Random Thoughts XI: Listen to the Whispers
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Shakespeare doth be hot garbage.
Zardnaar
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New world heritage sites
https://www.cnn.com/travel/article/unesco-world-heritage-list-2021/index.html
https://www.cnn.com/travel/article/unesco-world-heritage-list-2021/index.html
There is no way you'd get me climbing those stairs in the bell tower, with or without a handrail!
Especially when a couple of them look about ready to fall to pieces.
shadowplay
your ad here
Thunderfall isn't real. He's based on a fable of a Dutch saint named Tunter Fell.
Zardnaar
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Oh dear.
EgonSpengler
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And yet, it's somehow apropos, isn't it?
EgonSpengler
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Zardnaar
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Yep. Dick head.
Babylonians calculated with triangles centuries before Pythagoras
The ancient Babylonians understood key concepts in geometry, including how to make precise right-angled triangles. They used this mathematical know-how to divide up farmland – more than 1000 years before the Greek philosopher Pythagoras, with whom these ideas are associated.
“They’re using a theoretical understanding of objects to do practical things,” says Daniel Mansfield at the University of New South Wales in Sydney, Australia. “It’s very strange to see these objects almost 4000 years ago.”
Babylonia was one of several overlapping ancient societies in Mesopotamia, a region of southwest Asia that was situated between the Tigris and Euphrates rivers. Babylonia existed in the period between 2500 and 500 BC, and the First Babylonian Empire controlled a large area between about 1900 and 1600 BC.
Mansfield has been studying a broken clay tablet from this period, known as Plimpton 322. It is covered with cuneiform markings that make up a mathematical table listing “Pythagorean triples”. Each triple is the lengths of the three sides of a right-angled triangle, where each side is a whole number. The simplest example is (3, 4, 5); others include (5, 12, 13) and (8, 15, 17).
The triangles’ sides are these lengths because they obey Pythagoras’s theorem: the square of the longest side is equal to the sum of the squares of the other two sides. This classic bit of mathematics is named for the Greek philosopher Pythagoras, who lived between about 570 and 495 BC – long after the Plimpton 322 tablet was made.
“They [the early Babylonians] knew Pythagoras’ theorem,” says Mansfield. “The question is why?”
Mansfield thinks he has found the answer. The key clue was a second clay tablet, dubbed Si.427, excavated in Iraq in 1894. Mansfield tracked it down to the Istanbul Archaeology Museums.
Si.427 was a surveyor’s tablet, used to make the calculations necessary to fairly share out a plot of land by dividing it into rectangles. “The rectangles are always a bit wonky because they’re just approximate,” says Mansfield. But Si.427 is different. “The rectangles are perfect,” he says. The surveyor achieved this by using Pythagorean triples.
“Even the shapes of these tablets tell a story,” says Mansfield. “Si.427 is a hand tablet… Someone’s picked up a piece of clay, stuck it in their hand and wrote on it while surveying a field.” In contrast, Plimpton 322 seems to be more of an academic text: a systematic investigation of Pythagorean triples, perhaps inspired by the difficulties surveyors had. “Someone’s got a huge slab of clay… [and] squashed it flat” while sitting at a desk, he says.
Journal reference: Foundations of Science, DOI: 10.1007/s10699-021-09806-0
https://www.newscientist.com/articl...ay&utm_medium=email&utm_campaign=NSDAY_050821
The ancient Babylonians understood key concepts in geometry, including how to make precise right-angled triangles. They used this mathematical know-how to divide up farmland – more than 1000 years before the Greek philosopher Pythagoras, with whom these ideas are associated.
“They’re using a theoretical understanding of objects to do practical things,” says Daniel Mansfield at the University of New South Wales in Sydney, Australia. “It’s very strange to see these objects almost 4000 years ago.”
Babylonia was one of several overlapping ancient societies in Mesopotamia, a region of southwest Asia that was situated between the Tigris and Euphrates rivers. Babylonia existed in the period between 2500 and 500 BC, and the First Babylonian Empire controlled a large area between about 1900 and 1600 BC.
