I use this fact about hyperspheres getting smaller in my dissertation on Emergent Design athttp://about.me/emergentdesign.
I have created something called Schemas Theory which is the next level of abstraction up from Systems Theory but contains all the schemas like Facet, Monad, Pattern, Form, System, OpenScape (meta-system), Domain, World, Kosmos, Pluriverse. Then to kick things off I created a hypothesis that Schemas were related to dimensions by a rule that there were two schemas per dimension and two dimensions per schema. And so there are ten schemas ranging from -1 dimension to 9 dimensions. It just so happens that String theory starts at the tenth dimension, but is unschematized, in other words we have no natural organizing template of understanding to relate to it. Schemas are projected organizations by which we understand the things in a given dimension. They are the way that we project Spacetime and find things in it intelligible given Kant’s idea of a priori synthesis.
Then the question comes why are there only ten schemas and why do they stop at the ninth dimension, and I use the fact that bounded spheres as in the example given overflow the surrounding spheres at that point which is something that goes beyond our intuitions of how space itself works. I think it works as an explanation as to why we don’t have natural models of intuition beyond the pluriverse (i.e. the multiverse). The point in my dissertation is that we use schemas as the basis of all our design activities.
So I think this fact of the overflowing of the hyperspheres of their surrounding spheres is quite important for our understanding of how we project spacetime templates of understanding as a framework for understanding dimensional phenomena.
The other point that I make in my dissertation that is related is that hyperspheres get bigger in terms of surface and volume and then they get much smaller and the dimensions where they are the biggest are at 5 through 7 dimensions. I make the point that when we say that we can hold 7+/-2 things in short term memory those are independent things, and that means that conceptually we can do design up to the ninth dimension but that the optima is in the fifth through seventh dimensions where the space of possibilities is largest. So we actually hold in our minds higher dimensional objects and we design in spaces of higher dimensions but not too high, but the optimal height is 7+/2 dimensions which is where we have the most room to maneuver the possible schemas. However in terms of manipulation it is the fourth dimension that is best because in that dimension movement has perfect laminar flow without singularities. And it is interesting that this is the dimension where the middle sphere is the same size as the surrounding spheres.
Anyway, I just thought I would mention this because it is a theoretical use of this fact about higher dimensions that we do not see referenced very often which I think has lots of implications for how we think and how we design especially in Software Engineering.