Geometry.

Geometry is my favorite class this semester. In the geometry class I'm taking, the main focus is not on circles, triangles, and so forth. Our objects of study are isometries: space transformations that keep distances the same. For example, in plane geometry, some isometries are translations, rotations, and reflections.

We can combine two isometries of a space by applying one after another, resulting in a third. The result is that the set of isometries forms a group of transformations that act on the space. We can then study the abstract algebraic properties of the group of isometries on a given space: how isometries interact, the general form each one takes, what its fixed points are, whether it reverses orientation, and more. In learning about isometries, we find out about the geometric properties of the space.

For the first half of the semester, we studied Euclidean space. Rather than confine ourselves to two dimensions or maybe three, our class covered general n-dimensional Euclidean space (characterized by the well-known Pythagorean distance formula). Each Euclidean isometry has a part that "rotates" or "reflects" (kind of), followed by a part that translates (slides) the space. The rotational or reflectional part can be represented by a matrix, so we can deal with the Euclidean isometry group through linear algebra. (For those of you who know linear algebra: remember orthogonal matrices? The linear part of an isometry is an orthogonal matrix, so the algebra of orthogonal matrices comes up a lot here. Your knowledge is useful!)

The class applied these ideas to learn about symmetry groups of three-dimensional solids (polyhedra); a symmetry group consists of the ways that a solid can be rotated or reflected. In particular, each of the five Platonic solids has a rotation group which can be characterized by permutations of certain parts of the polyhedron. For example, the rotations of the cube are identified with permutations of the four long diagonals, so we find that each cube rotation is equivalent to a reordering of a set of four things. This was one of my favorite parts of geometry, because of these ideas; I find them exceedingly nifty and compelling.

We just started with spherical geometry, and this seems like it's going to be really interesting, too. The geometry of the sphere follows naturally from the geometry of Euclidean space, because we can consider the sphere as an object in Euclidean space. Later, we will learn about hyperbolic geometry and projective geometry. I have had some exposure to hyperbolic geometry before (it is pretty much the opposite of spherical geometry), but I don't know anything about projective geometry. I am looking forward to these exotic new lands with exotic new distance formulas!

P. S. The following math video is not quite relevant to geometry, but I figured I'd post it anyway because it's so beautiful. There are two parts.