Sierpinski Triangles

As I write this, the governor of Alabama has just canceled school for the remainder of the school year. 

Looks like I'm going to be spending a lot more time acting as my kids' teacher! At least I'm in the same boat as many of you.

I can already see myself in two weeks, half-mad from being stuck in my house, retreating to the bedroom for a one-person parent teacher conference.

Me: Thank you for taking the time, and I brought some boxes of Kleenex for the class. And I know you're busy, but I just feel like Joel isn't being challenged enough.

Me: I appreciate that, and I want you to know that I am trying to meet the needs of all my students. As you know, I currently am running a multi-age classroom with your son, a 5-year-old, and a toddler. 

Me: Yes, I wanted to speak to you about that as well. Our toddler has been throwing her food on the floor. Can't you do something about that at school? You're her teacher.

Me: ...


SCENE

Ok, so maybe I am going a bit stir-crazy already! But I know one thing that reliably calms me down: math art.

How to Draw a Sierpinski Triangle

I am a terrible freehand artist, but I love doing mathematical art. The whole process is about simple rules, precise measurements, and repeating patterns. It's a form of meditation for me.

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As a gateway, I wanted to share a drawing that just about any kid can do, to some level of sophistication. I'll give you the general outline and you can adapt it to your kids' abilities.

You'll want a ruler or straightedge, ideally. Start by drawing a big triangle on a piece of paper. The triangle can be any sort, but lots of people start with a classic equilateral triangle.

Then, find the midpoint of each side and mark it on your paper. Again, your kids can eyeball this, or they can use all sorts of informal or formal measuring techniques to find the midpoint. Make that part of the challenge of the project!

Once they've found the midpoints, they connect those to form a smaller triangle inside your original triangle. Now you should see four triangles: three that are facing the same way as your original, and one that appears upside-down. Let's call this "Stage 1"

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Now it's time to decorate! Color in that upside-down triangle however you want. Then turn your attention to the three remaining triangles. You know, they really look like smaller versions of the original triangle...

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So let's do the same process to each of them. Find the midpoints of their sides and connect them to form smaller triangles. Decorate the upside-down ones to finish Stage 2. Then take the rightside-up triangles and repeat the process! You've made a Sierpinski Triangle.

How long will this take? Well, if you want, it can literally take forever. The Sierpinski triangle is what's known as a fractal: an object that is infinitely similar to itself. Each small section of the Sierpinski triangle looks like a miniature version of the whole thing.

I find that it's fun to decorate each stage differently. This really helps bring out the contrast in the larger and smaller triangles.

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There is, of course, a ton to explore in the Sierpinski triangle. I'll just leave you with a few optional questions to explore while you enjoy your weekend:

  • What changes when you change the shape of the original triangle?

  • How many uncolored triangles are there after Stage 1? After Stage 2? Stage 3? Stage 10?

  • What fraction is colored in after Stage 1? After Stage 2? Stage 3? Stage 10?

Kent HainesComment