To quote a salient post from an earlier related thread:
The discussion here is strongly related to the not-quite-but-almost infinite threads we’ve had in the past asking if 0.999=1. Sorrowfully, the really major such thread seems to have gotten lost in the transition, as I can’t find it.
The whole question, at least at the level of theoretical mathematics, deals with the concepts of limits, and infinity, convergence and divergence, and sums of infinitely many terms. As almost always in these threads, nobody who understands these concepts really does much of a job explaining it to those others who aren’t really up on their advanced mathematics.
This particular threads seems to have gotten hopelessly bogged down in arguing over the theoretical possibility of measuring an irregular coastline with ever-smaller rulers, versus the practical, in-real-life impossibility of doing so.
The coastline of Norway is something that exists in the real-life universe. The snowflake curve, in its full infinite snowflakedness, exists only in the minds of those who contemplate it.
To explain how a finite area can be bounded by an infinitely long curve, and in particular the snowflake curve, requires some basic concepts from calculus (like limits).
And what are limits? And what can we possibly mean by “the sum of infinitely many terms (like 0.1 + 0.001 + 0.0001 + …)”.
As a matter of fact, yes we ARE all playing with definitions here. For starters, adding up an infinitely long list of terms isn’t just like adding a few numbers like you did in 2nd grade, because you can’t just add up infinitely many numbers. (Go ahead. Try it.) Instead, we must define a new meaning for addition that works well, specifically for adding up an infinite series. Only then can we say, with any meaning, what a sum like 0.1 + 0.001 + 0.0001 + … is, or 0.9 + 0.99 + 0.999 + … or the sum of all the sides of a snowflake curve, or the area of a snowflake curve.
Maybe I’ll try it again (but not in this post right now). I went through all that, in detail, in one of our lost 0.999=1 threads, the one I can’t find now.