How do we know that the ratio between the circumference and the diameter of a circle is preserved across radius sizes?

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• mushroommunk@lemmy.today ⁨3⁩ ⁨weeks⁩ ago

  There’s kind of a meek proof built of of work Euclid and Archimedes. There’s a paper that goes over it (I know skimmed) arxiv.org/pdf/1303.0904

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• al4s@lemmy.world ⁨3⁩ ⁨weeks⁩ ago

  You can approximate the length of any path (including circles) by adding the lengths of many small line segments that follow that path. Making a line segment bigger by some factor, will increase it’s length by the same factor. Therefore, scaling the circle by any factor, increases it’s circumference by the same factor. Scaling a circle is just scaling it’s radius so: Scaling the radius by some factor, changes the circumference by the same factor. That means the ratio between radius and circumference is always constant.

  I hope this is decipherable :D

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• can_you_change_your_username@fedia.io ⁨3⁩ ⁨weeks⁩ ago

  Shapes don't actually exist. They are abstract concepts that we use to describe and make predictions about the material world. We know that the ratio between the circumference and diameter of a circle are always the same because that's part of how we define a circle.

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• Devial@discuss.online ⁨3⁩ ⁨weeks⁩ ago

  Because circle all have the same proportions. You can take any circle, and just evenly make it bigger or smaller to make it perfectly overlap with any other circle.

  The ratios of shapes only ever change if their proportions change. That’s why every single square also always has the same ratio between it’s side and diagonal (√2).

  And the ratio of a rectangles side to it’s diagonal will always be the same, regardless of size, as long as the aspect ratio is the same.

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• TachyonTele@piefed.social ⁨3⁩ ⁨weeks⁩ ago

  The greeks figured this one out, and I tend to believe them.

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• solrize@lemmy.ml ⁨2⁩ ⁨weeks⁩ ago

  In math, it’s a theorem based on certain assumptions and definitions about the distances between points, and what length means. You start with human-made assumptions and follow them to wherever they lead.

  Those assumptions are pretty well justified based on local observations of the real world. Are they true on a bigger scale, say at astronomical distances? People began to wonder this in the 1800’s, in the era of Gauss and Riemann. There’s another theorem that the interior angles of a triangle add up to 180 degrees, and Gauss (an astronomer as well as a math whiz) actually proposed testing that on astronomical observations. I don’t know if hey tried any experiments though. A deviation from 180 degrees would mean that space was curved.

  Lo and behold, it turns out that space actually is curved, in the presence of gravitational fields. That was figured out by none other than Einstein, who became world famous when Eddington did an observation during a solar eclipse in 1919 and saw the apparent motion of distant stars when they got lined up with the edge of the sun. The eclipse was needed for the observation since otherwise the sun would have drowned out the distant stars. But, it was quite a sensitive experiment, maybe not possible in the era of Gauss.

  Anyway, the “big” answer to your question is that the ratio being constant is in the end an empirically observed fact, but that on a cosmic scale is only a close approximation, and (even Einstein didn’t foresee this) falls completely apart near very extreme ragions like black holes.

  Einstein’s theory (“general relativity”) was still an incredible work of genius. As the saying goes, they didn’t call him Einstein for nothing!

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• ThatGuy46475@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  There’s always a unit that makes the radius 1, 1cm, 1 foot, 1 furlong, one light year, 1 of something else. So the math for a circle of radius 1 would still hold.

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• Wolf314159@startrek.website ⁨2⁩ ⁨weeks⁩ ago

  You’re looking for proofs that Pi is constant.

  • Here’s one
  • Stack Exchange

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• Twinklebreeze@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  I think c/nostupidquestions is actually for questions that are kinda stupid. This is too smart of a question. /s

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• Reetsh@lemmy.ml ⁨3⁩ ⁨weeks⁩ ago

  We know the circumference of a circle is 2 * pi * Radius (c = 2 * pi * r). The diameter of a circle is 2 * Radius (d = 2 * r). Therefore the circumference of a circle is pi * Diameter (c = pi * d). The ratio between circumference and diameter is pi, which is a constant and therefore doesn’t change even when radius size changes.

  How do we know Pi? We have literally known about it for so long that no one has an historical account of who first conceived of it. The oldest example we have is from Babylon, and even then we don’t think they discovered it just that they were aware of it. www.britannica.com/science/pi-mathematics

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• captainlezbian@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  Because we use it predictively enough we’d hsve figured out if it wasn’t by now

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