Mansfield has been studying a broken clay tablet from this period, known as Plimpton 322. It is covered with cuneiform markings that make up a mathematical table listing “Pythagorean triples”. Each triple is the lengths of the three sides of a right-angled triangle, where each side is a whole number. The simplest example is (3, 4, 5); others include (5, 12, 13) and (8, 15, 17).
The triangles’ sides are these lengths because they obey Pythagoras’s theorem: the square of the longest side is equal to the sum of the squares of the other two sides. This classic bit of mathematics is named for the Greek philosopher Pythagoras, who lived between about 570 and 495 BC – long after the Plimpton 322 tablet was made.
“They [the early Babylonians] knew Pythagoras’ theorem,” says Mansfield. “The question is why?”
Mansfield thinks he has found the answer. The key clue was a second clay tablet, dubbed Si.427, excavated in Iraq in 1894. Mansfield tracked it down to the Istanbul Archaeology Museums.
Si.427 was a surveyor’s tablet, used to make the calculations necessary to fairly share out a plot of land by dividing it into rectangles. “The rectangles are always a bit wonky because they’re just approximate,” says Mansfield. But Si.427 is different. “The rectangles are perfect,” he says. The surveyor achieved this by using Pythagorean triples.
“Even the shapes of these tablets tell a story,” says Mansfield. “Si.427 is a hand tablet… Someone’s picked up a piece of clay, stuck it in their hand and wrote on it while surveying a field.” In contrast, Plimpton 322 seems to be more of an academic text: a systematic investigation of Pythagorean triples, perhaps inspired by the difficulties surveyors had. “Someone’s got a huge slab of clay… [and] squashed it flat” while sitting at a desk, he says.
Journal reference: Foundations of Science, DOI: 10.1007/s10699-021-09806-0
https://www.newscientist.com/articl...ay&utm_medium=email&utm_campaign=NSDAY_050821
Afaik the pythagorean triples (3 numbers which satisfy a^2+b^2=c^2) by the babylonians had zero theoretical grounding; they seemed to just play around with numbers and find (by chance) a few of the triples. For small enough numbers, this is easy, say 3,4,5.
The babylonians (or any similar civ) had no sense of even what a theorem is; their math was non-theoretical and used (in the case of triangles) just for approximations for building.
Maybe they could have noticed how to automatically generate them* - still not worth to be mentioned in the same sentence as Pythagoras, who is one of the most important humans ever
*googling presented this method, which is simple:
The babylonians (or any similar civ) had no sense of even what a theorem is; their math was non-theoretical and used (in the case of triangles) just for approximations for building.
Maybe they could have noticed how to automatically generate them* - still not worth to be mentioned in the same sentence as Pythagoras, who is one of the most important humans ever
*googling presented this method, which is simple:
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Much as Kyriakos says above, knowing that a set of numbers that obey Pythagoras’ theorem exist is very different from proving that it will work for any sides of a right angle triangle. The Babylonians were clever and all, but so was Pythagoras.
I'm pretty sure we do not know the thought processes behind the two Babylonian tablets. How do we know that the Babylonians had no theoretical grounding in their mathematics? Painting the Babylonians as ignorant savages to prop up Greece is pretty pathetic. Knowledge is gained and sometimes lost before it is rediscovered. Written records are often lost if they are very old. I think it is very likely that Pythagoras build on and formalized what was already known. Cuneiform was the dominate written language of the ME until about 250BCE. Tens of thousands of cuneiform tablets that have never been translated are in museums around the world. Too few people are able to translate them.
The more one looks at how knowledge advances, the more one can see that it usually has a longer history than expected and new advances in thinking have deeper roots than the "inventor". Greece gets credit for lots of things because their written records are well known.
The more one looks at how knowledge advances, the more one can see that it usually has a longer history than expected and new advances in thinking have deeper roots than the "inventor". Greece gets credit for lots of things because their written records are well known.
It has to be assumed, moreover, than in greek architecture of the time there wasn't much use of calculations to arrive at triples, given that (at least for relatively small areas - ie not the temple of Artemis at Ephesos or similar ) the architects would use circles and inscribed triangles. This was already known to always create a right-angle triangle, due to a theorem by Thales; decades before Pythagoras.
