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Self-avoiding and self-intersecting random surfaces are useful in the study and development of new materials. The self-avoiding and self-intersecting surfaces resulted from a simulation under various conditions of material growth at an atomic level, subject to various forces. From the point of view of the mathematician there were problems not allowing the object created to disconnect or its boundary to disconnect. Being able to predict all possible states in which a surface could be, and drawing the correct conclusions from experimental data, required applications of topological theory that were very interesting.
The oct-tree datastructure on the other hand was an application of methods known to computer scientists but not physicists. If you think of the material that has developed as a set of connected cubes, then if you have many cubes in the interior of the mass you have grown, they may be able to be condensed into one large cube. This saves space AND time because you have fewer objects to deal with. The straightforward simulations would simply use lots of cubes. Hanan Samet has a book on oct-trees if anyone is interested.