@futurebird

A sequence is a list of values that may go on forever. One special type of sequence is obtained by iterating a function, calling the same function on the output of itself. You can get values by taking one step at a time, or by building composite functions that make multiple steps at a time, which is what function exponents often mean when not doing trigonometry. (In trig, angles and ratios of lengths matter, so composition would mean treating a ratio of lengths as an angle which has no application in geometry, So exponents are powers of values and not composition of functions sin² 𝑥 + cos² 𝑥 = (sin 𝑥)² + (cos 𝑥)² = 1. )

𝑥ₙ₊₁= 𝑓(𝑥ₙ)
𝑥₁ = 𝑓(𝑥₀)
𝑥₂ = 𝑓(𝑓(𝑥₀)) = (𝑓∘𝑓)(𝑥₀)
𝑥₃ = 𝑓(𝑓(𝑓(𝑥₀))) = (𝑓∘𝑓∘𝑓)(𝑥₀) = 𝑓³(𝑥₀)

So what's the long-term behavior of these sequences of values?

Some eventually find a fixed point such that there is some n such that 𝑓(𝑥ₙ) = 𝑥ₙ. And sometime a fixed point like 𝑓(0) = 0 attracts nearby values like for 𝑓(𝑥) = 𝑥/(𝑥+1)².

Some loop. See 𝑓(𝑥)=1/(1−𝑥) where 𝑓³(𝑥) = 𝑥 when 𝑥 ≠ 0 or 1.

See also 𝑓(𝑥)=(1+√8)𝑥(1−𝑥) and 𝑥₀=0.15993 which is a root of 343 𝑥³ − (490+49√2) 𝑥² + (91+112√2) 𝑥 + 31−41√2 = 0.

Some grow forever. See 𝑓(𝑥)=1+2ˣ.

Some do none of these things for some inputs. See 𝑓(𝑥)=3.57𝑥(1−𝑥) and 𝑥₀=1/2.

This gives 0.5, 0.8925, 0.342519, 0.803963, 0.562655, 0.878486, 0.381093, 0.842024, 0.47488, 0.890247, 0.348814, ... which goes on forever never settling down or escaping from the box [0,1].

Consider 𝑔(𝑝,𝑞)=<𝑝ᵤ²−𝑝ᵥ²+𝑞ᵤ, 2𝑝ᵤ𝑝ᵥ+𝑞ᵥ> as a function of two points in the u-v plane and the series of points: \( z_0 = <0,0>, z_{n+1} = g(z_n, q)\) for some choice of q. If the sequence of z's stays close to the origin, forever, q is a point in the Mandelbrot set.

@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.

@futurebird

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."

@futurebird Jonathan Coulton has you covered - 4 min and you're back on track.

https://www.youtube.com/watch?v=ZDU40eUcTj0

@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."

@futurebird

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.