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 Originally Posted by OngBonga
The perimeter is a simple curve. Ok you construct the snowflake with traingles, ie straight lines, but one can never construct a perfect Koch snowflake because we can never replicate the fractal detail. Same with circles... we're essentially constructing a circle with a very large number of very small straight lines. One can never construct a perfect circle... find me a circle, and a good enough magnifying glass, and I'll prove it's not a circle. Eventually, at a small enough scale, the curvature will be broken.
We know a circle's circumference is directly proportional to its radius... so if the radius is finite, then so is the circumference. What makes the snowflake different? I'm going to confidently propose that the perimeter of a Koch snowflake is directly proportional to its radius (probably at a ratio related to pi), which therefore makes it very much finite.
You may be technically right about circles, but not about the line-segments.
Fields are continuously valued and exist at all points in spacetime.
At any rate, approximating things as circles is quite powerful and while it may not be a perfect description, the predictions are good, so we roll with it.
The Koch is different because it has no curvature at any location on its "curve." It lacks smoothness on any scale.
This explains it well:
https://www.youtube.com/watch?v=D2xYjiL8yyE
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