that people so closely link the Hallmark name with celebrations and special occasions, we can’t take credit for creating holidays. Congressional resolutions, proclamations, religious observances, cultural traditions, and grassroots leadership by ordinary people create these special days. It’s really the public who give occasions like Valentine’s Day, Mother’s Day and Father’s Day widespread acceptance as celebration events.

“…So if you think that special date on the calendar is a “just a Hallmark holiday,” think again. There are fascinating histories behind every holiday for which Hallmark makes cards…”

Fascinating indeed.

While some of these ‘holidays’ have unknown provenance, some of them have celebrated histories.

 For instance, Haaretz published the following headline a few years back:

“How a Jewish Mom Created a Global Ice Cream Holiday”

The article goes on to tell the story of mid 1960’s Rochester, New York housewife who decided one cold February morning to serve her six children ice cream for breakfast. The custom spread, and now several countries celebrate this ‘holiday’ on the first Friday in February.

However, if there is one international secular holiday whose date and reason instantly understandable it is the one from last week, March 14.

This is known as ‘National Pi Day’. No, not ‘Pie’ Day (that is on January 23^rd. Really!), rather Pi, the irrational ratio of the circumference of a circle to its diameter. This famous endless number begins, and is known as, 3.14, so March 14 (3/14) is the obvious choice.

While I am not sure how one is to observe this day –some mathematicians bake pies with the π sign! – this year a special opportunity arose.

On Shushan Purim the news of a fast-approaching storm was alarming, excepted to dump close to twenty inches on New York City.

Although while living in Buffalo this storm would simply be known as ‘Tuesday’, here in New York in anticipation of the storm Mayor De Blasio canceled all schools. Many chedarim closed as well.

Many parents would also be staying home from work, either to watch their kids or because their offices were closed as well.

A high school rebbe of mine once said that one can tell a true ben torah by how one acts on the second night of yom tov, when after arriving home we have to wait for the food to warm up. This is a time where one only has two options: waste time or learn.

I would apply the same to a ‘snow day’.

So, I decided that in honor of ‘Pi Day’ I would give a shiur in my shul before Mincha on the very subject of ‘Pi Day’ in Halacha. This would also hopefully give incentive for people to get to minyan on a day when parking was very difficult.

I was also betting on the title of this shiur awakening some curiosity. ‘What does Pi have to do with halacha?’

As it turns out, a lot.

Wilbur Knor, a professor of math history at Stanford University, asserts that Rambam was the first to lay claim that Pi is an irrational number (infinite, and can not be expressed in common integers).

While many cultures throughout history, and long before Rambam, had various and close approximations, this that Pi was irrational was only proven in the 18^th century.

The Rambam to which Proffesor Knor was referring was coming to explain a ruling in the Mishna (Eruvin 1:5 or 13b). The Mishnahthere teaches us about the dimensions needed for a koreh – a beam utilized for the fourth side of certain types of eruvin, and no longer practiced today (Rema). The Mishnah teaches that if the circumference of of koreh is three tefachim (handbreadths) than we know that its diameter is one tefach. In other words, they are teaching that the circumference of a circle to its diameter is simply 3:1.

The Rambam, in Peirush Mishnayos to this Mishnah is obviously bothered by this approximation (3:1) when the number is in fact irrational. “You must know that the ratio between a diameter of a circle and its circumference is not known. It is something whose measure is impossible to know precisely, this not due not to our lack of understanding rather to the ratio itself…”. His verboseness is likely due to not having a word for ‘irrational’.

As fascinating as this statement is, consider the Gemara itself.

Commenting on the ratio in the Mishnah (3:1), the Gemara asks how the Mishnahknew this; meaning what is its biblical source.

The Tosfos HaRosh (d. 1327) wonders why something based on math would need a scriptural source; can’t Chazal create a mathematical equation on their own?  He suggests that the Mishnah knew that 3:1 was not the exact value of Pi and yet rules that we may use an approximation for Halacha. This is what bothered the Gemara: from where does our Mishnah know that Hashem accepts our using an approximated Pi for Halacha?

Fascinating, as it would seem that both Rambam and Rosh accept that Chazal knew if not the true value of Pi than at least that it is irrational.

Even in our day, when we have proven Pi to be irrational, the Halachos affected by utilizing a mere approximation is unchanged. The Aruch Hashulchan makes this clear (siman 363) and mirrors the words of the Rosh, that this is precisely why Chazal wished to marshal pesukim, as they were aware that the number for Pi was far more complex.

On the hand, the Tashbeitz (chelek 1 #165) suggests the possibility that while Chazal were indeed only giving us the approximate for Pi, they did so not for halachic reasons, rather only to make Torah study simple for all. However, when calculating these measurements for Halacha l’maaseh one would have to embrace exactitude.

Let us end here with a final amazing idea. We mentioned that Chazal brought a pesuk to prove this ratio. Which pesuk would discuss this issue?

In Melachim 1:7:23 it describes the (round) yam shel shlomo hamelech, as ten amos in diameter and 30 amos in circumference. So there you have a navi utilizing the same approximation for Pi found in the quoted Mishnah.

But is it really only the approximation?

Many kiruv agencies have utilized the idea that follows. While many say this in the name of the Vilna Gaon it has have yet to be discovered in his name.

In the measurement of this bath’s circumference the pasuk utilized the term ‘kav’ or line. However, the word is oddly spelled with a hei at the end, but it is not recited this way. This is called a ‘kri’ and a ‘kesiv’- a word spelled one way in the Tanach and pronounced another way. When taking the gematriya of the word “kav” (106) and ‘kaveh’ (111), their ratio (111/106) is 1.0471698. This can be used to represent the ratio of the value for Pi to 3 (3.1415094/3 = 1.0471698).

However, we should point out that this only works by ignoring conjunctive vav, which if included would them make the figures 117/112, changing the math.

Others suggest that the kri/ksiv of this word is to hint to yet another hidden message. The ratio of the gematriyos of “v’chamesh” in this pasuk (the height of this circle in amos) and “v’kav” is 3.1415929…, within one part in a million of the exact value of Pi, and seemingly the closest any gematria would allow for.

However, we should point out that this last hint only works if one adds ‘one’ to each of the respective words. This is called ‘im hakollel’ and is something that is well cosseted in toras hagematria (see e.g. Baal HaTurim)

Either way, to discover this in the one pasuk in all of Tanach (repeated in Divrei Hayamim) describing diameters and circumferences, is pretty remarkable!

Hafoch Buh Sh’Kula Buh!