4-dimensional Imagination

The mathematics for 4 dimensional geometry are very well understood, and modern computing power is more than sufficient to simulate the multitudes of operations one could perform on virtually any 4 dimensional configuration.
It's a given that machines currently don't posess what anyone would truly consider an 'imagination', but for argument sake let's consider one that does. Could such a machine simulate a 4 dimensional environment, or at least be presented with one (this environment could even be something as trivial as a singular structure), and truly be able to visualise it, just as you can imagine an apple in your mind right now? 
Using the apple as our prime example, could this machine effortlessly and intuitively rotate a 4 dimensional apple about in its 'mind' and understand its form as easily as you see the 3 dimensional apple? And here I don't mean a 3 dimensional projection of a 4 dimensional shape, I mean actually comprehend some 4th dimension. For example we can simulate 3 dimensional space easily enough and create an interface for people to perform 3d work on a computer (albeit the interface is n-1 dimensions, so we have a 2d screen, 2-ish-d visual system, etc), but if we needn't present the output spatial coordinates to a screen per se, what's to say a sufficiently intelligent and aware machine couldn't comprehend and intuit greater dimensions?
To my mind I don't know of any limitations on this as a possibility (I would love to know more, or otherwise!). The greatest stumbling blocks in this scenario may be understanding descriptions given by the machine and/or proving that whatever the machine is telling us is indeed qualitatively different than number crunching only (which many mathematicians and programs already do with ease, with the aside that mathematicians also gain an intuition of what they are doing, the closest I think the raw human mind can get to 4d vision). With that being said, taking a slightly different approach, what about creating a system that could be connected with a person's brain directly? For that period of connection would they then be able to see 4d space in their mind (this reminds me of an experiment where colour blind monkeys were given the ability to see colour)? And in either case is there some limit to the number of dimensions?
I know this is all questions and no answers, but I simply find the idea fascinating and wanted to lay out some of the pressing questions that had come to me. If anyone reads this constructive feedback of all sorts is appreciated! More thought experiments to follow!