[NHC] Jewish Geometry
Aug. 18th, 2011 08:58 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
magidMy afternoon class at NHC this year was on the topic of Jewish geometry, taught by a Brandeis math post-doc, Adam Levine. Th class covered a number of discrete, though sometimes interlocking, ideas.
First up was the relationship between the diameter and circumference of a circle, ie π. In I Kings 7:23-26, there's a discussion of something being made for the Temple that is 10 cubits across, and 30 cubits around, which is obviously problematic in reality. There was a discussion of approximations of π in the ancient world; it was already known that 3 is not at all close, though if rounding to the nearest integer, it might do. Still, the Greeks and the Babylonians already had better approximations, so it's reasonable to assume that the composer(s) of Tanakh did as well. And if Tanakh is perfect, then what's going on here with 3 for π? There are a variety of explanations by commentators. Ralbag discusses inner v. outer circumference, as well as pointing out that it's just a very rough approximation, for instance.
Then to Mishna Eruvin 1:4-5, which discusses minimal dimensions of a beam, based on an assumptive square beam. However, if the beam is circular, the assumption is that a circumference of three handbreadths means a diameter of on handbreadth, ie, another π being 3 situation. The Talmud on the mishna (Eruvin 14a) brings in the source in I Kings as an explanation. The Tosafot haRosh comes to say that it's obviously an approximation, not exact. The Mishnat haMidot (a rabbinic math text found in manuscript in modern times) 5:3-4 discusses calculating the area, with a fairly good approximation of π.
In the Babylonian Talmud, Sukkah 7b-8b, there's a discussion of the minimum dimensions of a round sukkah, with a variety of rabbis weighing in on the minimum measurements, which, despite the square ones being in cubits, R. Johanan discusses by the minimum number of people who can sit around the circumference, which leads to a discussion of how much space a person takes up. and then this lead to the topic of inscribing a circle in a square, or circumscribing it. The Tosafot there has much to say on the topic, as well as the Tashbetz I.165. Interestingly, the Rambam (commentary on misha Eruvin 1:5) basically says that π is not able to be precisely calculated, ie, is irrational, and the rabbis were concerning themselves with a good enough approximation, for whatever value of good enough.
The last source was R. Matityahu haKohen Munk, "Three Geometrical Problems in the Bible and the Talmuc," Sinai 51 (1962) pp. 218-227, discussing why the I Kings source was so imprecise, with an interesting gematria. In Kings, kav ("line") has a ktiv and a kri form (written and pronounced). The written here is kuf-vav-heh, which is still pronounced kav. The gematria of the longer form is 111, while the gematria of the shorter form is 106. There's a complicated discussion of the ratio of the larger to the smaller number being a good apparximation for something to dowith the I Kings source, that seemed interesting at the time, but obscure now.
The second topic was square roots and the Pythagorean theorem, especially √2, since it's the length of the diagonal of a unit square. First, though, there was the general discussion of how math was all geometry then, not numbers, but lengths or areas.
The Tosafot on Sukkah 8a talks about how the Talmud approximation of 1.4 as the length of the diagonal of a unit square is not exact, with a proof showing that this so.
The Mishnat Hamidot 4:1 says that there are right, acute, and obtuse triangles, then defines a right triangle not by having a right angle in it, but by fulfilling the Pythagorean theorem.
The next sources were mishna and talmud (both Babylonian and Jerusalem) on Eruvin, on maximal spaces that one can carry within on Shabbat, with some abstruse wording that I find fairly incomprehensible, looking back now, about two beit se'ah (the area that is covered by a se'ah (volume measure) of seeds, ie, a particular acreage). The Rambam discusses how the diagonal isn't precisesly calculatable, though there are some decent approximations. .
The next topic was about the boundaries of the t'chum, 2000 cubits outside the city walls, beyond which one cannot walk on Shabbat. For a circular city, it's assumed to be a square with the diameter being the side length, then 2000 cubits added to each side, making a larger square, not a rounded corner square as one might think. The source there talks about how much extra one gets, and it's interesting to see how the rabbis increased this area twice, basically (if this law were being promulgated today, it would be fenced in and fenced in...).
Then there was a mishna in Kilayim 5:5, about planting two species together, which is forbidden. If another kind of vegetable is grown in a vineyard, how many of the surrounding vines are no longer usable? The Rash, the Rosh, and the Tosafot Yom Tove gave involved explanations, depending on how closely the vines were spaced, including an area centered at the offending other plant.
