Current Events > Circles confuse me.

(+) Yeah! (+)

When we first learn about how many sides a shape has, schools typically begin with triangles and go up from there. I don't know if you remember schools ever teaching you about circles and sides, but I can't really recall myself. I think, as a kid, if we did ever ponder the question, our intuitions told us one of two answers: a circle has zero sides or a circle has one side.

Zero would come from the thought that a circle cannot have two distinct points to mark the beginning and end, right? A circle is a curve that closes upon itself at the point in which it began (and to be fair, that isn't a unique property to circles as ellipses as a whole share that trait before I go on a "magic of circles" tangent). Additionally, a circle does not have a corner(s) to connect a line to another as its a continuous line from start to finish. On the other hand, there's the belief that we have as children that it has one side. We just decide that the entire shape itself is the side, it has one side that begins and ends.

As I (and others, I'm not special) have gotten older, though, we notice a pattern, and a bit of calc might perhaps mix our brains up too (not that you need calc to recognize the pattern; just the concept of limits might make us create "equations" in our heads that aren't actually valid). Let me say real quick before I get too far and the actual geniuses of CE come on by; I love math, was pretty good at it up until a point, but I'm no mathematician. Was really good at the HS math, and even Calc 1, but beyond that, I didn't really have a future. That's just to say, there's an obvious chance I'm just misusing the properties of limits that my distant memory of them has probably warped as well over the years.

The pattern I mentioned though has to do with regular polygons (all sides equal) as the number of sides increase. From a triangle to hexagon, perhaps you don't really notice all too much, but from a hexagon to a decagon, you start to notice how they become more circular in appearance. Eventually, you start to ask what a 100-sided shape gets you (https://en.wikipedia.org/wiki/Hectogon), a 1000-sided shape (https://en.wikipedia.org/wiki/Chiliagon) and shapes even beyond that (https://en.wikipedia.org/wiki/Megagon). At these insane-numbers, they appear indistinguishable from a circle. You start to think of the idea then: as x approaches infinity, what shape do you get? Based on the pattern before, would it not be unreasonable to believe that a shape with an infinite number of sides is a circle after all? Does that mean a circle has an infinite number of sides to fulfill this pattern that we have noticed?

I recently looked it up and came across this mathematician's article on this very subject: https://richardelwes.co.uk/2019/09/06/how-many-sides-does-a-circle-have/

I was hoping to have my worldview accepted on my very first try, to be applauded by someone of even higher level than me (Yay, PiOverlord, keeper of circles, you are indeed right. A circle does have infinite sides!!! That's why you named yourself PiOverlord, right?) He even starts with a poll where many agree with me to give me short-lived confidence before the line stating that those who believe it to have an infinite number of sides are the only ones objectively wrong on this subject. Oof. It turns out, our child-like intuition is closer to the truth than what we perceive as a logical conclusion based on patterns and the concept of limits that approach infinity. Depending on what you accept the definition of a side to be, you can get different answers to this question; yet none of those answers are infinity.

Still, it baffles me, and somewhat irritates me. Patterns! Come on man. You're telling me we go from perfect circle to triangle to square to less-perfect not-circles to indistinguishable-from-circles to still-not-a-circle at the end of the road?!? What kinda hogwash is that? You're telling me that if we had a shape that closes upon itself with infinitesimal small lengths of infinitely-numbered sides, that's still not a circle!!!! It just... doesn't sit well with me. It feels paradoxical in nature. It feels like the nature of sides is pointing to the creation of a circle. Yet, this is wrong? No, that can't be true, I, the non-mathematician must be right! Right?

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What is your point?

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yillin posted...

What is your point?

Circles confuse me.

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https://youtu.be/OLodW1nyZ4s

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They confuse you because you think they are taught first? You think they don't have infinite sides? help us out here. Focus your rambling.

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https://www.youtube.com/watch?v=9WGO7dAxjD8

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yillin posted...

They confuse you because you think they are taught first? You think they don't have infinite sides? help us out here. Focus your rambling.

According to mathematicians, a circle does not have an infinite number of sides. You can argue a circle has zero or one, or even that none of those answers apply, but you can't argue it has an infinite number of sides. Yet, on the other hand, we can clearly see that the more sides a regular polygon has added to it, the closer it gets to a circle-like appearance. If a circle has one side, for instance, imagine the pattern. You go from what is a perfect circle to what is far from one (a triangle). Then you start to get closer and closer to that perfect circle, but you'll never actually get there.

As an analogy, imagine you are shown the perfect bottle of wine. This wine cannot be surpassed, it is the pinnacle of wine. Wanting to recreate this wine, you examine a set of wine. You notice that wine that has aged longer appears to be closer and closer to this perfect bottle of wine. Upon seeing this pattern, you conclude that the perfect bottle of wine is the result of a bottle that has aged for an infinite number of years (for this example's purpose of course). Suddenly, you're told that is wrong. The perfect bottle of wine actually was a bottle that had not aged yet at all. In reality, you are told, a bottle of wine that has aged for an infinite number of years is still not-quite-the-perfect bottle of wine!

What you see is at the beginning point 0, you have a perfect bottle of wine. Suddenly, at point 1, you have the worst bottle of wine you can get. Slowly, as the years pass, it does start to resemble the original perfect bottle more and more, but it will never get there. It feels like a pattern with an odd-man-out within it. It does not seem to conform to the logical pattern you notice the set presents you with.

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yillin posted...

What is your point?

The point is that there are infinite points or no points.. Ya know

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When we first learn about how many sides a shape has, schools typically begin with triangles and go up from there. I don't know if you remember schools ever teaching you about circles and sides, but I can't really recall myself. I think, as a kid, if we did ever ponder the question, our intuitions told us one of two answers: a circle has zero sides or a circle has one side. We do remember circles and sides, but there is one of them. A triangle would have one side, but a circle has one side. There would even be one side for which there would be no side. So in that sense, triangles never are real. So if we could simply use an axiom, a triangle would have all one end, and a circle none.

And that's what you'll notice is, you cannot really make triangles exist when you think about triangles. For one thing, I do not think they ever existed. If you thought the world was a circle that looked like a circle, you couldn't imagine what one of our senses would tell us. Now you don't even think about triangles, they seem like they would be in a more natural category, but they aren't. And also, if we can't think about triangles as real things, then why not think more about how we think about them? That's the nature of science.

AMY GOODMAN: I want to thank you so much for being with us today, John S


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https://www.youtube.com/watch?v=S0-coASIjkQ

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https://www.youtube.com/watch?v=S0-coASIjkQ

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https://www.youtube.com/watch?v=S0-coASIjkQ

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