Only the diameter is fixed (but can be moved at an angle within the circle). The remaining vertex can be anywhere on the periphery of the circle.
Also worth noting that Euclid presented his own algebraic calculation for generating pythagorean triples.
@Birdjaguar : it's not like this is some hot take; greek math has very different traits from pre-greek math. This has been examined very extensively, and already was being examined in the ancient era. The key stages of its development are also well-documented, since it was the center of the greek world of thought. For better or worse, greek math (and consequent math) is axiom-based and theorem-centered. Possibly this was the only way to advance - and then paved the way for uncertainty and limits forced by axioms, which is another issue.
As for Pythagoras, he traveled mostly to Egypt. Thales had some ties to Babylon. Both Egypt and Babylon had specific priestly orders who dealt with stuff the ordinary mortals shouldn't be told, such as math. Then again, neither produced any memorable proof about anything, unlike the concurrent to them pythagorean proof of the existence of irrational numbers - maybe this happened because they didn't have a notion of proof in the first place, as Samson alluded to and you didn't quite pick up
) the architects would use circles and inscribed triangles. This was already known to always create a right-angle triangle, due to a theorem by Thales; decades before Pythagoras.Only the diameter is fixed (but can be moved at an angle within the circle). The remaining vertex can be anywhere on the periphery of the circle.
Also worth noting that Euclid presented his own algebraic calculation for generating pythagorean triples.
@Birdjaguar : it's not like this is some hot take; greek math has very different traits from pre-greek math. This has been examined very extensively, and already was being examined in the ancient era. The key stages of its development are also well-documented, since it was the center of the greek world of thought. For better or worse, greek math (and consequent math) is axiom-based and theorem-centered. Possibly this was the only way to advance - and then paved the way for uncertainty and limits forced by axioms, which is another issue.
As for Pythagoras, he traveled mostly to Egypt. Thales had some ties to Babylon. Both Egypt and Babylon had specific priestly orders who dealt with stuff the ordinary mortals shouldn't be told, such as math. Then again, neither produced any memorable proof about anything, unlike the concurrent to them pythagorean proof of the existence of irrational numbers - maybe this happened because they didn't have a notion of proof in the first place, as Samson alluded to and you didn't quite pick up
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Underestimating the skills and thinking of those came before us is a common human trait. After all, the Egyptians certainly could not have built the pyramids without alien help. Athena, the goddess of wisdom sprang fully grown from the forehead of Zeus! Europe emerged from its "Dark Age" because Europeans were smarter than the rest. Greece emerged from it's "Dark Age" because they were smarter than their neighbors. In both cases the ensuing rebirths came from expanding trade and contracts with neighboring cultures and nations. Classical Greece deserves credit for lots of things, among them, better written records, but how to think about problems is far older.
Depends on what you mean by "how to think about problems". When you apparently only use math for practical stuff, like the egyptians and babylonians seemed to, you can get by with approximation. After all, even today if you are a civil engineer you aren't likely to need to go beyond some degree of detail (which can still be not easy to access for a layman, but certainly is trivial for mathematical research). But approximation vs total accuracy is only a secondary issue here, and it stems from the primary issue, which was that Egypt/Babylon/similar simply had no framework of proof, making their math paradoxically something like physics, where you are dependent on trial and error, "experiment" by calculating stuff. But math is a higher order exactly because there is no experiment; framed as a system, you form proofs using axioms (albeit some things do remain obscure).
Now, anyone can speculate, without any proof, that maybe the egyptian and babylonian priest classes had some framework. But if that was so - and it is a huge "if" - it was certainly lost due to their secrecy. Contrary to what happened with most in the greek world (Pythagoras not so much; he formed his school as a cult and was very secretive), where knowledge would be available and allow more people to contribute and keep track.
Now, anyone can speculate, without any proof, that maybe the egyptian and babylonian priest classes had some framework. But if that was so - and it is a huge "if" - it was certainly lost due to their secrecy. Contrary to what happened with most in the greek world (Pythagoras not so much; he formed his school as a cult and was very secretive), where knowledge would be available and allow more people to contribute and keep track.
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