# We expanded our definition of geometry to encompass the geometry of a sphere and of negatively curved surfaces.

Our goal this block was to explore many facets of geometry not usually seen in high school or college classes, but which are important to many parts of modern mathematics. We started out by revisiting Euclidean geometry, but using the language of vectors. We saw that thinking about vectors instead of points and line segments could be a powerful new tool, which allowed us to succinctly state and prove theorems. Students tackled problem sets on vector geometry, solving and presenting all of the theorems themselves.

Next, we explored projective geometry. We began with a reading by Jordan Ellenberg on how to win the lottery with geometry. In it, Ellenberg describes using the Fano plane, a geometry with only seven points and seven lines, to win the lottery. The Fano plane is a small example of a projective plane, which is a plane where two lines always meet at a point. In other words, there are no parallel lines. Students came up with their own example of a projective plane with thirteen points, and we spent some time making and proving conjectures about what these tiny geometries look like.

We saw that projective geometry is also the geometry of artists who want to paint pictures with realistic perspective. Students tested out their understanding of projective geometry by making perspective drawings of tiled floors. They also figured out how to measure the height of a person in a photograph. We tested out this theory by taking a picture of one student, and seeing if we could accurately guess his height. Our estimates centered around the right number, but we were off by a couple inches-- pretty good!

In our final unit, we used paper triangles, taped seven around every point, to make models of the hyperbolic plane. We drew straight lines and noticed that given a line and a point, we could find infinitely many other lines which go through the point, and don’t intersect the line, negating Euclid’s Parallel Postulate. We discovered that the bigger our triangles were, the smaller their angle sum. Students built other models of the hyperbolic plane out of squares, taped five to a point, and annuli. By working with many physical models of the hyperbolic plane, we were able to develop intuition about what might be true in this seemingly paradoxical kind of geometry.

Over the course of the block, we experimented with new kinds of geometry, made art, proved beautiful theorems, and stretched our ideas of what geometry can encompass.

*--Sachi Hashimoto*