I’m so excited about the new project I’m working on with my son that I’m almost at a loss for words. Yesterday I saw the following note on Twitter from Steven Strogatz:

Around 20 minutes in to the first lecture is a quote, or rather an “evangelical plea”, from Bob May stating something like –

“We should stop teaching only linear math to our college students and our graduate students and show them that once you allow systems to be non-linear all bets were off and you could discover all kinds of things. It was time to stop lying to the students in the classrooms.”

This idea really struck me because it connected with a number of different things that I’ve heard over the last year – Conrad Wolfram’s talk about computers and math comes to mind, for example (see a link here: https://mikesmathpage.wordpress.com/2013/12/01/computer-math-and-the-chaos-game/ ). Anyway, we were about to end the year talking about sequences and series, but spending a few weeks playing around with the logistic map suddenly seemed like a much better idea, so off we went. I was also really excited to tackle this subject because I studied a little bit about the logistic map in high school in Mr. Waterman’s Enrichment Math class. It is always doubly exciting to be able to pass along stuff I learned from Mr. Waterman to my kids.

The first thing I did when I got home from work yesterday was sit down with my son and introduce the concept. Had I spent even two seconds thinking about what to do I probably would have started with an easier recurrence example – the Fibonacci numbers, say – but that idea didn’t occur to me until today.

In the first video we walked through the relation . We compute a few iterations in the case where and then finished off looking at a few points on the graph of I admit that this might not be the most exciting start to the topic, but it does lay the foundation and also allowed me to double check that the math behind the quadratics and the iterations wasn’t too far over his head:

After last night’s basic introduction we spent about 30 minutes this morning diving into the geometry of the logistic map. Seeing this connection between the geometry and the algebra in high school was absolutely amazing to me. Prior to our little five minute film we spent time studying two equations and and that allowed us to study one of the more complicated examples in the video:

Tomorrow we’ll look at some of the really baffling examples, though we got a little preview of that thanks to Alexander Bogomolny who saw some of my enthusiasm on twitter and alerted me to this section of his site:

For homework today my son read this page and played around with the applet – declaring it to be “awesome.”

Such a fun topic, and as I point out at the end of the second video, it really is cool to be able to introduce some relatively modern math to my son. Don’t quite know where this is going to go in the next week, but it looks like we are going to have a really fun time no matter what direction we end up going!

I finally got down to the post on the exponential function. (in passing, how do I follow your blog? can you page the posts and index or list them? and one or two more things!!!)

Anyway, I would start with doubling, ie y = 2^n, or y = 2^x for x a pos integer.

Examples from biology – growth of molds, bacteria, etc

and physics/chemistry – radioactive decay

Obvious choice of time interval will give the doubling function.

You can look at 3^n similarly.

Now look at the difference between 2^x and 2^(x+1)

2^(x+1) – 2^x = 2^x times 2 plus 2^x = 2^x

So the slope of the chord (crude estimate of the rate of change) is proportional (equal in this case) to the value.

Try with 2^(x+1/2) – 2^x

Diversion to argue that 2 to the power one half is the square root of 2

Do it with x + 1/4

Watch the constant of proportionality, as the chord moves towards the tangent this settles down (or at least appears to, which for the moment is good enough).

It is less than 1

Do it for 3^x and get a settled down value which is greater than 1

THEN

You can define e as the number for which the limit of the constant is 1

So, for later, you have “done” d/dx(e^x) = e^x