I came across this picture/sketch on the internet, however there is no any explanation about it:
What is "UNPREDICTIBLE"? "UNSTABLE"? "TOO SIMPLE"? "elliptic", "ultrahyperbolic"??
Some related questions here:
Is 3+1 spacetime as privileged as is claimed?
What makes four dimensions special?
Why are there 4 Dimensions and 4 Fundamental Forces?
Why doesn't the number of space dimensions equal the number of time dimensions?
The terms elliptic and ultrahyperbolic are technical terms used in the classification of partial differential equations, and going into their precise meaning wouldn't be very illuminating. Basically for these types of PDEs a set of initial data doesn't uniquely determine the evolution of the system and any particular solution, like the planetary orbits in the Solar System, wouldn't be stable (and without a stable Solar System we wouldn't be here :-). This is known as the initial value problem if you feel like Googling it further.
An aside: you might want to have a look at the paper On the dimensionality of spacetime by Max Tegmark. This is pretty accessible and gives a reasonably accessible description of the physic involved.
Back to the diagram: it should hopefully be obvious that spacetimes with zero time dimensions or zero spatial dimensions aren't physically reasonable. What is perhaps less obvious is that having more than one time dimension is also physically unreasonable because it allows for closed timelike curves and violations of causality (this is the initial value problem again). There have been proposals for spacetimes with two time dimensions by Itzhack Bars, and I believe F Theory has two timelike dimensions though this is far outside my area of expertise. However these are special cases and have extra structure to guarantee stability.
So if we restrict ourselves to one time dimension how many spatial dimensions are allowed? Again it should hopefully be obvious that less than three spatial dimensions is too simple and doesn't allow complex systems like humans. If we have more than three spatial dimensions central potentials like the Sun's gravity have no stable orbits so the Solar System wouldn't be stable and we wouldn't be here. This singles out three spatial dimensions as the only case in which humans can exist.
That's why one time and three spatial dimensions is the only case in which you could be reading this post.
"Unstable" has the meaning that the current understanding of physics does not support stable orbits, except for exact circular ones. Anything can knock a planet out of orbit.
"Ultrahyperbolic" means that there are several time dimensions, and our little mindkins can not deal with an acre of time.
"Too Simple" means that the space can not do things like supporting crossing things.
The sort of space that we suppose exists by radiating fields (ie fields where the source is supposed to radiate flux, and flux-density equates to field strength), produces all sorts of unstabilities, like planets breaking out of orbit, unless there is an inverse square law.
It works with then when one dimension of the four is related to the other three by 'i', which means that quarterions work. But our experience is three spacial dimensions, and hence one time dimension.
