1) AN APPLICATION OF "PI TIKRAH YORED V'SOSEM"
QUESTION: The Gemara discusses a plank that extends horizontally from the side of a building. If the plank is at least four Amos long by four Amos wide, we may apply the following method and view the plank as comprised of four Mechitzos that surround an area of four by four Tefachim below it: We view a square of four by four Tefachim as cut in the center of the plank, and we view the rest of the plank on each of the four sides of the center square as cut and descending downward, forming an enclosure of Mechitzos around an area of four by four Tefachim directly below the center square.
Why is such a complex mechanism necessary in order to consider the area beneath the plank as enclosed? The plank should create an enclosed area beneath it simply through the principle of Pi Tikrah Yored v'Sosem! We should view the sides of the plank as extending downward to form Mechitzos, making an enclosure around a Reshus ha'Yachid, even if the plank is not wide enough to construct Mechitzos of ten Tefachim on each side!
(a) TOSFOS (86b, DH Gezuztra) answers that the Gemara here is discussing a plank shaped like a pointed roof, which slopes downward at its sides. In such a case, Pi Tikrah does not apply, as the Gemara later teaches (25b, "Ki Urzila").
(b) Tosfos suggests another answer. Perhaps the principle of Pi Tikrah Yored v'Sosem applies only when the exposed edge of the board's thickness is a flat, smooth surface. Here, though, the side of the plank is jagged and full of dents and nicks, and thus Pi Tikrah does not apply.
(c) Tosfos cites RASHI in Sukah (18b) who says that Pi Tikrah applies only when the beam is at least one Tefach thick. Here, the plank is less than one Tefach thick.
(d) The RITVA and other Rishonim cite an opinion that Pi Tikrah does not apply to a board that hangs more than twenty Amos above the ground. Perhaps the plank here is higher than twenty Amos above the ground.