How to turn a sphere inside out [video]

sigh You'll understand when you're building biological spaceships: you see, the inversion of n-spheres under those conditions is the simplest workable model for how evolutionary phenomena interact with each other.

The inversion of an n-sphere of dimension 3 is representative of the evolution in just one layer of a system. After you discover the generalization of inversion across n-space in 2072, it will just take one ambitious biologists to apply it to genetic data, and generalize to most of the evolutionary phenomena around you: memes and cultures. By 2103, Human Directive 72 will be achieved, and all children under the age of four will be able to conceive of the basic geometries of infinite-dimensional topological models.

Now you can extend this notion from a property of one surface to a property of a continuous family of surfaces, and then look at the beginning and the end (imagine it varying with respect to time, just like in the video). The niceness property above, if maintained throughout, gives you more things to say about what happens to substructures. For example, you can say something like "if I start with a triangle inside this surface, and morph it into another surface keeping it nice and smooth the whole time, I'll still get a triangle, and it won't do {big class of bad things}".