The circle is a strange creature, and most definitely not as simple as it seems. In fact, you will see that a circle in the plane doesn’t enclose anything – that ‘inside’ and ‘outside’ are completely arbitrary and in the end, meaningless. Hold on to your seats!
Imagine a sphere upon which you’ve written the word “THIS”. On the opposite side of the sphere you’ve written another word, “THAT”. Around the word “THIS”, you’ve drawn a circle. Now imagine that circle as a kind of magical rubber band. Stretch that circle over the ‘equator’ of the sphere, and let it come to a rest around the work “THAT”.
When the circle was around this, was it ‘enclosing’ it or was it instead enclosing that? You didn’t do anything special to that circle other than gently increase its diameter so that it could slide over the other side of the sphere – but you didn’t turn anything inside-out. Or didn’t you? Play around with this in your mind – the whole ‘arbitrariness’ of sides… Yes, this is beside that, but can you really say which is inside or outside?
“Ok,” you may think, “but this is a sphere. Things won’t be so ambiguous in the 2D plane…” but let’s have a look:
Imagine a circle in an infinite, flat 2D plane – the basic ‘Euclidean’ plane. There are a few problems that come to mind:
If the plane is indeed ‘infinite’, then what is ‘outside’ is infinitely-larger than what’s ‘inside’ – how is the circle’s area not zero or at least not infinitely-small? Don’t get too ‘mathy’ here – I’m not talking ‘metric’ yet. Just that there is infinitely-more space ‘outside’ than ‘inside’.
We generally agree that because the circle is closed, it can be considered a kind of barrier or boundary which cuts the space into two – i.e. that there are now ‘two sides’ to this space; one ‘in’ one ‘out’.
Remember that magical rubber band that doesn’t break? Let’s stretch our circle out so much that the ‘enclosed’ space is now exactly as big as the ‘exclosed’ space. This may seem impossible, because you might think that you can’t ‘quantify’ the outside space – but we can quantify the inside space. We can say that it contains an infinite area (which hints at a continuous expansion of the circle).
How the heck is this still a circle?!? If the definition of a circle is that all points on a circle are equidistant from one single point – the centre – then, especially remembering the ‘arbitrariness of sides’ you experienced above, wouldn’t all those points also be equidistant from some point ‘out’side of the circle? If so, how could this circle circle? It would be a straight line (sort of) and yet it is still closed and so must ‘wrap around’ somewhere, somehow. Would this be where the third dimension arises – i.e. that the only place for such an equally-tugged circle to ‘curl’ is ‘off the plane’?
If, for the sake of argument, we did succeed in creating equal areas – which is enclosed and which is ‘open’?
Given the arbitrariness of sides, how can we persist in thinking that what’s ‘outside’ the circle remains ‘infinite’? Is it not also bounded by the circle?
Reimann Sphere and Plane:
There is this idea of a plane which corresponds to every point on the surface of a sphere. Imagine a sphere resting tangentially-close to a point, S, on a plane. Let’s call this the sphere’s South pole. Every point on the surface of that sphere can have a corresponding point in the plane by ‘projecting’ a line from the sphere’s North pole, N, through the sphere to the point on the surface, and then out onto the plane. But there is immediately a problem with this plane too: it does not have a regular metric. Points closer to the North pole, in the plane, move away from point S faster as you get closer to N. And because of the difference between curved and flat, the distance between any two points on the plane will always be greater than their original pairs on the surface of the sphere.
Artists have been playing with these distortions for a while now:
And it is called ‘anamporphic art’.
But let’s say we were correct and that there was such a thing as the ‘infinite’ flat Euclidean plane where one could go on forever without accidentally coming back to a point we’d already been. In such a plane, I no longer believe it is correct to say that a circle cleaves the space into ‘two’ spaces. Yes, one would be an enclosed space within the circle, but ‘that from which’ the circle was enclosed is not a space because it cannot be enclosed (otherwise it would not be ‘infinite’)
What do you think?