HCSSiM Workshop, day 12
We originally defined as and now we re-introduce it as the set of matrices of the form with the obvious map where is sent to:
After we reminded people about matrix addition and multiplication, we showed this was an injective homomorphism under addition and also jived with the multiplication that we knew coming from Overall our lesson was not so different from this one.
Then we talked about actions on the plane including translations and scaling and showed that under the above map, “multiplication by ” or by its corresponding matrix gives us the same thing.
We went over the 5 platonic solids from yesterday- we’d proved it was impossible to have more than 5, and now it was time to show all 5 are actually possible! That’s when we whipped out Wayne Daniel’s “all five” puzzle:
We then introduced the concept of dual graph, and showed which platonic solids go to which under this map. We saw an example of a toy which flips from one platonic solid (cube) to its dual (octahedron) when you toss it in the air, the Hoberman Flip Out.
Here is it in between states:
Finally, we talked about symmetries of regular polyhedra and saw how we could embed an action into the group of symmetries on its vertices. So symmetries on tetrahedra is a subgroup of . It’s a lot easier to understand how to play with 4 numbers than to think about moving around a toy, so this is a good thing. Although it’s more fun to play with a toy.
Then one of our students, Milo, showed us how he projected a tiling of the plane onto the surface of a sphere using the language “processing“. Unbelievable.
After that we went to an origami workshop to construct yellow pigs as well as platonic solid type structures.