The third topic was the great circle route to Jerusalem, ie, spherical geometry, particularly in relation to which way one faces when praying. We started with inflatable globes, playing with string to show great sircle paths and how much they weren't exactly east, then moved on to look at texts, starting with I Kings 8:44-45 and Daniel 6:11 which talk about facing the Temple/Jerusalem when praying. Then the mishna in Brachot 4:5 talks about which way to face or to turn one's intention. The Babylonian Talmud Brachot 30a has a hierarchy of where to face. A blind person or one who does not know the location intends the heart toward the Almighty. In Israel, face Jerusalem. In Jerusalem, face the Temple. In the Temple, face the Holy of Holies. In the Holy of Holies (!) faces the beit hakaporet And one who is behind the kaporet faces towards the front of the kaporet, saying that in this way, the whole nation directs their hearts to a single place.
The Babylonian talmud Bava Batra 25a-b talks about where one can put a tannery (smelly, which shouldn't be somewhere sacred), which lead to a discussion of which way one faces when praying. Amazingly, there's lots of opinions, including west (like the morning sun), south or north (to become wise or rich). By the time of the Tur and the Shulchan Aruch the expected 'face Jerusalem' opinion is paramount. However, the Levush (Orech Chayim 94) brings up the issue of how one shouldn't face due east (to avoid appearing to be idolatrous, worshipping the sun), and how instead, one should examine (complicated considerations that boil down to using the great circle path to Jerusalem, rather than the plumb line path, except that his reasoning wasn't precise, so the Emunat Chachamim brings more precise mathematical discussions, invoking trigonometry and great circles by name). The Netzach Yisrael also points out errors in the Levush's reasoning. The Shulchan Aruch HaRav (Orech Chayim 94:2) weighs in with how much one should build a synagogue facing somewhat south or north of east.
Interestingly, the teacher included an email from the rabbi of Congregation Or HaTzafon (great name!) in Fairbanks, AK, which ended up showing that it's likely people there came from the eastern US, given how the shortest route from there would be over the North Pole, and since it's more than 180 degrees west of Jerusalem, theoretically, they should face west. But facing east has the great power of evocativeness, which is the direction they face.
The last topic was the calendar, taught by Ben Dreyfus, about how the alendar is no longer determined by witnesses to the new moon, but by a fixed calendar with 7 leap years every 19 year cycle, designed to keep Passover in the spring. However, over thousands of years, this will also have drift unless there ends up being enough consensus to fix it. And there was more, but my mind somehow goes vague with calendrical discussion.
First up was the relationship between the diameter and circumference of a circle, ie π. In I Kings 7:23-26, there's a discussion of something being made for the Temple that is 10 cubits across, and 30 cubits around, which is obviously problematic in reality. There was a discussion of approximations of π in the ancient world; it was already known that 3 is not at all close, though if rounding to the nearest integer, it might do. Still, the Greeks and the Babylonians already had better approximations, so it's reasonable to assume that the composer(s) of Tanakh did as well. And if Tanakh is perfect, then what's going on here with 3 for π? There are a variety of explanations by commentators. Ralbag discusses inner v. outer circumference, as well as pointing out that it's just a very rough approximation, for instance.
Then to Mishna Eruvin 1:4-5, which discusses minimal dimensions of a beam, based on an assumptive square beam. However, if the beam is circular, the assumption is that a circumference of three handbreadths means a diameter of on handbreadth, ie, another π being 3 situation. The Talmud on the mishna (Eruvin 14a) brings in the source in I Kings as an explanation. The Tosafot haRosh comes to say that it's obviously an approximation, not exact. The Mishnat haMidot (a rabbinic math text found in manuscript in modern times) 5:3-4 discusses calculating the area, with a fairly good approximation of π.
In the Babylonian Talmud, Sukkah 7b-8b, there's a discussion of the minimum dimensions of a round sukkah, with a variety of rabbis weighing in on the minimum measurements, which, despite the square ones being in cubits, R. Johanan discusses by the minimum number of people who can sit around the circumference, which leads to a discussion of how much space a person takes up. and then this lead to the topic of inscribing a circle in a square, or circumscribing it. The Tosafot there has much to say on the topic, as well as the Tashbetz I.165. Interestingly, the Rambam (commentary on misha Eruvin 1:5) basically says that π is not able to be precisely calculated, ie, is irrational, and the rabbis were concerning themselves with a good enough approximation, for whatever value of good enough.
