The Koch Snowflake – a fascinating fractal shape

This post is prompted by a discussion on about the Koch Snowflake. It was part of a long discussion about infinity. Here is the link. (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’s the proof in Google docs form:
And here in pdf form.

Here are the diagrams to which the proof refers.

Andrew Kirk

Bondi Junction, 2012

Hilbert’s Hotel

Theologian William Lane Craig, in his resuscitation of the Kalam Cosmological Argument, relies on a proposition that ‘an actual infinite cannot exist’. His primary argument in favour of this proposition is a reference to Hilbert’s Hotel and the apparent paradoxes it creates.
Hilbert’s hotel is a thought experiment devised by the early 20th century German mathematician David Hilbert. It is a hotel with an infinite number of rooms, all of which are full. A new guest arrives wanting to stay and is told all the rooms are full. But the hotel manager makes room for the new guest by moving the guest in room N to room N+1, for N=1 to infinity, and putting the new guest in room 1. Craig adduces the apparent absurdity of this as a compelling reason why an actual infinite, by which we presume he means an infinite set of objects each comprised of a finite, nonzero amount of matter, cannot exist.
But why is the outcome absurd? It is unusual, but there are many things in the universe that are strange and unusual. Is the sticking point that an empty room appears to have been created where none previously existed? If so then that problem is resolvable, by considering the process of transition to the state in which all the guests, including the newcomer, are accommodated. To do so we need to consider the process by which the hotel manager effects the transition, bearing in mind that we are dealing with a hypothetical physical situation here, so the laws of physics must be obeyed.
One possible such process is as follows: the manager writes an order, addressed to all guests, stating that, on receipt of the written order, each guest should:

  1. pack up and move out of their room
  2. go to the room numbered one greater than theirs and deliver the letter to the guest in that room
  3. when the guest in that next room has moved out, move into that room.

In this scenario, there will always be one guest without a room, as they wait for the next guest to vacate. This situation persists forever because the manager’s message only proceeds at a finite – indeed a very slow – pace. So it would appear that no extra room has been created: we had one person without a room as soon as the new guest arrived, and we still do, in perpetuity. It’s just that the person without a room changes every few minutes.
OK then, but what if the manager asks all guests to move at the same time – does that remove the need for all this infinite waiting outside rooms? Maybe. Let’s see.
How can the manager ask all the guests to move at the same time, bearing in mind that this is a physical hotel in a physical world, and hence must obey the laws of physics? Well, he could transmit his order by email so that it flashes up on a large electronic screen prominently situated in each hotel room. Let us assume the message is transmitted at the speed of light – physics dictates that it cannot travel any faster – from the manager’s office adjacent to room 1. The message says ‘To all guests: on receipt of this message please immediately pack up and move to the room with number one greater than yours. The occupant of that room has been instructed to move to another room. Thank you for your cooperation. By order of the manager.’ Seeing the message, all the guests start packing up and moving. On arriving at the next room, they have to wait while it is vacated. The waiting time, including the time taken for the guest to travel form the door of her room to the door of the next room, will be a minimum of d/c where d is the distance between rooms and c is the speed of light. It will be longer if the guest in the next room is out, or if they weren’t paying attention to the screen and so did not see the message at the earliest possible moment. If the guest’s walking speed is v (which must be less than c) then the guest will spend time of at least d/v outside a room, longer if they have to wait to be admitted to the next room.
As in the previous scenario, the room shuffle will never be completed, as there will always be an infinite number of rooms at which the manager’s message – travelling at the fast but finite speed c – has not yet arrived. At any point in time there will be a very large number of people outside their rooms as they walk from one room to the next. As the typical time taken to move from one room to the next is d/v and the delay from one room receiving the message to the next room’s receiving it is d/c, there will typically be (d/v)/(d/c) = c/v people in the corridor at any time after the first few minutes. Based on a typical walking pace of 1m/s this means there will always be about 300,000,000 people in the corridor. Clearly this situation is much worse than with the hand-written message.
In fact, however you try to arrange the transition, it will always involve at least one person being without a room for an infinite amount of time.
Although it is possible for all the existing guests plus the newcomer to be accommodated in the hotel without having to create new rooms, the transition from the existing state to a new state where all are accommodated involves an infinite amount of homelessness. This seems unsurprising, given the hotel was full to start with. So, when we take account of the process of transition, there are no miraculous or absurd consequences that cause us to conclude that an actual infinity cannot exist.
Hilbert’s thought experiment can be extended to allow an infinite number of new guests to arrive, or even an infinite number of coaches each carrying an infinite number of new guests. As with the single new guest, these new guests can be accommodated in the existing set of rooms together with the existing guests. But, as in the above scenario, the transition will still involve an infinite amount of homelessness.
A further twist on the hotel ‘paradox’ is the situation where a guest leaves. In this case all the guests move down a room, thereby moving towards a state in which all the rooms are occupied again, despite the departure. This version is easily understood by inverting the logic above and thinking of a room without a guest in the same way as we previously thought about a guest without a room. Hence the transition will involve one or more rooms being empty for an infinite amount of time.
Craig’s assertion that an ‘actual infinity’ is impossible relies on being able to deduce an absurdity from the existence of such an actual infinity. That absurdity must be something other than our scepticism that a hotel with an infinite number of rooms could exist, otherwise the argument becomes circular: ‘an actual infinity cannot exist because I consider the existence of an actual infinity to be absurd.’ If, when the new guest arrived, the manager were able to wave a wand and make all the guests instantaneously and simultaneously teleport, with all their luggage, into the next room, then we would have an apparent absurdity via the immediate accommodation of the newcomer despite all the rooms being full (that is the absurdity adduced by Craig as proof that an actual infinity cannot exist). Such an action can be contemplated in mathematics, but Craig has no objection to mathematical infinities (‘potential infinities’). It is actual, physical infinities that he claims are impossible. But by considering the problem of transition, which only exists for physical infinities, we see that it is precisely the constraints imposed by physics that prevent the absurdity from arising.
One might object that this is a nitpicking point about transition and that, after the transition is complete – i.e. after an infinite period of time has passed – everyone including the newcomer will be accommodated and we will have a physical absurdity. This objection is easily dismissed on the grounds that:

  1. we will never get to the point when an infinite amount of time has passed
  2. after a finite period of time all the guests will have died (or disintegrated by slow radioactive decay even if the guests are androids and hence not vulnerable to death); and
  3. after an infinite period of time, nothing should surprise us – think monkeys typing Shakespeare

The conclusion is that, whatever arguments might be made against the possibility of an actual infinite, Hilbert’s Hotel cannot be one of them.