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The representation of self-interesting shapes and surfaces is very challenging. One way to achieve it is using optimal geometry,which uses topological techniques. Dr. Sullivan introduced the concept of manifolds as part of this concept. These are spaces which locally look like R^n. They are divided into 0-nfld (a point), 1-nfld (line segment or circle), and 2-nfld (Euclidean plane or a torus kline). The interesting characteristics of the manifolds are used to ellaborate optimal surfaces. On these, the bending enery is minimized. One important aspect of these surfaces is that the optimal metric (intrinsic shape) should be symmetric, meaning that points and directions are equivalent, locally. Also, there is an intrinsic idea of curvature in manifolds. On a plane it is equal to 0, while for a sphere is greater than 0, and for an hyperbolic surface it is less than 0.

There are interesting examples of manifolds in different dimensions. 0-nfld and 1-nfld are not particullary interesting. Tori, Klein Bottles, and Mobius Surfs are good examples of 2-nflds. They behave peculiarly. For example, the torus doesn't have a completely symmetrical behavior, because of its spheric and hyperbolic components. Other important fact about manifolds is that there is always the possibility of finding a metric for any kind of manifold: Euclidean space, Spherical space, Hyperbolic Space, Sphere X Reals, Hyperbolic-plane X Reals, or three others related to Lei Groups.