Why are there Three Dimensions?

 Gimbal locking, plus relativity.  (Alternatively, gimbal locking plus holographic principle, if the idea that relativity implies 2+1 dimensions, rather than 3+1 dimensions, doesn't seem incredibly obvious to you.)

Locally, at least, all dimensions eventually collapse down to 2+1.

Assuming the mechanism I'm thinking of works - it's not quite gimbal locking, but it's kind of the same principle, in which two rotational dimensions become locally dependent - any finite number of spacial dimensions should reduce to two spacial dimensions in practice for the oriented matter, over enough time, at least locally.  I don't think the last two dimensions can lock, because you need at least one non-time orthogonal dimension (because some interactions create orthogonal forces), but my thoughts are very vague right now.

This explanation probably will not actually work in conventional physics, because the distribution of mass/energy doesn't work out nicely, as those extra dimensions never actually go away or become compact, they just become dependent; I do not believe you would get the inverse square law under these conditions.

A Random Thought

 If we consider the extra dimension(s) on which the amplitude of the wave function given to the Schrodinger Equation, the wave function instead defines a topology-like geometric object.  If the topology can be evaluated over time by some alternative mathematical construct, that alternative mathematical construct may form the basis for a more powerful (in the sense of describing a wider range of potential phenomena) physics, because it should be constructable in such a way as to not possess the limitations of the Schrodinger Equation that the function returns a value for the entire dimensional space under consideration.

The amplitude of quantum waves are geometrically limited in a way that the topology shouldn't be; quantum waves have an extent from 0 to the amplitude, whereas a more general topology should permit discontinuous extents.  The amplitude of the probability would be equivalent to a thickness, or measure, in the topology; the exact position of the topology relative to the dimension of amplitude could vary (this variance is why I describe this as a topology).  These values could potentially matter for the generalized version of the Schrodinger Equation, which would describe the special case where these values don't matter, where the measure is continuous, and where the topology is defined for all of the non-amplitude dimensions.

I suspect there is a class of such generalizations, which make distinct predictions.

Time Travel

 If you want to travel backwards in time, the steps are simple.

First, travel away from your current location faster than the speed of light.

Second, catch up with that period of time.

Third, convert yourself destructively into a gravity wave propagated in the direction of travel, with all the transformations and distortions that the original gravity wave experienced, such that you precisely imitate the gravitic wave that your existence at that time would have produced, had the wave also traveled alongside all the others; if you get any of the transformations or distortions wrong, it won't work.