[Avodah] Sukkah GT 20 amot high

[hankman]

RAM wrote:
Geometrically defensible? No way. Sitting in the sukkah's shade is 
dependent on the angle at which the sun is shining, combined with 
both the height of the s'chach and also its horizontal edges. The 
simplest example is that if one is anywhere north of the equator, 
sitting against the southern side of the sukkah is *never* in the 
sukkah's shade. (Similar arguments can be made by Chanuka: Height is 
irrelevant. The angle is significant, and maybe the distance too.)

CM responds:
Actually, the interesting and more revealing cases are in the far north. I assume there is no p'tur on the mitzvah of Sukkah in the far north (except maybe mitztaer :-). So let's take someone living at the Arctic Circle in Alaska and a converted Eskimo living at the North Pole. Since Sukkos comes out reasonably close to the equinox and the trig functions at the values we need vary reasonably slowly for our parameters near the equinox the error will be reasonably small for this simplifying assumption.

So we can consider our Jewish Eskimo first. He goes out and throws some sechach over a roofless igloo he made. At the equinox, what he sees is the sun on the horizon for the full 24 hour day (for the start of the 6 months  of night) or close to it depending on how many days Sukkos is removed from the equinox. So the sun will be near the horizon or slightly BELOW it, thus the angle is near 90 degrees to the wall or the sun has already set for six months. Thus the entire Sukkah, no matter how low the walls are, and no matter how large the LxW dimensions are, will never have tzel sechach AT ANY TIME and not just immediately south of the wall. Rav Zeira will not be able to move to the North Pole.

Our Jewish Alaskan on the Arctic Circle has it somewhat better. For him the sun is at an altitude of 23 1/2 degrees at noon (since the sun is then overhead at the equator) and he will have close to a 12 hour day or reasonably close to these numbers if Sukkos is a couple of weeks removed from the equinox. Thus,  if our Alaskan builds his Sukkah to the max kosher height of just under 20 amos, then at local noon his wall will cast a shadow of approx. y=20 x tan (66.5 deg.) = 20 x 2.3 = 46 amos. Tan function at these values for the parameter changes very slowly so that the approximation for a couple of weeks removed from the equinox will be pretty close. Thus our Alaskan friend will have to build his 20 amos high sukkah AT LEAST 46 amos long before he gets ANY shade from the sechach just at the far wall. Tough for Rav Zeira's shita.

Interestingly, if our Alaskan friend builds a minimum height sukkah of 10 tefachim, then the shadow cast at noon by the wall will be y=10 x tan (66.5 deg) = 10 x 2.3 = 23 tefachim, which is just short of 4 amos, which is the minimum dimensions for a sukkah according to Rebbi. I am not sure if this is just coincidence or something significant in Rebbi's shita.

But for RAM's post, I would not have thought to make these calculations. I hope I have not made any serious mistake. I think it is pretty straight forward.

Kol Tuv

Chaim Manaster


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