The Community
General Category => Matters of Life and The Universe => Topic started by: Zzzptm on May 30, 2018, 08:06:40 AM
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https://en.wikipedia.org/wiki/Pascal%27s_theorem
If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum.
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What makes you think you can connect the same 6 points in 60 different ways and still have a hexagon? This is not correct.
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Rather, there are 60 different ways of connecting the points that make the hexagon on the conic, corner to corner. The original sides of the hexagon are not considered in the operation.
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But the 60 lines you first mentioned are not associated with the Hexagrammum Mysticum Theorem. This theorem states that if you extend the 3 opposite sides of the hexagon, they intersect at 3 different points, and these 3 points form a straight line.
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Gotta keep going... there's a whole theorem, here.
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Only took 1 year of geometry in High School, aced it but we never talked about this kinda stuff... over my head. :think:
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I've heard it said that there are two books that have had more impact on human society than any others:
The Bible
Euclid's Elements
Hrmm...
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I could agree to the Euclid one. I once had a course in Modern Geometry that took Euclid's Elements and then applied them to non-flat planes. While parallel lines went off the rails depending on the type of space one was in, the other theorems held up well. It's like Euclid was able to put into words some of the most fundamental geometric foundations of our universe.
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Consider this, about dear Euclid's magnum opus:
He didn't merely introduce geometric concepts, he introduced the basis of mathematical logic. Using a previously proven fact to thereafter support proving a heretofore unproven fact. It seems simple only because Euclid wrote it down thousands of years ago. The very way we think and make decisions is influenced by Euclid's process.
On a side note, there probably wasn't even a guy named Euclid who wrote a book called The Elements. As with much of ancient Greek authorship, it's very cloudy. But, his name is on it, and whoever put all of the theorems together changed the world.
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That's a great point, how one big thinker would have loads of work attributed to him by his students, followers, and admirers. They figured if the guy didn't say it, well, he *should* have said it and then put his name to the document.
Funny enough, it's something that also happened in China, in a tradition totally separate from the Mediterranean.