This post is prompted by a discussion on philosophyforums.com about the Koch Snowflake. It was part of a long discussion about infinity. Here is the link.
http://forums.philosophyforums.com/threads/cantor-and-infinites-of-different-sizes-53862-5.html (scroll down to post #174 by Banno).
An animated version of the snowflake is included.
In between a bunch of other things about aspects of infinity, a discussion ensued about this snowflake, including questions of whether it even existed – “it” here being the limit of the boundaries of the iterations of the snowflake.
It was agreed after a while that the limit existed but then alleged that maybe it was just a ‘smattering of points’, an allegation with which I agreed, and even offered (what I thought was) a proof. By a ‘smattering of points’ we mean that the points were not connected to one another. The important concept of connectedness here (there are many different concepts of connectedness) was that of path connectedness: whether there is a continuous path that traces its way through all the points and arrives back where it started, without any breaks or gaps.
What made the path connectedness seem unlikely, to me, was that the length of the boundaries of the finite iterates of the snowflake increases without limit. This suggests that the length of a boundary of the limit would be infinite and hence, arguably untraversable.
One mathematical poster pointed out a flaw in my reasoning and came up with a very nice example that gave intuitive hope that maybe the limit of the boundaries (to be precise, the closure of the limit of the boundaries) may be path connected. The mathematical poster suggested an approach using topological methods to prove the convergence and continuity of a potential function defining a path around the boundary. The obstacle was proving surjectivity of the function though – does it actually trace out the entire boundary of the snowflake?
After wrestling with this intriguing problem on and off for a couple of weeks, I have finally got a proof that I think is valid, that the boundary K* of the snowflake exists and is path-connected. Further it is equal to the closure of the limit K of the boundaries of the finite iterates of the snowflake (which I call A0, A1, A2, etc – the boundaries being K1, K2, K3, …).
The links to the proof and the accompanying diagrams are at the end of this post.
It’s a bit long (there are seventeen lemmas). I daresay it could be made a lot shorter by wheeling in some heavy machinery from topology, but it’s so long since I did any of that that I wouldn’t trust myself to operate the machinery safely.
A couple of interesting points I’ve noticed from doing this are:
- Since the set K* is path connected and contains an uncountable number of points that are not in K, the limit of the boundaries, it seems likely that K is not path connected, and probably not even connected. That seems to make sense because I would expect that taking out the points in K*-K would “leave gaps”. But I have not attempted to prove that.
- Although the ‘closure points’ in K*-K seem to plug gaps in K to make it path connected, it is very hard to visualise how that happens. It is certainly nothing like imagining the closure of an open disc by adding the circular boundary.
- I find it remarkable that there is a curve that can traverse the infinitely long boundary of A in a finite time – implying an infinite speed amongst other things. This casts an interesting light on the arguments some people try to make about infinities, such as the argument that the past cannot be infinite because an infinite past cannot be ‘traversed’.
I would be grateful for any comments anybody may be able to offer about the proof, including any errors, suggested simplifications, or parts that are not clear. Given however, that it is eighteen pages, I won’t be offended if nobody reads it. I’m happy just to have finished it and to have resolved this question for myself (I think, although the proof may turn out to be erroneous).
Here are the diagrams to which the proof refers.
Bondi Junction, 2012