.ח Easy as Pi

אמר ליה רב אסי לרב אשי לעולם גברא באמתא יתיב ורבי יוחנן מקום גברי לא קחשיב כמה הוו להו תמני סרי בשיבסר נכי חומשא סגיא היינו דלא דק ולחומרא לא דק

A "mathy" guy like myself can't help but be curious to examine the precise math behind this גמרא. The results are quite fascinating.

Let's start with what the precise הלכה would be:

If we do, in fact, require a circle that encompasses a 4x4 square, the radius of that circle would be √(32) by the Pythagorean Theorem. That would make the circumference of that circle √(32)π which is approximately 17.7715 אמות circumference. 

Now, let's examine ר' יוחנן's statement according to רב אסי. The גמרא assumes based on its loose interpretation of π that if we made a circle of 24 men, the circle inside of that ring would be 18 אמות circumference. We could end right here and declare that the margin of לא דק is even more narrow than the גמרא suggested. But let's try to get a little more precise. A circle of 24 אמות circumference would have a diameter of 24/π. The men take up exactly one אמה so the inner circle would then have a radius of 24/π - 2. The circumference of that inner circle would then be (24/π -2) x π which comes out to 17.7168 !!! So ר' יוחנן's statement is off by only 0.0547 אמות !!! The only issue, of course is that it is לא דק לקולא but at that minuscule amount, that should be allowable.