Just for illustration of my previous point from physics. It is well known that a 3D Kepler problem (which describes many interesting situations, including a relativistic analog of hydrogen atom) can be exactly represented (and quite successfully treated) via the model of 4D oscillators with a specific constraint. The latter is responsible for a "hidden" dynamical symmery of the initial Kepler problem. Therefore, studying a very simple 4D oscillator model under different constraints, one can actually gain valuable information about other important 3D systems (not just 3D oscillators).

In conventional physics 3nD (or 6nD) has to do with n particles and 4 instead of 3 stands for treating time as a coordinate. In this form, studying higher dimensions than 3 is indispensible.

Dear Pawel, I've just come across an extremely interesting book entitled "Emerging Health Technology" (edited by Kristian Wasen). You are probably aware of this fascinating book. Personally, I was strongly impressed with the results described in the second chapter of this book entitled "The Visual Touch Regime: Real-Time 3D Image-Guided Robotic Surgery and 4D and “5D” Scientific Illustration at Work" by Kristian Wasen and Meaghan Brierley. Simply to get the discussion started, let me just post here the Abstract from this chapter (the PDF file of the whole book is attached):

***Emerging multidimensional imaging technologies (3D/4D/“5D”) open new ground for exploring visual worlds and rendering new image-based knowledge, especially in areas related to medicine and science. 3D imaging is defined as three visual dimensions. 4D imaging is three visual dimensions plus time. 4D imaging can also be combined with functional transitions (i.e., following radioactive tracer isotope through the body in positron emission tomography). 4D imaging plus functionality is defined as “5D” imaging. We propose the idea of “visual touch”, a conceptual middle ground between touch and vision, as a basis for future explorations of contemporary institutional standards of image-based work. “Visual touch” is both the process of reconciling the senses (human and artificial) and the end result of this union of the senses. We conclude that while new multi-dimensional imaging technology emphasises vision, new forms of image-based work using visual materials cannot solely be classified as “visual”.***

Dear Ulrich - thank you for answer. So with respect to added time to "equation" - do you know any insight gained in 5D which was useful in 4D problems?

Dear Sergei, thank you for the book - I wasn't aware of it. I had a look but it's not exactly what I'm looking for. It's 4D/5D imaging of e.g. heart, but it's still 3D heart we are working on. I'm rather looking for answer to bit different question: if we e.g. would investigate (theoretically) "4D heart", would insight into it's mechanics result in a greater insight into it's 3D version?

Though more of an application but still mathematically valuable, graphics has benefited greatly from constructions in the 4th dimension for dealing with 3D graphics. A very simple example is homogeneous coordinates for vectors and points in the plane. Other great examples are in coding theory where n-dimensional space is always considered on a regular basis to deal with fixed dimensions.

Dear Pawel, I think so, provided that a "4D heart" image would entail the visualisation of all the relevant "interactions" of the "3D heart" with the rest of our body. In physics, any extra dimension is always responsible for an extra "interaction" or "hidden" symmetry which is not usually and/or easily seen on 3D level.

Just for illustration of my previous point from physics. It is well known that a 3D Kepler problem (which describes many interesting situations, including a relativistic analog of hydrogen atom) can be exactly represented (and quite successfully treated) via the model of 4D oscillators with a specific constraint. The latter is responsible for a "hidden" dynamical symmery of the initial Kepler problem. Therefore, studying a very simple 4D oscillator model under different constraints, one can actually gain valuable information about other important 3D systems (not just 3D oscillators).

@Sergei,

? Misprint or did I miss something?

@Ulrich, I mighta been a bit sloppy, and this is probably called "The Kepler problem in Dirac theory" more correctly but the meaning is still the same. Take a look.

@ Sergei: Thank you - finding symmetries by adding a new dimension is something what I'm looking for. Will give it more thought.

I found an interesting paper regarding this example of hydrogen atom and 4D oscillators (more specifically 2 coupled 2D oscillators: http://physics.gmu.edu/~isatija/Phys701/HatomSym.pdf)

@ Daniel: Could you please elaborate more on " coding theory where n-dimensional space is always considered on a regular basis to deal with fixed dimensions."? I'm not sure I can follow you.

@Pawel, thank you, amigo. You did my homework for me (I hope Ulrich is happy now). Here is a more elaborate (hence, more interesting) example about 3D-4D relationship. This time I am talking about condensed matter physics. There is an old (and still hot) problem in this area related to strong electronic correlations in disordered systems. It was found (sometimes even proved) that an EXTRA dimension plays a crucial role in understanding the 3D picture. I refer to the phenomenon known as quantum Hall effect (QHE) in some semiconducting structures. Recently, the QHE has been generalized to four spatial dimensions. Why the study of 4D case is important in this case? This is because at the edge (or better, boundary) of this four-dimensional quantum liquid it is possible to induce a purely topological and dissipationless spin current by an electric field in the physical, three-dimensional space. This is a non-trivial example. For those who are interested in further details regarding this very REALISTIC physical theory, I attach one of the key papers.

Thanks Sergei - this is exactly what I'm looking for! I couldn't follow this paper in detail but I got general idea (brilliant) and found links to another works.

And I found something that I can relate more: http://en.wikipedia.org/wiki/Hopf_fibration

My background is biological physics so I could relate more to mapping between circles/spheres (in general area of biological oscillations). Such topological peculiarity with increasing dimension such as Hopf fibration is also a good example of qualitatively new insight coming from increasing dimension.