h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Squirrel Dilemna - A Puzzle - A16967659
Author: AlexAshman [!] - U566116

Post 101

Alex, I've had a long think about this entry.

I think you are describing some sort of partial exercise in the use of mathematical modelling techniques, but there is a fundamental problem with the way you have presented it, which has confused some of us, including myself for a while.

Mathematical modelling is a rigorous technique used in problem solving, but the first step is to state the problem in real terms, and I don't think you do this. You subsequently dip in and out of a number of modelling systems, but if you don't have a real problem in the first place, they won't generate a real solution for you at the end. Other reviewers said something similar earlier in this PR thread.

Now it's intuitively obvious to most people what's going on here, the man and the squirrel are both moving around in circles, of different sizes, and in opposite phase, and we can soon see that he will never catch it. The man has a problem with the squirrel, but that isn't the problem you describe - your problem is understanding what is going on in modelling terms, but you have to be precise as to what you are trying to model and why.

We see examples of motion systems in everyday life - a car slowing down at traffic lights, a reflector on a bicycle wheel, etc, and to a degree we can explain these though modelling them and understanding the mathematics which supports the model. Why do we do this? It helps us understand the mechanics of the world around us, and predict the behaviour of these dynamic systems. We can then answer questions like "When will the car come to a halt?" and "How high will the reflector be after a period of time?".

At one point you ask the question "Does the man encircle the squirrel?" I'm not sure why you asked this in practical terms, but I think it may be because you assume "if he encircles the squirrel, then he must at some point catch it". Just saying the man encircles the squirrel's locus is correct, but apparently insufficient. We need to describe the motion more precisely, but in simple terms, and in a way which supports the problem we are trying to solve.

You go on to present a couple of ways of modelling the motion: first from the tree's viewpoint (and you give us some maths which shows us that the participants will remain the same distance from each other). You then use the rigid bar model and explain something about angular velocity. At the end, we have discovered some interesting ideas, but they appear to be by-products rather than the solution you set out to find. If you had originally asked the question "Does the man get any closer to the squirrel as they move around the tree?", then you would have solved that one.

In the end, the man doesn't go home because he's getting no closer to the squirrel. He goes home because the squirrel is never in his line of sight, which is a different problem altogether.

Icy

Any better now?

I've had a quick look at the entry now.

I'm not at all happy with it. You seem to be presenting my statements without discussion, simply as an amusing example of how strongly some people are convinced they are right, but not presenting any of the opposite camp's arguments.

Instead you go into detail on a mathematical description of the relative movements of the man and squirrel. I don't think anybody disputes the relative movements - we all know that the man and squirrel are stationary relative to each other, if you adopt a rotating frame of reference for the squirrel; but understanding that doesn't throw any light on the problem whatsoever, so I really don't think you have succeeded in your entry Alex.

Does the question really remain unanswered? As far as I can see, only Fizzy and Rudest Elf think that the man does not circle the squirrel. Their only argument seems to be that the squirrel never has his back towards the man, something that the rest of us think has no relevance.

It's better, Alex.

I think the concluding paragraph needs looking at.

<<...the two characters are not moving relative to one another. This would suggest that the photographer is not encircling the squirrel...>>

<>

I don't think you've sufficiently covered these ideas in the body of the article in order to make these conclusions.

You initially conclude that because the participants remain in the same relative positions in a rotating environment then one doesn't encircle the other. I don't think this is intuitively obvious. If the tree was infinitely small for example, then the man would very obviously encircle the squirrel, as the squirrel would be motionless.

The train of thought which indicates the opposite conclusion isn't explained, either.

Finally, I quite liked the silly bit you had there before about why the photographer gives up and goes home - just don't link it to a formal point of mathematical logic.

Icy

Put simply, one side takes the view that just one circuit of the tree by the photographer is sufficient to prove encirclement, whereas I (we?) argue that the squirrel cannot be rounded on any single lap.

I hate the squirrel and everything it represents



PS. I still think its impossible to circle it given the restrictions imposed.

Rudest Elf and Fizzymouse's postings convince me that there is a guide entry in this. I think Alex has to find a way to articulate and then model their perceptions.

I also believe there is an issue that deserves to be explored in an entry. But I still think it is a purely linguistic one, and amusing though it is to think about squirrels strapped to rotating rods, it doesn't solve the problem.

I think it goes beyond a linguistic argument, Gnomon. I think it could be something in the field of spatial awareness.

Possibly. I'd to explore the possibilities.

In which of the following situations does the man circle the rabbit?

a) Squirrel sits motionless on a spot, man walks in a circle centred on the spot.

b) Squirrel stands on a spot. Man walks in a circle centred on the spot. Squirrel turns on the spot so that he always faces the man.

c) Squirrel sits motionless on a spot. Man walks in a circle centred on a different spot but with a big enough radius that his circle includes the squirrel's spot.

d) As c except that squirrel turns on the spot and always faces the man.

It's a trick question, as he doesn't circle a rabbit at all, does he?

Ignoring the mammalian slip for a moment, I think I'm going for a,b as definites, and c & d depending on whether or not we are defining "to circle" to mean "to describe a circular orbit around".

The rabbit/squirrel slip was a genuine mistake.

In all four scenarios he circles the mammal/squirrel - at least according to my knowledge of mathematics, English and physics. The only argument against that so far is that 'I don't think he does if he is facing it', which doesn't make sense. If you applied the same motion to an object without a face, or a spinning object, you'd have to conclude that it was circling it.

Isn't the squirrel also circling the man? I mean, following a more-or-less circular (with little loops maybe) trajectory around him?

I don't quite understand how the direction the squirrel/rabbit/other cuddly woodland or grassland creature is facing can be used to definitively prove anything, but the crux of this idea is that the motion of the squirrel negates the motion of the man, with the fact that they face each other all the time being a by-product of this.

I've rewritten the last paragraph - is it ok, and can I have comments on how to change the other bits instead of sweeping statements on people's visions of what it should be? Pretty please?

Hmmm. Rather than justify the opposing views you've completely avoided mentioning them.

I guess you read what Gnomon and I were discussing yesterday - what's your view on it? Is there a perception/spatial awareness thing which prevents some people from seeing the man encircle the squirrel?

Hmmm... that's like asking 'does that person over there see colours differently?' Besides, I don't think it's spatial perception in just one direction - I think spatial perception may also explain why some can't see that the motion of the squirrel s the whole thing up.

I'd like to see the topic discussed so that we all understand it before you write the entry about it. I think we're all stuck in not being able to see the other person's point, so it makes it difficult for you to write the entry and for us to comment on it.

What are your own views on the cases given in posting 111, Alex?

In which of the following situations does the man circle the (rabbit)?

a) Squirrel's circle has radius of zero, so squirrel = tree.

b) Squirrel's circle has radius of zero, so squirrel = tree.

c) Squirrel's circle has radius of zero, so squirrel = tree.

d) Squirrel's circle has radius of zero, so squirrel = tree.

So none of these situations properly model the situation we're looking at - it is very easy to make the man circle the squirrel if we turn the squirrel into a tree.