[ sorry this is written up quickly – lots of running around with the kids today]
A few weeks ago we did a fun project looking at the surprising relationship between three Platonic solids – an icosahedron, a dodecahedron, and a cube:
I was looking for a follow up project since that prior one was so much fun and pulled this book out of my bookshelf:
My thought was that it would be fun to see if we could make some of the stellated icosahedrons, but a statement in the beginning of the book caught my attention – you can inscribe an icosahedron inside of an octahedron. I do not have a lot of background in 3 dimensional geometry, so this statement was quite a surprise to me.
Before dinner last night the boys and I played around with our Zometool set to see if we could figure out how to construct this arrangement of the octahedron and the icosahedron. This project was pretty challenging, but we eventually figured it out! After we finished building that shape last night it seemed like a fun project for today would be comparing our old shape (the icosahedron / dodecahedron / cube combination) to the new icosahedron / octahedron combination.
This project proved to be a little bit more challenging that I was expecting because we didn’t have enough large blue Zometool pieces to build big versions of both shapes we were studying. We were able to improvise, but it is just a little bit harder to see the symmetries in the smaller shape. Time to get a few more Zometool pieces, I guess!
Before we started comparing the rotations of the two shapes, we took a close look at the shapes. The video below shows our shape from last week. We attached a few extra pieces to the cube to help make it easier to see where that cube is hiding. Hopefully this all shows up well on camera (the easiest time to see the shape is at the end of the video when we have it rotating):
Next we looked at the icosahedron / octahedron combination. Again, it is probably easiest to see this shape when it is moving:
Next we went to the living room floor to see the amazing rotational symmetry these shapes have, and the surprising relationship between the octahedron and the cube in the two shapes.
The first symmetry we looked at was the five fold rotational symmetry of the icosahedron that comes from rotation around one of the vertices:
the second symmetry we looked at was the three fold symmetry of the icosahedron that comes from rotating around one of the triangle faces. Again we find that these rotations turn the cube and the octahedron in the same way (sorry my hand is blocking part of the view of the smaller shape):
The last symmetry we looked at was the hardest one – the rotational symmetry around an edge of the icosahedron. It is harder than it seems to rotate these objects around an edge and return to the same shape. Playing around and figuring out exactly how to do this was one of the most interesting parts of this project. Hopefully that helped developed a little bit more intuition about 3D geometry.
This was a fun project, though one of our more difficult ones. Figuring out how to build these structures out of the Zometool set was fun, but challenging. Studying the rotations was also a lot of fun and required a lot of work ahead of the videos. It is also really interesting to see that there are 5 cubes inside of the dodecahedron and 5 octahedrons outside of the icosahedron that are related to each other. The action of the rotations on those cubes / octahedrons help you understand the symmetry group of the icosahedron. That bit is obviously more advanced than what we looked at today, but even just studying these shapes superficialy as we did today shows some really fun geometry.