Coding With Processing, Part 3

I just learned about for loops in Processing, and it occurred to me that they are perfect for illustrating iterated function systems. The classic example of an iterated function system is where you start with three vertices of a triangle, labeled 1, 2, and 3. Then perform the following steps:

  1. Plot a point (x0, y0) anywhere inside the triangle.
  2. Use a random number generator to generate 1, 2, or 3 randomly.
  3. If 1 comes up, plot the midpoint between vertex #1 and (x0, y0).
  4. If 2 comes up, plot the midpoint between vertex #2 and (x0, y0).
  5. If 3 comes up, plot the midpoint between vertex #3 and (x0, y0).
  6. Make your new point (x0, y0).
  7. Repeat the process.

Even though the locations of the points are being generated randomly, a very interesting picture emerges. Here is after 100 iterations:

And here is the result after 1000 iterations:

And here it is after 10,000. It’s pretty clear that we’re getting the Sierpinski Triangle!

Here’s the code:

float x1 = width/2;
float y1 = 0;
float x2 = 0;
float y2 = height;
float x3 = width;
float y3 = height;

float x0 = width/2;
float y0 = height/2;

size(600, 600);
background(0);
stroke(255);
frameRate(5);

for (int i = 1; i < 10000; i = i + 1) {
int roll = int (random(0, 4));
if (roll == 1) {
x0 = (x0 + x1)/2;
y0 = (y0 + y1)/2;
point(x0, y0);
} else if (roll == 2) {
x0 = (x0 + x2)/2;
y0 = (y0 + y2)/2;
point(x0, y0);
} else {
x0 = (x0 + x3)/2;
y0 = (y0 + y3)/2;
point(x0, y0);
}
}

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The Wonderful World of Conics

My Algebra Two students are in the thick of learning about conics – parabolas, ellipses, circles, and hyperbolas. I’ve just finished recording a series of screencasts about them, and I am posting them below for your enjoyment.

Why are they called conics? Well, we can construct them using cones, as illustrated by this demo, courtesy of  Irina Boyadzhiev via GeogebraTube (I wish I could embed it, but WordPress won’t allow it):

https://ggbm.at/T8TV2JqG

Here are my screencasts:

Conic Basics

The Parabola

The Ellipse

The Hyperbola

Light, Math, and Color – 2017 Projects

I just finished teaching another Light, Math, and Color minicourse. Twenty students researched a math topic and illustrated it with a stained glass window. Their projects this time around are really spectacular, especially considering these are first tries. (Note: If you hover over a picture, the math topic it illustrates will show up.)