From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Fundamental region of a point group
Date: 29 Nov 1997 06:55:51 GMT
In article <6586a0$spj@mozo.cc.purdue.edu>,
Richard Kingston wrote:
>A wish to define the fundamental region of a symmetry group (specifically
>a point group).
>
>By fundamental region I mean any region that under transformation by
>the members of the group just covers the space without overlapping and
>without interstices. Crystallographers refer to this as the asymmetric
>unit of a symmetry group.
>
>I need to define the fundamental region in terms of a three-dimensional
>rotation space (not in Cartesian coordinates) i.e. I need to describe the
>bounds of the fundamental region in Eulerian or Polar angles or something
>similar ...
Is this then a discrete subgroup of an orthogonal group, and thus finite?
Is what you seek a fundamental domain for the action of that group on the
unit sphere? (A fundamental domain for the action on all of R^3 is then
found by taking the cone on this region in the plane.)
If I've thus understood the problem correctly, a simple procedure for
finding the fundamental domain is to pick one point in the sphere, and
compute the set of all its translates under the group (should then be a
finite set of points in the sphere). Then you need only look at the
region of points which are closer to the first point than to any of
its translates. (You can draw this by drawing a great circle between
the original point and each translate, to separate the points nearer to
one from the points nearer to the other.)
People do this sort of thing all the time in computer graphics; see e.g.
http://www.math.niu.edu/~rusin/known-math/index/68U05.html
and look for Voronoi cells or Delaunay decomposition. It's all geometric,
and independent of the choice of coordinate system used for the computations.
But -- there aren't that many point groups in R^3; surely there is a
crystallographers' handbook which already illustrates a fundamental
domain for each, in language better suited to your discipline? (I was
under the impression all 230 crystallographic groups were given a fixed
notation -- even a canonical ordering on some list.)
dave