Naia Faraji, Olivia Lewis, Eve Louvet--Monsanglant, Janna Stewartson, and Lauren Rabe

In Section One, “Basics of Surfaces and examples” we learn that a surface is something that has width and length. Then, we learn about dimensions, which is the number of mutually perpendicular directions one can travel. The dimension of a point is 0. The dimension of a line is 1 dimension. The dimension of a flat plane is 2 dimensions. The dimension of a human is 3 dimensions. Also, apparently time is the fourth dimension, but how? Objects with dimensions are classified as orientable or non-orientable. Orientability is based on if you can picks consistent upwards direction everywhere on the surface. In lesser words, this is described as If I travel around the surface, when I return back to the same spot will I face the same direction I started in?

In Section Two, “Euler Characteristic”, we learn more about invariants in surfaces. The formula for the Euler Characteristic is V-E+F. After this, we discovered that shapes with the same Euler Characteristic were homeomorphic, which meant that they had the ability to be bent or morphed into one shape into another. We also explored the torus, and realized that due to its hole (genus), it has a different Euler Characteristic. We found a pattern where when another torus was added to the first torus, 2 was subtracted from the Euler Characteristic.

In section three, titled “More on Surfaces”, we learned about how to classify interior and exterior points in both surfaces and shapes. This primarily consisted of finding certain points on the surface, and seeing whether or not a circle drawn around that point fit within that surface (with the surrounding circle being infinitely small enough so that points right next to the boundary could still count as interior points). Similarly, points in shapes can be surrounded by spheres to conclude whether or not the points are considered interior or exterior. Another interesting topic of discussion we came across was punctured planes and punctured shapes. This essentially means that in a plane, there is a single point that has been “poked” or punctured so that it is an exterior point, and therefore not part of the plane. In order to be a closed surface, the boundary must be a part of the surface. Strangely enough though, the infinite plane is considered a closed surface despite not having a boundary at all- which we concluded was because the non-existent boundary would theoretically be contained in the plane since the plane itself is infinite.

In our last and final section, titled “Advanced Cutting and Pasting”, we started off by learning about how to get the “word” of a shape. You start off by choosing an orientation to transverse the shape’s perimeter which can either be clockwise or counterclockwise. We keep recording the labels until we return back to the starting point. The resulting collection of letters with exponents is called the “word” of the polygonal region (ex. ab^-1a^-1b). Interestingly enough, a single shape can have multiple gluing patterns and multiple words. For example the torus (a donut-shaped object) has the words abca^-1b^-1c^-1/ab^-1a^-1b. Although they look totally different, we learned how to translate these words into gluing patterns and then use the Euler characteristic to discover what physical shapes these were without having to physically make these shapes.