I am a teacher and am currently tutoring middle/high-school students for research initiation, I gave small groups of them topics and let them advance by trials and error with essentially no guidance (they work one hour a week on it). I tried to find topics both doable and interesting, letting open many directions to go further, so that everyone can be sure of finding things but also won't get bored if they understand too fast.
One of these topics is the following one: we want to tile a wall, but minimizing the ratio perimeter/surface (I added a fancy background to the problem, but this is it). That would mean they should understand what polygons can tile the plane or not and to do some optimization on the possible polygons, maybe also to understand also that getting closer to the circle is more efficient, and trying non-regular tessellations.
However, the group with this topic already understand/know most things (they still picked that topic): they right away thought hexagons are the best and know that there aren't much possible tillings. They of course still need to prove and compute a bit, but I am fearing I won't be able to keep them occupied the whole year with it.
I am searching for ideas to go further in this direction. I thought of:
- consider non-regular (or maybe with two polygons only),but I fear this will be worse than the hexagons
- letting them consider the boundary problem, and to wonder about the exact cost taking into account small pieces of tile to finish the boundary of the wall
- making them prove it is optimal even among non-regular tiling (I don't know whether it is true or not, nor have I any idea about how to approach this)
- can I ask the analogous question on the sphere? Is the answer the same and the problem not much more interesting? (or too hard?)
Any less trivial idea or any lights about the above ideas would be of great help!