Anybody on board with enough high level math expertise to help flesh ideas involving geometric transformations (translations, rotations, etc.) in higher-dimensioned spaces?
The applicabilty is both to questions regarding physics and models of cognition, specifically of imagination. I can’t teach myself this stuff. I’ve tried.
oops. I meant this to go on GQ! Can I request a move?
Rotations are usually represented by orthonormal matrices. Translations, in matrix form, are handled by adding a “dummy dimension” to keep the transformation linear.
It might help if you had a more specific question. As The Ryan suggested, coordinate transformations are usually handled by matrices. Since (presumably) we’re talking about R[sup]n[/sup], all you have to do is transform the basis vectors and that’ll give you enough information to transform anything.
I saw the thread title and I had to wonder what the heck there was to debate about n-dimensional transformations. 
Off to GQ.
bibliophage
moderator GQ
I believe that orthogonal matrices with determinant 1 correspond to rotations, and orthogonal matrices with determinant -1 correspond to flips + rotations. There’s also the concept of Euler angles. I don’t know anything about these, however. But I do know that in 2 dimensions, only one angle is required to describe a pure rotation, and in 3 dimensions, you need 3. (Two to describe the axis of rotation, and one to determine how much you rotate by.) I notice that each of these numbers is n(n-1)/2, the number of independent components of an n×n antisymmetric matrix. I would speculate, but not presume, that this holds true in higher dimensions?
In 3+1 dimensional Minkowski space, you need six parameters for a full “rotation” (Lorentz transformation): Three Euler angles, and three components of velocity. I think that this should be the same number of degrees of freedom as 4 dimensional space, consistent with Achernar’s formula. Toss in the n=0 and n=1 cases (zero degrees of freedom, either way), and the formula starts looking prety good.
Well, here are the two subjects that I’d like help dealing with in a manner more rigorous than my mathematical training allows. They are issues I can visualize but not definitively express. I hope that I don’t just sound like a nutcase espousing something like Phi Theory!
This I’ve posted on before. This should be straight forward enough. I’m trying to play out the speculation that matter and antimatter are present in equal amounts but in mirror symmetric forms such that they have different projections in the three extended spatial dimensions. I’d like to model out taking some random n-dimensional shape placed in a thick brane of, say, 11D with 3 extended dimensions, and its mirror symmetric partner, and see what different slices of 3D projections of each looked like as they were rotated through the space. My hypothesis is that many possible rotations exist where each would appear to have different amounts present in different slices of the 3D projections with subtle asymmetries between them.
The other is harder to express. It a modification of an idea of Peter Gardenfors called “Conceptual Spaces: The Geometry of Thought.”
The short version of his idea (hard to condense a book to a paragraph synopsis) is to express concepts as n-dimensioned objects in a concepual space of various domains - colour, shape, frequency, etc … The nature of dimensionality varies between different domains (the colour domain is spindle shaped, for example). He places this conceptual space between neural networks and symbolic logic.
My proposed modification of his idea is to consider these conceptual objects as probabilty wavefunctions (sort of a way to effect the fuzziness of categorization) and to consider the effects of geometric transformations of such wave functions. If such a transformation is applied to a data set and finds a surprising fit then a metaphor results which leads us to make other inductions about this other dataset.
I’d love to have some help developing this speculative model in a more rigorous mathematical manner. Anyone interested can respond e-mail or by the board. Thanks.