Extensions beyond Euclidian Geometry for Secondary students

My secondary geometry class is really amazing and is likely going to finish everything I have for the curriculum with a few weeks left in the school year. I was thinking to use these last few weeks to look into some interesting Non-Euclidian geometries, such as hyperbolic, affine, elliptic etc. Unfortunately, I do not know much about these different geometries (my training is in computer science) so...

1. What are some good resources to get a good overview of non-euclidian geometries?
2. What are some fun/interesting activities involving non-euclidian geometry? It will be the end of the school year so i don't to get into anything too rigorous but interesting would be great!
3. Are there any other extensions to either non-euclidian or euclidian geometry that are not in the typical secondary curriculum but which would be fun and interesting for my students?
• "My secondary geometry class is really amazing and is likely going to finish everything I have for the curriculum with a few weeks left in the school year." ....???!!! This is so outside my realm of imagination that I have difficulty phrasing a question that probably could be asked here. – Xi Yu Mar 18 '15 at 17:18

You are probably looking for something light and accessible. I would stick with spherical geometry as it is easier for students to conceptualise. Discrete spherical geometry makes it even easier and more hands-on.

1. Introduce basic graph theory and use Euler's formula to calculate the possible regular polyhedra (Platonic and Archimedean solids) which are topologically equivalent to a sphere. Spherical geometry can be studied from these polyhedra. You will also encounter 2 infinite series.

2. Draw graphs of some of these polyhedra and create nets to construct physical models. Some graphs are quite difficult, but can be made easier if you start at a vertex for the tetrahedron and a side for the most of the others. You want to finish drawing at a side, not a vertex. Practice drawing them yourself first. Grow the graph symmetrically around the starting figure. Look for forced lines (one line left at a vertex) and keep track of filled vertices (maybe draw a circle around them when they are full). Don't worry about curved lines, especially around the outside of the graph, as curves don't matter topologically.

3. Notice that the graphs of spherical surfaces become highly distorted. Draw polyhedra on a spheres (oranges/mandarins). Discuss strategies that map makers have used to address the distortion of spheres projected onto flat surfaces. You could peel and flatten an orange skin as a demonstration. Students can do it outside during recess.

4. Calculate the angle defect of each vertex of a polyhedra. Use Descartes theorem to link to Euler's characteristic.

5. Look at possible symmetries of a sphere. Discrete symmetries (of Platonic solids) correspond to discrete groups, another interesting and accessible topic. Continuous symmetries correspond to only a few Lie groups. You can make a hierarchy of symmetries based on experimentation on just Platonic solids.

6. Find underlying similarities between dual pairs of Platonic solids based on possible symmetries. Or transform graphs of any solid to its dual.

7. You can do group arithmetic on solids. Some group arithmetic is commutative, some is not! The single rotations of a prism compose with nice modular arithmetic, but see what happens when you allow a flip. Your students might even handle some simple problems using the 3 rotations of a cube. Keep in mind the full state of a cube with no reflection is the front face and orientation of the face - up/down/left/right - leading to 24 possible states, so you will not want to create a full Caley table, just do some simple arithmetic. Be very specific on your rotation directions.

8. Decorate solids to demonstrate the discrete spherical symmetries.

9. Extra for experts: create graphs and maybe nets of (parts of) regular discrete hyperbolic surfaces (full hyperbolic surfaces are infinite). Graphs will collapse into something like a Poincare disc rather than exploding like a sphere. Will take some creativity to make nets that can be stuck together.

I've used Geogebra for creating nets as it is easy to draw joined polygons, but it is not necessary for simply polyhedra. Some regular polygons to trace around maybe? It was often useful to draw graphs of surfaces first before doing the net.

I hope a few of these ideas are at the right level of your students.

This might give you ideas...

From sarah-marie belcastro's Knitted Hyperbolic Surfaces, a knitted pseudosphere, a surface of constant negative curvature:

Taxicab Geometry!!

Show your students how to pose problems in mathematics. Get them posing their own geometric questions. What if I change that definition/axiom, for example?

For example: http://www.jstor.org/stable/27970540 [Taxicab Conics!] Or do plain old conics from a purely synthetic approach! The Dandelin Spheres are a beautiful geometric argument that they could appreciate if they got to mess with some conics first.

Do some math history with the them! (Math History Wildberger on youtube)

The most important real-world application of noneuclidean geometry is in relativity. If you use GPS, then relativity is part of your everyday life. General relativity has a reputation for being difficult and highly mathematical, but it doesn't need to be. Here are some resources that let you get at the subject at a fun level:

I also have some lecture notes of my own that might be of interest.

If you can get hold of all of it, Not Knot gives an introduction to 3D hyperbolic geometry in the context of knot theory.

The the various programs by Jeff Weeks are also fun, if focused more towards topology. Playing pool on a hyperbolic surface does at least teach you it's different to Euclidean geometry.