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ר׳ זירא says that a Sukkah that is lower than 20-אמות, the סכך is the one blocking the sun and creating the shadow, not the walls, and therefore the Sukkah is kosher, like it says in the פסוק

״וסוכה תהיה לצל יומם״

In the season of תשרי (that when the yom tov Sukkos is) the sun is away from the zenith in Israel 32 degrees, and if you do the calculation, a wall that is tall 20-אמות = TAN (32) x 20 = 13-אמות, meaning to say that the surface of the Sukkah needs to be 13-אמות wide, to have the shadow of the סכך not the wall?

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  • Hello @Yid and welcome to Mi Yodeya. While not answering your question, this Ritva may prove helpful.
    – robev
    Jul 18, 2021 at 19:57
  • I thought we may go after the highest altitude of the sun in Yerushalayim at the summer solstice (81.67 degrees). This helps a bit (you get 2.9 amos), but you still don't get 7 tephachim.
    – pcoz
    Jul 18, 2021 at 22:56
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    The issue isn't necessarily the shade from the walls, rather it is the lack of shade from the schach. So even if the walls are providing shade, as long as the schach are capable of giving shade in the theoretical absence of the walls, then it's considered shade of schach. So when the sun is overhead, even at 32 degrees, the schach would still provide some shade if it's 7 tefachim (according to that shita in the gemarah) if there's no walls, therefore the shade of the wall isn't counted to passul. (Pnei Yehoshua, Aruch Laner, and others based on Rashi that only sunlight is mavatel shade)
    – Chatzkel
    Jul 18, 2021 at 23:18
  • @Chatzkel You should make your comment into an answer.
    – N.T.
    Jul 19, 2021 at 8:17

1 Answer 1

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The math works out almost exactly if you make slightly different assumptions than you're making.

Instead of Yerushalayim on Sukkos, look at Jebel Musa, the mountain commonly identified as Har Sinai, on the summer solstice. This is the place and approximate date when Sukkos was first mentioned at the end of parshas Mishpatim.

Also, we're going to want to rotate the minimal sized sukkah so that the diagonal is north to south. That means the length of the sukkah is 7sqrt(2) tefachim ~= 1.65 amos.

The latitude of the peak is about 28.5 degrees north, and at the time of Matan Torah the Tropic of Cancer was at approximately 23.9 degrees north, so the angle of the sun at noon on the summer solstice is about 4.6 degrees. tan(4.6 degrees) = 0.0804, which is within 3% of (7sqrt(2) tefachim) / (20 amos). That's well within my errors in identifying the exact spot on Har Sinai and in reading the plot from Wikipedia.

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  • Why do you think the gemarra is telling you the limit on the dimensions of a sukkah is based on the location and date the mitzvah was first introduced? Compare your assumptions with the Ritva I cited above who says we measure based on Tishrei, when the mitzvah is observed.
    – robev
    Jul 19, 2021 at 14:16
  • @robev I saw the Ritva but I don't understand what he's saying. He says the sun is directly overhead at noon on the summer solstice, but that's only true at the tropic of cancer. The Mosad Harav Kook footnotes don't say anything. I used Har Sinai because it makes the math work out.
    – Heshy
    Jul 19, 2021 at 14:51
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    @Heshy if you look at the Ritva in the piece beforehand he says that the schach does not have to give shade, rather it needs to be POSSIBLE to give shade if the walls would be removed. So even if the walls are actually giving shade, as long as when the sun is at noon, wothout walls, the schach would give some shade that is sufficient.
    – Chatzkel
    Jul 19, 2021 at 15:14
  • @Chatzkel I see, so the reason he says there's no shade is something like this? scienceabc.com/eyeopeners/… and I guess Chazal can estimate how diffuse a shadow can be and still count as צל. That makes a lot of sense. I've been really struggling with that Ritva. Thank you!
    – Heshy
    Jul 19, 2021 at 20:43
  • @Chatzkel Rashi seems not to say like the Ritva (he talks on 2a about how הצללין מגיעין זה לזה) so my answer still works according to him.
    – Heshy
    Jul 19, 2021 at 20:52

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