WordPress has been running extremely slowly for the past half day, so I have no idea when I will actually be able to put out this post. But I will write it up this morning – Saturday – as per my usual schedule, and I will put it out when WordPress permits.

Let me lead with a picture: this is called Heighway’s Dragon:

Let me start, however, with a different subject.

I suppose that some of the characters on the TV show “The Big Bang Theory” might not realize that this weekend is the Super Bowl. I’ll admit that if the Pittsburgh Steelers were not in it, I myself might not realize that this weekend is the Super Bowl.

But they are… and it is… and so I will not be doing mathematics tomorrow afternoon. We’ll see if I manage to get a technical post ready for Monday.

In the meantime, I have been distracted by two pieces of software.

Someone asked about “phase portraits” on the Mathematica USENET newsgroup… and one of the replies was to point out a Mathematica package here.

I’ve been executing the sample notebooks that come with it. So far, I’ve run through (not mastered, run) 20 tutorial notebooks and seven basic notebooks… I have yet to look at the set of 12 advanced notebooks, and three folders of notebooks on population, disease, and epidemics.

I haven’t actually decided if I’m going to learn to use this package, or whether I’m going to bypass it and use Mathematica directly. The package takes care of some administrative overhead for looking at things.

Let me show you three screenshots of output from the package. If you’ve looked at dynamical systems even a little bit, you’ve probably seen all of these. I’m delighted to have created them myself so easily.

Here’s a 3D phase portrait of the Lorenz equations…

Here’s the limit cycle for the Lorenz equations…

Here’s a bifurcation diagram of the logistic map…

And that’s one piece of software.

One of my alter egos, the kid, picked up Gerald Edgar’s “Measure, Topology, and Fractal Geometry”, a Springer UTM (undergraduate texts in mathematics). It’s a fun book. (And well written. I recognized his name from the USENET sci.math newsgroup, and bought the book because I respected his posts. I’m quite pleased with the book.)

I’ve looked through it before. This time, when he said he was using the programming language LOGO to draw interesting curves, I decided to look for LOGO on the Internet. I found one here, and it seems to work fine on my Mac. (LOGO was the language for the well-known “Turtle Geometry” by Abelson & diSessa. Yes, of course I own a copy. I’ve just never read it!)

I’ve learned how to define and save functions… and I’ve made my own drawings of…

the Koch curve…

and Heighway’s Dragon… which is what I opened this post with.

And I just realized how easy it is to have LOGO do 3 Koch curves in sequence to create the snowflake curve:

As always, I love generating these things myself.

Now I’m going to hit Dummit & Foote’s “Abstract Algebra” for an hour or so, and then I’ll start on a post for Monday….

February 6, 2011 at 9:39 am

It appears that the dynamical systems package runs in version 7 – which is my default – but not in version 8. Well, it says it runs in version 7, so that’s fine with me.

February 12, 2011 at 10:38 am

Here’s Edgar’s code for the Koch curve:

if :depth = 0 [forward :size stop]

[

koch :depth – 1 :size / 3

left 60

koch :depth – 1 :size / 3

right 120

koch :depth – 1 :size / 3

left 60

koch :depth – 1 :size / 3

]

Here’s my code for the Koch snowflake:

hideturtle

cs koch 4 150

right 120 koch 4 150

right 120 koch 4 150

And here’s Edgar’s code for the Heighway dragon:

make “factor 1/sqrt 2

if :depth = 0 [forward :size stop]

[

left :parity * 45

heighway :depth – 1 :size * :factor 1

right :parity * 90

heighway :depth – 1 :size * :factor (-1)

left :parity * 45

]

and the drawing was created with the command

cs heighway 7 150 1