Its early August, and the beginning of the next academic year seems an infinite amount of time away, but the two months will go quickly. This summer I am doing what I wanted to do last summer but didn’t have time for, which is to add a couple of optional assignments to my second-year computing teaching.
I enjoy teaching computing, but one challenge is a very wide spread of computing abilities. This means that that the assessed assignments don’t really stretch the students who are really good at computing. So this year I will provide a couple of optional ones, that they can do for fun.
Above is what is called a fractal, I produced this from the second of these optional assignments. It was easy to do, the program only had 39 lines of code.
The course is on doing numerical maths on a computer, and part of this is solving equations on a computer. One of the basic features of solving an equation on a computer is that you always need to start with a guess at the value of x, which if it is reasonably close to the value of x that solves the equation, is improved by the program until you have a very accurate solution.
Usually, unless the equation is nasty, a reasonable first guess at x is easy to find. But sometimes things can be a bit more fun. The fractal above shows what can happen in some cases. It is just a plot in which at each point, if you use it as a guess for x, then the pixel is coloured red if then you go to the correct solution, and coloured white if the guess is no good, it does not lead to a solution. [It is a 2D plane as here x is a what is called a complex number, so has two bits to it.]
The fact that it is a fractal means that for many (infinitely many in fact) points, you can pick a pair of guess for x which are very very close to each other, but one guess will work and the other won’t.
I think there is an educationally useful message here: think about the initial guess you make for x. Also, a very pretty picture, which also helps.