by Paul » Wed Sep 22, 2004 12:06 pm
Hello all,
I have a vague idea of a geometric possibility for the ordering of the kD facets of a nD (regular) polytope (n > k >= 0). For now, at least, I wish to restrict our polytope to being regular.
Perhaps... again, I'm not sure this will go anywhere helpful...
Put the centroid(? not sure this has a well-established definition, but I suspect you know what I mean...) of the (regular) polytope at the origin and inscribe the polytope in an nD-hypersphere.
Then, using a Euclidean coordinate system (x,y,z,w,... p,q), as usual marking off distances on orthogonal 1D lines from the origin, choose the last coordinate, here q, and traverse the hypersurface from the maximum q of the nD-hypersphere inscribing the polytope to the maximum -q also inscribing the polytope.
Of course, the course I suggest will find us setting out a (n-1)D hypersurface on the nD-hypersphere as we traverse from q to -q.
Continue proceeding to set out the (n-1)D hypersurface traversing the nD-hypersphere until a kD facet of the polytope is intersected by the (n-1)D hypersurface.
At this point, form an (n-2)D hypersurface orthogonal(? I'm not entirely sure...) to the (n-1)D hypersurface at the point of intersection between the (n-1)D hypersurface and the kD facet of the polytope... and traverse this similar to how we are traversing the nD-hypersphere.
Continue forming and traversing (n-t)D hypersurfaces until one gets to a 2D-hypersurface, which I believe will be a circle.
Traverse the circle from the last point of next highest hypersurface intersection, in a counterclockwise direction, using the maximal coordinate (the rightmost) aribitrarily choosen ordering of coordinates in our Euclidean coordinate system (x,y,z,w,... p,q), to define a perspective to assign clockwise directions.
The first kD facet encounted will be numbered the first, or r+1, and then, the next one, the second, or r+2.
After the circle has been completely traversed, increase dimensionality by traversing the series of intersecting hypersurfaces reverse of the direction used for getting to the circle from the nD-hypersphere.
Once you've returned to the nD-hypersphere again at the original hypersurface point of intersection, continue down the nD-hypersphere, continuing to sweep out a (n-1)D hypersurface, until another point of intersection with a kD facet of the polytope is encountered,... and continue in the same fashion until the nD-hypersphere has been fully traversed.
Please... be kind. This is only a very rough idea of a concept which may not work at all. And probably still needs much greater description to be employed.
Hopefully someone here can clarify whether this method of geometric description... makes any sense. If not, might something similar make some sense...? Or, either way, perhaps someone can improve the description?
Thanks