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What would happen if a four dimensional sphere was inverted
If a 4D sphere could be inverted, it would have to be represented with a 3D object. For example, an orange peel could represent the entire contents of a 4D sphere. The peel represents the topology and geometry of the original space.
Now we can imagine that this 3D object, or peel, is turned inside out. Normally a solid object cannot undergo such a transformation since there are no physical means of turning the inside into the outside (however, with clever use of quantum phenomena it is possible to do so).
Now, if the peel were to be turned inside out, what would it look like? It would probably form a singularity at its center. For example, as one approached the orange peel from any direction there would only be an ever increasing amount of space between you and the central singularity point.
However, there would be no singularity since the orange peel is only 3D. For that to happen, a 4D object must be turned inside out so as not to create a singularity point.
However, if the 4D object is turned inside out, it could create a new singularity--one that would be stable. This means that the old 3D singularity point, which acted as a black hole would be greatly enlarged and thus become visible to one who was looking at it.
At this point, the newly inverted 4D object would be divided into two 3D singularities. These dual singularities could eventually join back together and become one; however, if they did not then we would have a situation where the original sphere was separated into two disjoint parts.