Curves and Surfaces

by Moira Chas and Karyn Lundberg

WISE 187 , Spring 2003

The main goal of this project consists in exploring one of the most amazing branches of mathematics: topology. Here, the concepts reach beyond numbers and formulae into the realm of visualization and spatial intuition. Topology is "rubber sheet geometry", in which objects are equal if they can be deformed one into each other by stretching or compressing.

We will start by traveling through the world of curves and surfaces. A surface is a space of two dimensions: length and width. A curve is a one-dimensional space. We will make a first approximation to the concept of dimension.

By making concrete constructions with paper and scissors we will study properties of surfaces such as orientability, genus and number of boundary components. We will see how words in certain alphabets can be used to encode properties of surfaces and curves. Finally, we will investigate how surfaces appear in the "real world" as the configuration spaces of certain machines.