Even if we couldn't perceive shapes across a 4th dimension, we would still perceive things moving through the 4th dimension like we see in 4D toys. In reality we don't ever see anything like this (spatially anyway). Is that correct?
Ehrenfest (1917/1920) studied the hydrogen in n dimensions and concluded for n> 3 that neither classical atoms nor planetary orbits can be stable, because the inverse square law of electrostatic and gravity becomes an inverse cube law. When n > 3 there are no stable orbits to the two body problem: an incoming light body attracted by a heavy one would either escape to infinity or get sucked into collision.
For n = 3 we get stable elliptic orbits or non-bound parabolic and hyperbolic orbits.
Collision only occurs when the lighter body heads directly towards the heavy body within 2R (R being the heavier body's radius), ie. the impact parameter is zero [2]
[1] https://doi.org/10.1002/andp.19203660503
[2] https://en.wikipedia.org/wiki/Impact_parameter
edit: grammar
It is perfectly possible to formulate theories of physics in higher spatial dimensions. In fact, many high-energy theories like string theory require many spatial dimensions for mathematical consistency. It is something we actively look for signatures of in experiments.
What we have found is that our observations are incompatible with 'extended' additional dimensions. Extended here means that you can go a sizeable distance. It's not that the other dimensions would just stop, but they'd be more like Pac Man---these dimensions might have periodic boundary conditions, making them circles. If the circle's radius (or circumference, equivalently) becomes very large, such a 'compactified' dimension starts to have a lot of space in it, much like the familiar 3 dimensions. So, we can put an upper bound.
My most recent recollection. We may yet have missed dimensions as large as 1mm in [diameter, circumference, I forget]. The most sensitive probes with the fewest assumptions about the structure + particle content of the universe tend to be gravitational, and gravity is very hard to measure precisely on very short length scales.
edit to add: the latest summary of extra dimensional searches from the Particle Data Group http://pdg.lbl.gov/2019/listings/rpp2019-list-extra-dimensio...
even better, here's the PDG's review, rather than their technical summary: http://pdg.lbl.gov/2019/reviews/rpp2018-rev-extra-dimensions...
But our universe as a whole is almost entirely empty space, with visible objects appearing to make up only about 5 percent of the universe's total volume. Noticing 4D things flit in and out of our 3D universe might be difficult at that scale.
String theories require extra spatial dimensions (to total of 10, 11 or 26), but those dimension are so small and/or looping so that we cannot detect them.
Other approach is to treat our 3D space as slice of higher dimensional space.
https://en.wikipedia.org/wiki/String_theory#Extra_dimensions
String theories have a lot of problems (biggest ones is that they are currently unverifiable using experiments and they do not give any meaningful predictions about our 3D view of universe), but many theoretical physicists are working on them.
Makes me appreciate that relativity and quantum mechanics are verifiable experimentally.
Nothing (in the classical, non-quantum) setting is solely determined by position coordinates.
Recently there has been super interesting research on using these nurbs as ansatzfunctions e.g. for FEM stuff. They got a lot of applications.