Euclidean and Non-Euclidean Geometries

Euclidean Geometry seems like a heavy topic for ninth graders, but it just means Geometry on flat surfaces. There are only 4 students in my IM1 right now, so I had each of them illustrate the basic axioms:

1. There exist at least 3 distinct points
2. Not all points lie on the same line
3. For each 2 distinct points, there exists a unique line on both of them
4. For every line there exists at least 2 distinct points on it

The illustration for axiom 2 is slightly off, although I completely see where they’re coming from. They drew each point having a unique line, not one unique line though both of the points. I think that one is on Euclid/ whoever rephrased it in my textbook, although I do wish I had noticed this before it went up.

We talked about why Euclid might have thought it was convenient to create a Geometry for just flat surfaces, and the fact that the world isn’t actually made out of just flat surfaces. That’s where the non-Euclidean part comes in. I’ve been pretty excited about the idea of hyperbolic planes ever since one of my colleagues showed me this TED talk about how coral is a hyperbolic plane, and how the best way to represent those is through crochet. You should watch it!

I crocheted several of these hyperbolic planes over the summer (cool sentence, right?), so we did this investigation about parallel lines on 3 types of surfaces: planes (blue), spheres (green), and hyperbolic planes (pink):

I had the students figure out the parallel postulate in each of these geometries: “on a given point not on a given line, exactly how many lines can be drawn parallel to the line?”

[SPOILERS] It was quickly clear that on a flat surface, there is only one parallel line through each point. We’ll be making the connection from this fact to the coordinate plane next week. It was almost immediately obvious that making parallel lines on a sphere is impossible. Creating any type of line on a hyperbolic plane was unfortunately where this investigation broke down. Most of them threaded their string through the middle of the hyperbolic plane, even though they had a vague sense that it didn’t count.

On Monday, before we bring it back down to just the Euclidean stuff, I’ll show them the TED talk because she has stitched lines onto her example. This might be a time that it’s better to not be hands-on. We’ll see.