Rotate normal vector onto axis plane in 4D

Question

The problem I need to solve is to rotate a 4-simplex given in 4D on the hyperplane with normal vector (1, 1, 1, 1) so that I can draw it in 3D. For instance I need to know the rotation for the regular one having vertices e_i (that is the coordinate vectors), and all its sub-simplices after division.

In order to understand the problem, let's go one dimension back. If you have a 3-simplex in 3D on the hyperplane with normal vector (1, 1, 1) like here (http://upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/150px-2D-simplex.svg.png), one can follow the idea of Nosredna to the question

Rotate normal vector onto axis plane

It works fine in 3D, but in 4D there is no cross products, so I cannot extend this answer to my question. On the other hand using rotation matrices I managed to rotate the simplex around the x axes by -45 degree, then rotating around the y axes by around 35 degree (atan(sqrt(2)/2) using the coordinate rotation matrices (http:// upload.wikimedia.org/math/2/8/5/2851c9dc2031127e6dacfb84b96446d8.png).

I also tried to calculate a rotation matrix from axes rotations like in http://ken-soft.com/2009/01/08/graph4d-rotation4d-project-to-2d/ but I could not find out what should be the angles to use. So I used R=rotXU*rotYU*rotZU with the angles pi/4, -atan(sqrt(2)/2, and -pi/6, which looked good, but somehow the result wasn't ok.

Sorry, I could not put the images directly as I'm a newbie...

Thank you for any answer!

Answer 1

There are no rotation axes in 4D, for the same reason there is no cross product: the group of 4D rotations is 6-dimentional, and the space you are rotating is 4-dimentional. Imagine, for example, a simultaneous rotation in XY plane and ZT plane: it has no non-zero stationary vectors, and therefore no axes.

The most appropriate thing to do is to work with the usual transormation matrix, which is applicable in any dimention N:

[ a11 ... a1N d1 ]
...
[ aN1 ... aNN dN ]
[ 0 ...   0   1  ]

Here d1 ... dN represent translations, and the NxN submatrix aIJ represent rotations, dilations, projections, and mirroring. To limit to rotations only make this matrix orthogonal: its product with its own transpose should be the unit matrix. This is a common practice for N=2 and N=3, you just do the same for N=4.

To find the appropriate rotation matrix in your case write down add the requirement that the entire 4th row of an orthogonal 4x4 matrix is zero, and that would give you a bunch of solutions, each being an acceptable answer to your question.