## Most random masterclass question ever

Would your DNA be long enough to weave into a tea towel?

Yesterday I was in Stirling giving my usual knots masterclass when I got asked the most unusual question ever! I was explaining how DNA is like a very very long piece of string sitting inside each of our cells. (It’s actually about 2-3m in each cell!) One pupil then put their hand up and said “If you took all the DNA from all the cells in my body, would it be long enough…to weave into a tea towel?” I was so taken aback by this that I just laughed, but actually the answer is probably yes! Depending on the size of tea towel that he wanted of course. And it would be so thin that it wouldn’t be very good at drying dishes, but that is beside the point.

It is always a pleasure to be surprised by children. And I was especially impressed at all the questions I was asked in Stirling. It helped that the masterclass was much smaller than usual – 15 pupils compared to about 70 in Edinburgh and 40-50 in Glasgow. It meant that I was able to talk to everyone while they were doing exercises, and it was much less intimidating for people to stop me and ask questions during the presentations. But even given those things, I was genuinely taken aback by the insightful questions I was asked and the interest that they showed in what I had to say.

My other favourite moment was when another child pondered “I could take a one-dimensional piece of string and weave it into a 2-dimensional object…”. (I assume this was inspired by the previous question about tea towels.) Now, I had not mentioned the word ‘dimension’ once in my whole masterclass. Most people don’t understand the concept of dimension. Even the undergraduates I teach would have trouble understanding why a knot is inherently one dimensional and not three dimensional. And here is a 12-year old child explaining to me how you could use one dimension to fill out 2-dimensional space!

In case you still don’t understand why this bowled me over, you should go and Google “space filling curve”. (Wikipedia is not a bad reference, but is a little technical.) In the mid-19th century, mathematicians had the idea that a curve could be drawn inside a square so that it went through every single point of the square. This is counter-intuitive, as it seems like the square is ‘bigger’ than the curve, so how could the curve fill it all out? Cantor had showed in 1878 that the infinity which is the number of points in a line segment is the same as the infinity which is the number of points in a square, but it was not until 1890 that Peano came up with this geometrical argument that demonstrated it.

How to construct a space-filling curve iteratively. After infinitely many iterations, it will fill out the whole square.

It is still very counterintuitive to mathematicians that this curve is continuous (i.e. can be drawn without taking your pen off the paper) but is nowhere differentiable (i.e. every point is a ‘corner’, so the curve is always changing direction) and is everywhere self-intersecting (every point on the curve touches another point on the curve). Maths is full of these great examples that challenge our assumptions and intuitions and I hope that I can teach this to my undergraduates later in the semester.

So although the masterclass pupil probably didn’t have infinities and deep thoughts in mind when he made the comment about weaving, it’s exactly questions like this which got mathematicians discovering such things over 100 years ago. I hope that his teachers continue to encourage this wonderful imagination and willingness to ask questions, however silly they may seem at first.