# The Squirrel Proposition Refuted

A photographer arrives in a forest, looking for a squirrel to pose for a photograph for a nature magazine. However, he soon realises that not only do squirrels not pose, but also they actively try to stay out of sight, making it impossible to get a good shot. Seeing a squirrel on the side of a thick-trunked redwood, he tries to sneak round to take a snap, but the squirrel simply darts round to cling to the opposite side of the trunk to that which the photographer is facing. Being the typical furry sadist, the squirrel stays at the same level on the tree trunk but moves at the same time as the photographer, keeping exactly opposite him as he circles the tree trying to catch up. Naturally, the photographer never catches up with the squirrel no matter what he does, but the motion of the two characters is much more interesting. The actual puzzle has nothing to do with the photographer's wish to take a picture of the squirrel, but is actually this - does the photographer circle the squirrel as they move round the tree?

### The Problem

So, if the squirrel will always stay level with and opposite the photographer, no matter what happens, does the photographer move in a circle around the squirrel? The situation can also be modelled quite nicely by spinning your hands around each other, or by looking at the motion of a pair of figure skaters (A short online video of figure skaters is available after a quick download.). The reason we ask this question is not because we are interested in whether the poor photographer ever has a chance of catching the squirrel, but because we are interested in the strange form of movement involved. However, the motion is in fact so strange that those looking at it usually fall into two camps with differing opinions:

• Camp One - the photographer encircles the squirrel by encircling an area which at all times includes the squirrel - the motion can easily be resolved if time is ignored. In a given amount of time, the photographer will have described a circle, which can be plotted in two dimensions, as will have the squirrel. The squirrel's circle will lie within the photographer's, and therefore the photographer will have encircled the squirrel.

• Camp Two - the photographer cannot encircle the squirrel due to its motion - in order to encircle something heading clockwise, you must pass to the left of it, then behind it, and then to the right of it1. If the photographer tries to move so that he passes to the left of the squirrel, the squirrel will move to negate the action and thus prevents the photographer from ever passing to the left of the squirrel. This statement is in a way similar to Zeno's Paradox, and so we will refer to this argument as the Squirrel Proposition.

### Semantics - To Circle Or Not To Circle

Before looking at the details of the puzzle, it is important to remember that languages can be funny things, and the use of semantics can allow multiple viewpoints to be correct, provided that the correct wording is used. For instance, if we were to ask 'does the photographer move around the squirrel?', then we would have to ask ourselves carefully what we meant by 'around'.

The first step is usually to visualise the problem, which leads to the question but what exactly do you mean by 'circle'? For starters there are the two obvious answers - circling meaning to orbit the squirrel as it goes around the tree, and circling meaning to move in some other sort of circle around the squirrel. The former is obviously not possible, but the latter would seem possible provided that the describing of a circle by the photographer around the area in which the squirrel is moving is sufficient for the photographer to 'circle' the squirrel.

If the squirrel were to simply sit still on the tree, we could happily state that the photographer was circling him, but the fact that the squirrel stays 180° around the tree from the photographer is enough to ruin the possibility of simply rejecting the Squirrel Proposition without some debate.

### Going Nowhere Fast

However, there would seem to be one piece of information that still supports the Squirrel Proposition. Imagine viewing the situation from the squirrel's point of view as it races around the tree, and then imagine that the tree is see through. From this point of view we can see the tree rotating as the squirrel remains still, and despite all the action taking place, the photographer isn't actually moving any closer to or further away from the squirrel - his motion relative to the squirrel is zero. Even if he takes a step towards or away from the squirrel in order to follow a circle with a different radius, he's still just going to be standing the other side of the tree if we look at his motion relative to the squirrel.

However, if the squirrel were to be rotating on the spot, thus describing a circle with zero radius, the photographer could easily have no motion if we look from the squirrel's point of view, and so it would seem that a lack of relative motion is not sufficient to prevent circling.

### Relative versus Absolute

We can easily look at the absolute motion of the squirrel, as would be seen standing on the ground next to the tree, and see that the photographer encircles the squirrel's area of operations. However, in the previous section we looked at the relative motion, as seen from the point of view of the squirrel as it circles the tree, which led to the confusing statement that the photographer can circle the squirrel without actually moving relative to the squirrel.

The problem is that in order to look from the squirrel's point of view, we must use a rotating reference point to look at what is taking place. Imagine trying to plot the motion of a squirrel on a piece of paper, which spins around a pin in its centre at the same rate as the squirrel circles the tree. You know that you're moving the pencil around in a circle trying to draw the squirrel's path, but the paper keeps up with your pencil so that you end up with a single dot. Attempts to draw the path of the photographer would lead to a similar result, and it becomes apparent that although both characters are circling the tree, on this special piece of paper they simply appear to be standing on either side of the tree without moving. It is still possible that the photographer is circling the squirrel, but that the point of view we are using is cancelling out this circular motion as well as that of the characters circling the tree. Likewise, it is possible to end up looking at circular motion that isn't really there, as would be seen if we kept the pencil still whilst the paper rotated. Keeping the pencil still and watching the paper rotate produces a circle around the pin - does this mean that the pencil has circled the pin without moving? Does this mean that circling can happen irrespective of the relative and absolute motion of the participants? In the end we must accept that the photographer has circled the squirrel, as the way we think of circular motion demands that we must.

'We lay there without moving. But under us all moved' - Beckett, Krapp's Last Tape

### And Finally

Another squirrel who thinks a lot about his movements is Tufty, who has been used to teach UK children road safety since the 1950s. He starred in his own public information film of the 1970s, as you can see on the BBC News website.

1In other words, the photographer would have to pass around the squirrel from his point of view, the first step being to move to the right hand side of the squirrel's field of view, thus placing him to the left of the squirrel.