You've learned the formula for the cross product, but where does that formula come from? We'll see how the cross product naturally pops out when we look at rotations and angular velocity in three dimensions. Then, we'll put on our 19th century hats and visit the cross product's parents, the quaternions. We'll learn what quaternions are and see that, just like 2D rotations can be described with complex numbers, 3D rotations can be described with quaternions, making quaternions useful for computer graphics. If we have time for dessert, we'll talk about what happens to these ideas in higher dimensions, how the determinant expression for the cross product is more than just a mnemonic device, or why 3D rotations are the same as lines in 4D space.

Take any number whose last digit is either 1, 3, 7, or 9. Raise it to the 21st power. The last two digits of the answer will be the same as the last two digits of the number you started with. Want the last three digits to match? Raise it to the 101st power instead. Want to know why this works? Come to this class.

In a group of nine people, in how many ways can you pick three of them? In a group of four people, in how many ways can you distribute six $1 bills between them? (Giving nothing to someone is allowed.) Why are the answers to these two questions the same? In this class, we’ll explore a triangle of numbers called Pascal’s triangle, and we’ll see how it can answer the questions above. After that, we’ll see that Pascal’s triangle has many cool things hidden inside it, including the Fibonacci numbers, a pretty fractal called the Sierpinski triangle, and much more.

You're driving around in the xy-plane, and, being mathematically inclined, you decide to drive so that your velocity is a linear function of your position. More specifically, you decide that, as you drive over the point $$(x,y)$$, your velocity in the x direction should be $$x+3y$$, and your velocity in the y direction should be $$-4x-2y$$. In this class, we'll find out whether you get back home to the origin or instead get a speeding ticket.

David is setting up ten chairs around a round table for a party. He has two kinds of chairs to choose from: wood or metal, and he can use as many of each kind as he likes (including zero). How many ways are there to set up the chairs? Two setups are considered to be the same if one can be rotated to match the other. To start solving, you might point out that at each of the ten positions, you have two choices, for a total of $$2^{10}=1024$$ configurations. For any such a configuration, you could rotate it to get a total of ten configurations that correspond to the same setup. The answer should thus be $$1024/10=102.4$$. Uh oh. Alas, we would then resort to casework: There is 1 setup with no wooden chairs, 1 setup with one wooden chair, 5 setups with two wooden chairs, and so forth. But there is a better way! In fact, the first method that gave us $$102.4$$ setups is not hopelessly flawed. Once we figure out what is wrong with it, we'll be able to apply some clever ideas about symmetry to fix it up and solve the problem. But then, Cathy shows up to the party and sees the reflection of the table in a mirror. She notices that the reflected setup is the same as the original: The chair setup has an axis of mirror symmetry! Knowing this additional information, how many possible setups are there now? With the powers of group theory that we will learn how to wield, we'll see that this problem is actually not that much harder than the first one.

What is symmetry? What makes us call things like kaleidoscope patterns, regular polyhedra, and umbrellas "symmetric"? What do asymmetric objects like a left shoe and the letter G lack that makes us think of them as not having symmetry? Is there a way in which we can extend the notion of symmetry to non-geometric objects? For instance, is there a context in which it makes sense to say that the word "llama" is symmetric? What about the formula $$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2+xy+yz+zx)$$? In what way is that formula symmetric? We'll see how questions like these led to the study of group theory, which gets to the heart of the idea of symmetry and lets us see what things are true about all symmetric objects, no matter what kind they are. If you're familiar with group actions, you probably won't see many new things in this course. However, if you've seen group theory but haven't seen group actions, you'll get to see group theory from a new and useful perspective. No knowledge of group theory is required for this course.

In a group of nine people, in how many ways can you pick three of them? In a group of four people, in how many ways can you distribute six $1 bills between them? (Giving nothing to someone is allowed.) Why are the answers to these two questions the same? In this class, we'll explore a triangle of numbers called Pascal's triangle, and we'll see how it can answer the questions above. After that, we'll see that Pascal's triangle has many cool things hidden inside it, including the Fibonacci numbers, a pretty fractal called the Sierpinski triangle, and much more.