Discussion
How would you explain the mandelbrot set to a sixth grade student who is trying to distract you from teaching your substitute lesson on solving linear equations?
(I swear to god these kids pass around lists of which topics distract which teachers most easily. They know the ants won't work, it's too obvious. But this was a good one.)
Said student also has some peers convinced that the mandelbrot set will "eat you" and I haven't done much to clear up this rumor.
You could just show the little smart alecs the Julia set and really freak them out?
@futurebird "OK, to truly understand the Mandlebrot set and whether or not it can eat you, first you need to know how to solve linear equations. So...."
@futurebird "Tell you what. If we get through linear algebra with time enough to spare, I'll tell ye about the Mandelbrot set"
@futurebird "Have you ever punched in on a calculator, 'X = X/2 + 1' or something, and run it again and again to see what number it eventually turns into?
With the Mandelbrot set we found a process like that, "X = X squared plus C". And we discovered depending on which C we chose, this process either blows up and goes infinite, or it turns into a number.
So we made a graph. We picked different complex Cs, and at each point we checked if X=X^2+C blows up, or if it becomes a number."
@mcc @futurebird A digression from the original didactic question: are all the attractors of the Mandelbrot set either fixed points or infinity, or are there orbits in there as well?
(I just thought of this given your wording.)
@futurebird "And when we made the graph we found out the shape was REALLY weird. It had a basic bubbly shape but when we zoomed in close around the edges we found all these copies of the basic bubbly shape. And we don't know why. It doesn't seem like there *is* a reason why. Just there's a lot of very strange shapes in math if you can graph a problem the right way."
This is maybe not a short description, and it has the problem they won't get it if you haven't explain complexes yet
@futurebird I also don't know if the calculators you have can do that kind of "repeat operation over and over by mashing a button" thing. The TI-8Xes could do it easy. X=X/2+1 isn't very interesting, that just converges to two. The one I always used to play with is X=1/X+1 . That one converges on the inverse golden ratio, and at that point you can actually see that the golden ratio is ϕ = 1/ϕ - 1. I think that one is "spooky" in the same way that the mandelbrot set is spooky.
@mcc @futurebird
You can just tell them you did it with a number and if it's enough, then fine. If they ask "how come these numbers have an x and a y instead of just being a line?" they're ready to have their mind expanded.
@futurebird Jonathan Coulton has you covered - 4 min and you're back on track.
YouTube
@futurebird wait!!
wait!!!
are mandelbrots like mandible bros? or like mandible bratwurst? is this really an ant thing?
@futurebird
they can use a pocket calculator to plot successive values of iterated points. Its iterations of z^2+c
you can use it to introduce complex numbers. just tell them complex numbers are pairs of point in the plane... theyre represented by (a,b) = a+bi so i is just a symbol that means "on the y axis" and theres a special rule that i*i = -1
otherwise you do math the usual way (a+bi)(c+di) = (a+bi)c + (a+bi)di
you can have them learn to iterate in scheme via @spritely hoot!
@futurebird Just tell them it is forbidden knowledge.
That should fix it so they go learn everything about it, and do it on their own time.
@futurebird
Maybe...
"The Mandlebrot set is a nonlinear equation. A very nonlinear one. It's what you get if you take x, square it, and add a number to it, then make that x and repeat again and again until it either blows up or goes to zero. That's not the sort of thing that can be solved, you need to just run experiments and use lots and lots of computer time.
Linear equations, on the other hand, CAN be solved. It really helps to get a handle on these before you go to the wilder stuff."
Its like.... okay fine rocks fall and everyone dies. Back to linear equations.
@futurebird (My attempt)
If you have a function, you can take the output of it and feed it back in. This is called iteration. If you keep doing this, one of three things will happen: it'll head off to infinity, or it'll get closer and closer to one point, or it'll just bounce around within a range of values forever. We call these attractors. There's a simple formula, f(z)=z^2+c, where the z and c are what's called complex numbers. You can pick a value of c and iterate this formula. 1/2
@futurebird "Oooh, that's a really interesting question...let's write it down over here on the parking lot board, and if we get through today's material in time, we'll dig into it!"
@futurebird It's day-glo pterodactyl, so could conceivably eat you.
@futurebird I was that kid and certain teachers did think of me as a distraction.
And because the teacher blew me off, I blew off education and curriculum.
Best thing that happened to me.
If you don't want their prizes, you don't have to play their games.
What value is a gold star? 🤣
We lost lehrfreheit and lernefreheit somewhere along the way in education institutions that pit teachers against curious students in antagonist ways.
@futurebird While I'm here... I should write up all the little mostly-graphical maths-based programming toys that we came up with at school. Mandelbrots, Julia sets, iterated functional systems, but also (mathematically) simpler things like fractal landscapes and Lissajous figures.
There's a lot of fun you can have with some of this stuff, if you're mathematically minded...
I got a lot of inspiration from @gleick 's book, Chaos.