Penrose tiles and some simple 3D variations

This weekend we spent a few hours making shapes with Penrose tiles and playing around with shapes that can fill up 3 dimensional space.  This was another weekend Family Math project inspired by an Evelyn Lamb tweet:

Although what gave me the  idea for this project was the rhombic dodecahedron blog post that Lamb linked to, I thought that it would be easier to play around in 2D first.  Penrose tiles seemed like the obvious place to start.

It wasn’t surprising to see that someone had already posted a template for Penrose tiles on Thingiverse.  Actually, there’s probably more than one, but here’s the one that we used:

We printed 8 kites and 8 darts and the boys spent 20 minutes or so making various shapes with them before we got going with the movie:

Following that, we began talking about 3 dimensions.   The easiest idea of a shape that could fill up 3 space was a cube, so we stacked up some 2x2x2 Rubik’s cubes at the beginning.  Following that, we talked about an incredible pair of shapes that can form a special cube – Iwahiro’s “Apparently Impossible Cube” – that we’d found by coincidence on Thingiverse earlier in the week:

The boys had really enjoyed trying to solve Iwahiro’s puzzle (which may be more difficult to get apart than it is to put together!).

After this little 3D intro we moved on to what had inspired Evelyn Lamb’s tweet:  Laura Taalman’s (aka @mathgrrl ‘s) post about printing a rhombic dodecahedron:

We printed 4 of the small rhombic dodecahedrons and then spent about 30 minutes building the same shapes out of our Zometool set.  It was a little hard to get the printed shapes to stay stacked together, so the Zometool set was actually a big help.  Here’s our talk about these shapes:

Playing around with all of these shapes made for a really exciting afternoon of math.   It is so cool to see all of the different ways that the Penrose tiles can fit together, and it is amazing to be able hold these special 3 dimensional shapes and see first hand how they  can fit together to fill up 3 dimensions.  Definitely a fun day.