Surfaces and Homeomorphisms.
So far, we have encountered a few geometric objects including the Sphere, the Cube, and the Circle. There is a rough distinction between the first two objects and the last one. Imagine an ant walking on the sphere. From its point of view, the object it is moving on is a curved 2-dimensional plain, while an ant walking on a circle has the feeling that it is walking on a line. This property refers only to a small "area" of these objects and is therefore called the local property of the object; all geometric objects that locally behave like the 2-dimensional plane are called Surfaces. Below are examples of some surfaces:
It is impossible of course to map a sphere to a circle since locally these objects are distinct. A person on earth can walk in all direction of a plane while on a circle he is bound to walk in two directions only. Could an immoderate Geographer create a map of earth on any surface? Will she succeed if she tries to use the torus as a mapping surface? No matter how hard she tries and what sophisticated tools she will use, she will always end up with unexceptional results where for example France would be torn away from the shores of England and would have borders with Japan. Her failure is not a coincidence, for it is mathematically impossible to map a sphere to a torus. We leave the proof of this fact to our next lesson, using the Euler Number
As for the valid maps of earth on the cube and dodecahedron, they are examples of the fact that mathematically it is possible to map the sphere to these surfaces.
Surfaces that can be mapped to one another are called Homeomorphic Surfaces.
We can divide surfaces into classes; each class contains surfaces, which are homeomorphic to one another, while two surfaces from two different, classes are not homeomorphic to one another. The Cube, the Sphere and the Dodecahedron are members of such a class and the Torus is a member of another (distinct) class. To a topologist the interesting objects are the various classes of surfaces, in our discussion, when we will study a surface we would only query to which class of surfaces it belongs. In this context, there is a unique description of planet earth, only members of the same class of the sphere could take valid maps of earth. Likewise, mappings of configuration spaces will be unique only in this manner. It is only natural to query which are the various classes of surfaces, on the next lesson we will give a partial answer to this. For now, we will only give a few examples of (members of) distinct classes:
|Sphere(oriented surface of genus 0)|
Oriented Surface of genus 2
|Oriented Surface of genus 5|
Next, we will investigate another simple machine's configuration space.
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