The last source was R. Matityahu haKohen Munk, "Three Geometrical Problems in the Bible and the Talmuc," Sinai 51 (1962) pp. 218-227, discussing why the I Kings source was so imprecise, with an interesting gematria. In Kings, kav ("line") has a ktiv and a kri form (written and pronounced). The written here is kuf-vav-heh, which is still pronounced kav. The gematria of the longer form is 111, while the gematria of the shorter form is 106. There's a complicated discussion of the ratio of the larger to the smaller number being a good apparximation for something to dowith the I Kings source, that seemed interesting at the time, but obscure now.
The second topic was square roots and the Pythagorean theorem, especially √2, since it's the length of the diagonal of a unit square. First, though, there was the general discussion of how math was all geometry then, not numbers, but lengths or areas.
The Tosafot on Sukkah 8a talks about how the Talmud approximation of 1.4 as the length of the diagonal of a unit square is not exact, with a proof showing that this so.
The Mishnat Hamidot 4:1 says that there are right, acute, and obtuse triangles, then defines a right triangle not by having a right angle in it, but by fulfilling the Pythagorean theorem.
The next sources were mishna and talmud (both Babylonian and Jerusalem) on Eruvin, on maximal spaces that one can carry within on Shabbat, with some abstruse wording that I find fairly incomprehensible, looking back now, about two beit se'ah (the area that is covered by a se'ah (volume measure) of seeds, ie, a particular acreage). The Rambam discusses how the diagonal isn't precisesly calculatable, though there are some decent approximations. .
The next topic was about the boundaries of the t'chum, 2000 cubits outside the city walls, beyond which one cannot walk on Shabbat. For a circular city, it's assumed to be a square with the diameter being the side length, then 2000 cubits added to each side, making a larger square, not a rounded corner square as one might think. The source there talks about how much extra one gets, and it's interesting to see how the rabbis increased this area twice, basically (if this law were being promulgated today, it would be fenced in and fenced in...).
Then there was a mishna in Kilayim 5:5, about planting two species together, which is forbidden. If another kind of vegetable is grown in a vineyard, how many of the surrounding vines are no longer usable? The Rash, the Rosh, and the Tosafot Yom Tove gave involved explanations, depending on how closely the vines were spaced, including an area centered at the offending other plant.
The third topic was the great circle route to Jerusalem, ie, spherical geometry, particularly in relation to which way one faces when praying. We started with inflatable globes, playing with string to show great sircle paths and how much they weren't exactly east, then moved on to look at texts, starting with I Kings 8:44-45 and Daniel 6:11 which talk about facing the Temple/Jerusalem when praying. Then the mishna in Brachot 4:5 talks about which way to face or to turn one's intention. The Babylonian Talmud Brachot 30a has a hierarchy of where to face. A blind person or one who does not know the location intends the heart toward the Almighty. In Israel, face Jerusalem. In Jerusalem, face the Temple. In the Temple, face the Holy of Holies. In the Holy of Holies (!) faces the beit hakaporet And one who is behind the kaporet faces towards the front of the kaporet, saying that in this way, the whole nation directs their hearts to a single place.
The Babylonian talmud Bava Batra 25a-b talks about where one can put a tannery (smelly, which shouldn't be somewhere sacred), which lead to a discussion of which way one faces when praying. Amazingly, there's lots of opinions, including west (like the morning sun), south or north (to become wise or rich). By the time of the Tur and the Shulchan Aruch the expected 'face Jerusalem' opinion is paramount. However, the Levush (Orech Chayim 94) brings up the issue of how one shouldn't face due east (to avoid appearing to be idolatrous, worshipping the sun), and how instead, one should examine (complicated considerations that boil down to using the great circle path to Jerusalem, rather than the plumb line path, except that his reasoning wasn't precise, so the Emunat Chachamim brings more precise mathematical discussions, invoking trigonometry and great circles by name). The Netzach Yisrael also points out errors in the Levush's reasoning. The Shulchan Aruch HaRav (Orech Chayim 94:2) weighs in with how much one should build a synagogue facing somewhat south or north of east.
Interestingly, the teacher included an email from the rabbi of Congregation Or HaTzafon (great name!) in Fairbanks, AK, which ended up showing that it's likely people there came from the eastern US, given how the shortest route from there would be over the North Pole, and since it's more than 180 degrees west of Jerusalem, theoretically, they should face west. But facing east has the great power of evocativeness, which is the direction they face.
The last topic was the calendar, taught by Ben Dreyfus, about how the alendar is no longer determined by witnesses to the new moon, but by a fixed calendar with 7 leap years every 19 year cycle, designed to keep Passover in the spring. However, over thousands of years, this will also have drift unless there ends up being enough consensus to fix it. And there was more, but my mind somehow goes vague with calendrical discussion.