Mathematical Cabbage

Math - It's what's for dinner
Math – It’s what’s for dinner

It’s not every day you taste a new vegetable. Especially not one that defies the laws of space and mathematics. But yesterday I found just that at our local organic farm, Green Earth.

They have some exotic stuff there from time to time, even new potatoes that taste like – and I know this is hard to believe – new potatoes, but I was taken aback to see that they had fractal florets, chaotic kale, or to give it a name people actually call it, romanesco broccoli.

Fractals are a phenomenon of nature of course, and you come across them in things from fern fronds to snail shells. But you rarely see them so clearly in three dimensions. Or I should say, more than three. Imagine a wiggly line drawn on paper. It’s an idealised line, so it has only one dimension – length without width. Now we zoom in. Normally when you do that, the section of line you focus on will look straighter than the whole wiggle because you’ll see fewer twists and turns, or even none. But our line is strange. We find that when magnified it still looks every bit as wiggly as it did on the larger scale. It has wiggles within wiggles, smaller-scale twists and turns in between the big ones. This is called self-similarity, and it too is a natural phenomenon. A coastline is still wiggly whether you see it from space or look at where the water meets the sand through a magnifying glass.

If the line is more wiggly than it looked from a distance, that means that it’s also longer than it looked. So if you could somehow keep looking closer and closer forever, you’d find it was always longer. Isn’t that a bit weird? It’s just a line on a finite, two-dimensional sheet of paper, yet somehow it’s infinitely long. That leads to the idea that shapes like this wiggly line, similar on all scales, must somehow be more than one-dimensional – though still less than two. It’s one-and-a-bit-dimensional. Fractionally dimensional. Fractal.

Just as there are wiggly lines that are a bit more than one-dimensional, there are flat patterns that exist in more than two. And there are solid objects – like the romanesco in my hands – that occupy more than three. Of course it doesn’t really extend into some invisible extra space.┬á The fraction of a dimension is just a clever way of quantifying the self-similarity. Yet looking at it, I feel like I’m wearing 4D classes┬╣. The symmetrical complexity is fascinating and beautiful. Its spires are made up of spirals made up of spires, spiralling into spire upon spiral spire. Whorls within whorls. Amen.

And gently steamed for about fifteen minutes, mathematics is delicious. Especially organic mathematics.

  1. One red lens, one blue lens, one green lens.

An easy intro to fractals.

Leaving Us Confused

Histogram of sepal widths for Iris versicolor ...
This distribution of people who nod as if they know what graphs like this actually mean

Today is the day that bad decisions are made.

For it is the day that the results of the School Leaving Certificate Examination (the “Leaving”) come out. Today the students make bad decisions by getting drunk while still mostly underage. And politicians, by making promises.

Ten percent of students failed maths at ordinary level. In a knowledge-based economy that is simply not good enough, etc. Something must be done. Teachers must be fired, students must be fired, schools should be closed, opened or set fire to. Lessons must be made harder, exams easier, students must work more and take more time to rest. Draw your own headless chickens.

But… Isn’t the whole point of exams that some people fail them?

I don’t really think that ninety percent is so terrible a pass rate for an exam that, you know, is actually testing something. And not merely basic numeracy; the Leaving Cert ordinary-level paper is essentially a qualification to enter university, as almost all courses require it. So are we really in trouble if only ninety per cent of the population qualify for third level education? Less than sixty percent actually avail of it.

Could it be that the reason the public panic over standards in mathematics is that they don’t understand some basic mathematics? Because if they don’t… Wait.