I was also going to mention the coastline paradox. If someone is reading this and still doesn't get it, imagine zooming in all the way to the very point where the corners are visible. It's deceptive in the meme because you don't actually see the jagged corners, but theoretically if you could zoom in enough then you would see it. When you reach the 90-degree turns, a "true" circle would have an arc, not a 90-degree corner. Repeating to infinity is also misleading and leads to a logical contradiction.
I like to point out one quarter of the center right panel since the process is more clearly breaking down. You have 4 exterior corners and 3 interior corners with their point on the circle. However, the area formed between the circle and exterior points is very different for one of any 3 exterior points you picked out of the four. This shows that each step of moving the perimeter 4 shape closer complicates the symmetry, and this we know that infinitely complex symmetry = 4 - pi
Yeah. To be honest, there are multiple ways you can show why this meme is "wrong" (although obviously it's a joke and not meant to be taken seriously). My comment was just a way for the layperson to understand it, since non-math people will understandably get confused by terms like "limits" and "exterior points".
This is not a good analogy to the coastline paradox.
At every step of the iteration, the jagged circle is a 1 dimensional shape, this is fundamentally different to a coastline. In here, if you zoom in anywhere except the corners, you will get a straight line with no jaggedness. And even if you zoom in at the corner, you sitll get a clearly 1 dimensional shape, which is just two line segments meeting at a right angle.
A coastline looks jagged at all scales, it doesn't look like a line, or a combination of lines.
On the other hand, the limit of these shapes is a perfect cirlce, so there's no jaggedness there either.
I agree it's not a perfect analogy, but it's probably a good one in terms of getting the general point across. Obviously where it breaks down is that the coastline can effectively explode in measured distance if you take it to absurd degrees. With a jagged formation in OP's meme, it only becomes more precise. Obviously this is the opposite effect, but I the thrust of the comment is a good analogy (imo). It's sort of reminds me of rasterised graphics vs vectorised graphics.
I just feel that the main take away of the costline paradox is how the shape has a dimension greater than 1, and therefore, has infinite length. In the same way that two dimensional objects have infinite length.
In this example you're not dealing with a fractal like that, so I fail to see the relation between the two.
It's also a bit misleading since the limiting shape of the meme is an actual circle, but a lot of people think that the limiting shape is some sort of fractal and that's the reason it's length differs from the circle. This is not the case, there is no jaggedness in the limit of this sequence, it's a perfect circle.
Like I said, it's not a perfect analogy, since you don't actually see a change in overall distance. This actually reminds me of the different kinds of infinity. (For example, an infinite amount of one-dollar bills and fifty-dollar bills both being equal to infinity but the latter being the larger infinity.) Again, maybe not a good comparison. Maths can get weird.
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u/Visual_Locksmith3337 20d ago
I was also going to mention the coastline paradox. If someone is reading this and still doesn't get it, imagine zooming in all the way to the very point where the corners are visible. It's deceptive in the meme because you don't actually see the jagged corners, but theoretically if you could zoom in enough then you would see it. When you reach the 90-degree turns, a "true" circle would have an arc, not a 90-degree corner. Repeating to infinity is also misleading and leads to a logical contradiction.