Describe infinity in 500 words? It shouldn’t be too hard: infinity, though large, can fit nicely into any sized circle.

Nothing in this physical universe is infinite. Not even were you to count every quark and fleeting particle that had ever existed would you come close. Infinity would still be very, even infinitely, far off. For no matter how small a thing is, if it has any size, weight or energy at all, then this universe cannot contain an infinite number of them. And yet in another sense, a mathematical sense, there are many infinities in our universe, some larger than others and all of them can be encompassed by the human mind.

It is one of the strange aspects of our universe, among all possible universes – perhaps its defining strangeness – that ours has been able to evolve beings who can conceive of the infinite. We cannot count to infinity but we can encompass it. Not only that but manipulate it, do calculations with it, puzzle ourselves with its paradoxical qualities. In fact physicists use infinity rather regularly to make their models of this finite universe add up and work.

Imagine a line with numbers marked on it at regular intervals. To reach infinity the line would have to be infinitely long. A line without end could not fit inside the universe. Yet bend this line into a circle so that it must have an ‘end’ and it will still contain an infinite number of sides or points. The trick, of course, is that those sides must be infinitesimally small.

Now shrink this circle. Each circle you make, smaller than the last, however small, will still have an infinite number of sides to it. Though the infinity of the smaller circle must necessarily be smaller than the infinity of the larger. Even the very smallest circle, one that is only just infinitesimally larger than the single naked point at its centre, will still have an infinite number of sides. The single point at the centre, of course, will have no sides at all.

Of course, you may well object that this is just mathematical trickery detached from anything you can actually see. Or that if circles are all that we have to show for the infinite then let us leave them and it alone. But if you have ever seen a Mandelbrot set or a Julia set then you have seen just two of the many shapes in which you can go exploring the true wildness of the infinity of the circle. Because the circle, it turns out, is just one particular Julia set; the Julia set where the numbers are tuned just so. Change the numbers and the Julia set reveals the still largely unexplored nature of infinity.

There are an infinite number of wildly different Julia sets. And each one corresponds to a single one of the infinite number of points which make up the Mandelbrot set, itself infinitely complex. All in 500 words, or slightly less.