Post Amhb0zoPvLtXtIOFto by llewelly@sauropods.win
(DIR) More posts by llewelly@sauropods.win
(DIR) Post #AmhafsEgtgX4JQbjyi by futurebird@sauropods.win
2024-10-05T16:47:03Z
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Certainly the circle divides the plane into two regions, but which one is inside of the circle? Which one is without?On the surface of a sphere you have a real problem. (OK call the one with smaller area "inside" done and dusted.) On an infinite plane, maybe you can say something about finite vs. infinite to make the distinction. But this question has always made me uncomfortable.
(DIR) Post #AmhawhvIUdLVFbYzh2 by richpuchalsky@mastodon.social
2024-10-05T16:49:42Z
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@futurebird Why is this problem specific to round shapes? If you don't like the common undeerstanding of inside and outside, it's going to happen with any shape
(DIR) Post #Amhb0zoPvLtXtIOFto by llewelly@sauropods.win
2024-10-05T16:50:50Z
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@futurebird a non-coplanar parallel plane is without the circle.
(DIR) Post #Amhb2qSPfzcnSLyo4G by mcc@mastodon.social
2024-10-05T16:51:00Z
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@futurebird extend the plane with a "point" at infinity and then you can just do (az + b)/(cz + d) and everything's simple, the plane is a sphere https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
(DIR) Post #Amhb6XpirVL0qxe9h2 by futurebird@sauropods.win
2024-10-05T16:51:48Z
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@richpuchalsky It's not. The circle is just the example.
(DIR) Post #Amhc5AZjwr5MNw4yA4 by Wharrrrrrgarbl@an.errant.cloud
2024-10-05T17:02:51Z
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@futurebird if the circle is defined as the set of points a specific distance from a center point, then it seems natural to me to say that the center is inside the circle.Conversely, English metaphors like an inner circle don't usually refer to a space of possibilities that stretches to the horizon and excludes only a small group with specific properties (almost the exact opposite, really) .
(DIR) Post #AmhcHEwzVY8HQei6Fc by futurebird@sauropods.win
2024-10-05T17:05:01Z
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@Wharrrrrrgarbl On a sphere two points meet that definition. On an infinite plane the point at infinity meets the definition if you are willing to take it seriously.
(DIR) Post #AmhducgLJBqYHLGZ16 by catselbow@fosstodon.org
2024-10-05T17:23:14Z
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@futurebird If you walk around the circle counterclockwise (but let's say "widdershins"!) we could define the interior as the part that's on our left-hand side. I think you always need to think about walking the circle.
(DIR) Post #AmhhISwTTL44sqs9Oy by algebraicyclist@mathstodon.xyz
2024-10-05T18:01:15Z
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@futurebird no, this is a great thing to be uncomfortable about! it’s actually a real theorem that a closed curved divides the plane into two regions (Jordan closed curve theorem)
(DIR) Post #AmhiXB1uS5LPcxzeAi by Wharrrrrrgarbl@an.errant.cloud
2024-10-05T18:15:08Z
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@futurebird is it just one point at infinity? Not an infinite number of them? That is, the direction you set off in from the center point in the smaller region takes you to the same place regardless? (Everything I learned about infinities I learned as an adult, non-rigorously, from YouTube)
(DIR) Post #AmhjMqAVG8Nvz9eEkq by MishaVanMollusq@sfba.social
2024-10-05T18:24:26Z
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@futurebird oooouuu you do Topology too.Wonder if you ever read back in the day
(DIR) Post #AmhtQYLuRKXvQiONH6 by richpuchalsky@mastodon.social
2024-10-05T20:17:09Z
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@futurebird Good, then it's a consistent problem. In that case if you think of these as geometric entities then it's not that the inside is merely smaller than the outside, it's that the inside has a finite amount of space and the outside has an infinite amount of space (within the number of dimensions that the shape is in).
(DIR) Post #AmiRCLjTxaCgg4oiMC by daviddlevine@wandering.shop
2024-10-06T02:35:33Z
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@futurebird I believe that the definition of "inside" (in a plane) is that, given a point, if EVERY line drawn through the point intersects the circle exactly twice the point is "inside." But if even one line drawn through the point intersects the circle 0 or 1 times, the point is "outside."