Pentagon Tessellation

I have joined the local makespace in order to have access to their laser cutter and while I was looking through interactive NRICH resources that need updating I came across a perfect problem- Bow Tie.

The pieces needed for the problem are a pentagon that tile the plane. There are only 15 types of pentagon that have this property, a result that has only recently been proved. More information about this discovery can be found here.

The Pentagons for this problem have the following properties: angles B and E are right angles, Angles A, C and D are 120° and the side lengths AE, DE and CD are in the ratio 1:1:2.

I decided to cut these pieces from acrylic as I thought that tiling the plane would be a nice Roadshow problem. I also wondered if there were other interesting ideas that could be explored. My first noticings included:

You can make lots of hexagons:

You can make lots of different shaped holes:

There are lots of options for different types of symmetry:

The tessellation of the plane is also really beautiful. There is a new interactive here now which you could use to explore.