From: jmount+@CS.CMU.EDU (John Mount)
Newsgroups: sci.math
Subject: Fundamental Tilings of the Plane
Keywords: tile
Date: 21 Jan 93 22:29:36 GMT
Article-I.D.: cs.C186HE.LC2.2
Followup-To: sci.math
Organization: Carnegie Mellon University

HI,

I was thinking about the 5 different types of crystallographic groups
(a crystallographic group is a subgroup G of the group of orientation
preserving isometries of the plane- such that there exists P, a
connected compact subset of the plane called a tile, such that G the
images of P under the action of G on the plane cover the plane exactly
once (except on a set of measure zero), two groups are of the same
type if they are conjugate) and their fundamental tilings (tiles such
that for h,g in G:
   interior(h(P)) intersect interior(g(P)) nonempty -> h = g).

I was wondering how many generators were required to present these
groups.  For four of these group I could find two generator
presentations of the groups- for the last I have not been able to find
such a presentation.

Does anyone know if such a presentation exists for the 5th group or
how to prove there isn't one (I guess it comes down to proving the
monoid over a 2 character alphabet isn't free over this group)?

I am going to represent orientation preserving isometries of the plane 
as 3x3 matrices: [  cos(a) sin(a) h ] acting on vectors of the form [ x ]
                 [ -sin(a) cos(a) v ]                               [ y ]
                 [      0      0  1 ]                               [ 1 ]

Then representatives of the five group types are:

        [ 1 0 1 ]       [ 1 0 0 ]
    a = [ 0 1 0 ]   b = [ 0 1 1 ]
        [ 0 0 1 ]       [ 0 0 1 ]

        [ 1 0 1 ]       [  0 1 0 ]
    a = [ 0 1 0 ]   b = [ -1 0 0 ]
        [ 0 0 1 ]       [  0 0 1 ]

        [ 1 0 1 ]       [        1/2  sqrt(3)/2  0 ]
    a = [ 0 1 0 ]   b = [ -sqrt(3)/2        1/2  0 ]
        [ 0 0 1 ]       [          0          0  1 ]

        [ 1 0 1 ]       [       -1/2  sqrt(3)/2  0 ]
    a = [ 0 1 0 ]   b = [ -sqrt(3)/2       -1/2  0 ]
        [ 0 0 1 ]       [          0          0  1 ]

and the one I don't know a 2 generator presentation for 

        [ 1 0 2 ]       [ 1 0 0 ]       [ -1  0 0 ]
    a = [ 0 1 0 ]   b = [ 0 1 1 ]   c = [  0 -1 0 ]
        [ 0 0 1 ]       [ 0 0 1 ]       [  0  0 1 ]
        
which, if I have it right, is the symmetry group of the following
marked tiles (ignore the aspect ratio):

------------------------------------------------------------------------------
|**    | *****|**    | *****|**    | *****|**    | *****|**    | *****|**    |
|*     |     *|*     |     *|*     |     *|*     |     *|*     |     *|*     |
|***** |    **|***** |    **|***** |    **|***** |    **|***** |    **|***** |
------------------------------------------------------------------------------
|**    | *****|**    | *****|**    | *****|**    | *****|**    | *****|**    |
|*     |     *|*     |     *|*     |     *|*     |     *|*     |     *|*     |
|***** |    **|***** |    **|***** |    **|***** |    **|***** |    **|***** |
------------------------------------------------------------------------------
|**    | *****|**    | *****|**    | *****|**    | *****|**    | *****|**    |
|*     |     *|*     |     *|*     |     *|*     |     *|*     |     *|*     |
|***** |    **|***** |    **|***** |    **|***** |    **|***** |    **|***** |
------------------------------------------------------------------------------
|**    | *****|**    | *****|**    | *****|**    | *****|**    | *****|**    |
|*     |     *|*     |     *|*     |     *|*     |     *|*     |     *|*     |
|***** |    **|***** |    **|***** |    **|***** |    **|***** |    **|***** |
------------------------------------------------------------------------------

-- 
--- It is kind of strange being in CS theory, given computers really do exist.
John Mount: jmount+@cs.cmu.edu               (412)268-6247
School of Computer Science, Carnegie Mellon University, 
5000 Forbes Ave., Pittsburgh PA 15213-3891
Newsgroups: sci.math