Sounds like you want

manifolds, in particular sub-manifolds. This stuff is part of

differential geometry. I know a little bit, but probably not enough to know what I'm talking about if I had to explain it.

To understand diff geo, you first need to know open sets, and I imagine knowing about compact sets would be helpful, basically you need to know a bit about

real analysis and

topology.
If you're looking for something simpler, then there are

Linear subspaces, you only need linear algebra to understand these.

How are you going with physics and abstract algebra? I've thought a little bit about physics while walking my dogs but I haven't sat down and done anything.

Thanks! Appreciated!

I used to know Differential Geometry from a Physics perspective (multivariate/vector calculus applied to curved 3D/Minkowski Spacetime) - but I've not used it in 30 years so it is beyond rusty. But the Differential Geometry I used to know would not help me here! (I'm not trying to represent one coordinate system in another curved space - e.g. translate between spherical and euclidian coordinate systems. I'm trying to bridge the gap between "Flatland" and 3D Land...)

I also got pretty proficient ~ 10 years ago in linear maths and vector spaces (was building electronics cad systems for fun and needed to use some pretty advanced stuff in the linear maths space...)

But, I think I'm looking for something probably closer to Topology...

I'll both explain and answer your question at the same time: I think I'm almost on the edge of being able to visualize the relationship between closed sub-universes with one set of dimensions - say ours with 3D - existing in a larger universe, say, with 8D.

Until this morning I couldn't conceptualize how a sub-universe could exist within one of larger dimension (e.g. 2D Flatland within our 3D) - but now I see them everywhere - based on hypercomplex numbers. (I finally figured out why they exist! They aren't just failed vectors! They were never vectors! They are SO much better! They define Universes, and vectors exist within those universes, obeying Geometric Algebra...)

What I am trying to get to now is to understand if there exists a branch of maths that could explain how a translation of a particle in the sub-universe that has components in the external universe, could generate a wave in another that extends through all of that sub-space...

I can actually see the transformation in my head but am trying to understand how to express it mathematically - or else I will sound like a typical pseudo-scientific idiot...

To make it real: An electron moving in 4D energy space generates an E/M wave that extends throughout all possible paths through our 3D space and does not experience time or the dimensions of our space. (But we in 3D Flatland CAN see it and we DO experience the wave taking time...)

Meanwhile, an electron physically moving in 3D Flatland space generates a <Gravity?> wave that extends throughout all possible paths in another external space, and those waves DO NOT experience time OR the dimensions of that space... (But I'm willing to bet that an observer in that space DOES experience time...)

I am trying to express the mathematical relationship that describes a translation between a 1 D movement in one universe into a wave in another. I see it happening, but I want to know mathematically why e=hc/λ...

It is not a bow-wave. The wave

**is** the dimensional representation of the movement through the other dimension and automatically exists throughout all of the subSpace. But why a wave with a specific wavelength?

I see it, but cannot put it into words...

Meanwhile Time gets interesting, because it doesn't exist unless observed by consciousness existing within the world in which the wave exists... (It's not a Quantum Mechanics outcome that needs an observer to exist, it is Time itself that only exists if observed by consciousness...)