Midpoints and Sierpinski Triangles (programming class)

We started the new year with some exercises that refresh several of the concepts we've been covering:

1. loops
2. arrays
3. coordinates

We haven't done very much with coordinates, so this is probably the part that feels the most "new." By this point, loops should feel reasonably familiar and, in the code today, we aren't doing any complicated or nested loops. The use of arrays is still difficult; these objects are a tricky beast for the kids to fully understand.

Midpoint jumps

Our first challenge is to get the turtle to jump halfway from where it is standing to a target point. For this, we recall the getxy() function and use the xy-coordinates for the screen (the home where the turtle starts is (0, 0) in this system). Here's an example:

Sierpinski Triangle

Using this basic code, we can reorganize things a bit to play the Chaos Game and generate an amazing picture. In each iteration, we randomly select one vertex, then jump halfway to that vertex, leave a dot, then continue:


This version already has a nice little twist with the random colors on each new dot. Nice!

Homework

Finish the challenges from the class (including writing explanations of the lines of code we highlighted) and then modify the last program to create your own picture. Here's an example of something I created:

A slightly different version is here: <a href="https://jgplay.pencilcode.net/edit/Math/Fractal3">more Fractals.</a>

Math is all around (proof without words)

Ok, I can't resist writing a little explanation. These are pictures of activities from the last 2 months that never quite made it into blog posts or, if they did, I just like the pictures and wanted to show them again.

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A fractal tree made from rummy king tiles (part of our Natural Math/Moebius Noodles Multiplication explorers course):

Another fractal tree:

Is it an alien or a self-similar bug? How many body segments would we need to draw for the next smaller level?

Family Tree fractal:

I think this was a substitution fractal, again, Multiplication Explorers:

Halving sequence, another activity from the Multiplication Explorers course:

This little piece was 1/16536th of the starting square (2^-14)

How many animal hybrids can we make?

Many ways to make a whole. How many ...?

Even in the round:

A version of Pascal's triangle made into a puzzle. I think we made all the cells mod 10. Hand cut!

Some pictures we saw at an exhibit in one of the halls at NASA in Houston. We did our own calculations and didn't agree with their claims about how many stamps or how many coins were used.

Finally, a couple of nice magic squares, fraternal twins or identical? I can't take credit for this, but don't remember where I saw it (prob on twitter):