actually, is what it is, is ultimately a lot of algebra, and so we're lucky to have with us today rosa orellana, professor of mathematics at dartmouth college, who uses algebra all over in her work. so, rosa, thanks for coming today. >> thank you for having me, dan. >> well, we've been seeing a lot of beautiful geometric objects and trying to make sense of them algebraically, but let's sort of make it a little bit more rigorous with some good examples, shall we? >> we should start with, like, a rectangle and try to explain the ideas, because it's easy to see it there. so actually, let's have a square. >> okay, squares are good because squares are highly symmetric. >> so now the idea, dan, is that we would want to have a motion in space that brings us back to the exact configuration that we have. >> okay. >> so, for instance, here is the square. you close your eyes and i do something to it, and you open it and you tell me if i did anything. >> okay. >> i can rotate it by 90 degrees, and this brings me actually back to the same configuration. >> that's right, that's right. occupies the same space