Open course ware

PROFESSOR: Examine every possible means for combining the symmetry at once, but there is a seemingly paradoxical trick that we can yet pull. And let me indicate what is true here for 2mm. OK, so this is a twofold axis. Has mirror planes perpendicular to it. If one of these is a mirror plane, the other one has to be a mirror plane as well. So there's no way we could make one a mirror plane and one a glide plane. OK, that requires a net that is exactly rectangular. So let's put in the twofold axis. And I add one to the corner of the cell. As we well know we have to have twofold axes at all of these other locations. We want to put a mirror plane in the cell. We could pass it through the twofold axis, and that would be the same as P getting to P2mn back again. But why do we have to put the mirror plane through the twofold axis? We have to have the twofold axis left unchanged when we add the mirror plane, because if we created a new twofold axis we create a new lattice...

ERIK DEMAINE: So with every sort of advanced class that I teach there's a final project. And the goal of the final project is for students to somehow get their feet wet with the material and sort of experience it at a more researchy level. In general, this can be things like surveying papers that I didn't cover in the class because there's only so much you can fit in one semester. So they'll go and read other material and kind of aim to teach that to the students. So there's a written project part, and then there's also a presentation in class. So this is an opportunity for students to learn more. But actually most students in this class aim for a particular type of project, which is to solve an open problem. So actually advance the field and then present their solution. Now, of course, not everyone succeeded. And that's fine for a project. You don't have to solve the problem that you set out to solve. But at least you try different approac...