Mathematical serendipity and the Apollonian Gasket.

Maths is vast. Sometimes, looking at how much has been done, I wonder at the mathematical Universe that has been created by the collective endeavour of all Man’s generations of mathematical thinkers. Now, this vastness has consequences which Ainan discovered recently.

Some time ago, Ainan drew what seemed to him a very interesting shape of tangential circles within a circle, echoing fractally on. He drew it because the form of it appealed to him, aesthetically. He liked its mathematical beauty.

Some time later, Ainan stumbled on something on Wikipedia: the Apollonian Gasket. It was the figure Ainan had drawn...the fractal shape of tangential circles. Ainan had rediscovered something invented, initially, by Apollonius of Perga, the Greek Mathematician. How odd...and how revealing of the vastness of mathematical creation, that a random creation of a child, playing with mathematical ideas, should, itself, have already been the random creation of an adult, long ago, in Greece.

Ainan is inventing much mathematics these days. He invents mathematical problems, schemas, conjectures, propositions and theories, for himself to play with. No doubt, on occasion, they will be something that has been done before – like the Apollonian Gasket – yet, at other times, I am sure, they are something that has never been done before. The problem with that is that I am not a mathematician and so I am not in a position to advise Ainan as to the merit, utility or value of anything he does. So, he just creates in mathematical isolation, playing with maths, as other children might play with the sand, at the beach: happily, joyfully, for the very purpose of immersing oneself in the environment – in his case, an abstract one – in the other’s, the beach.

Ainan is presently alone in his mathematical pursuits. There is no-one to share them with – though I listen to him and watch, with a sprinkling of wonder, at what he does. Yet, I cannot be the perfect audience to his mathematical work, because my primary domains are elsewhere, other than this.

Ainan spins on, growing, becoming, morphing, each passing year, into a being, ever more rarified, ever more unexpected. My role is to do what I can to foster that growth – even if it be in areas that are not my primary strengths. I enjoy it, but I wish, for his sake that I could be all the differing intellects he requires to be properly supported. I can be most of them – and am – but I cannot be all of them. Yet, what I can be, is appreciative of his intellectual growth, in whichever direction it meanders, no matter how far it wanders...I can appreciate it...and I do.

I hope that is enough.

Posted by Valentine Cawley

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