


A
euclidean crystallographic group
G is, by definition, a group of affine transformations of ndimensional
euclidean space whose subgroup of translations is free abelian of rank
n. One says that G is Bieberbach
if each nontrivial transformation has no fixed point. If G is
Bieberbach then the quotient M=R^{n}/G is a flat manifold. Tha GAP package Cryst contains the list of 219 threedimensional space groups. The following commands from the ACLIB package show that 10 of these are Bieberbach. 

gap>
3dBieberbach:=[]; gap> for n in [1..219] do gap> if IsAlmostBieberbachGroup(Range(IsomorphismPcpGroup( SpaceGroup(3,n) ))) then gap> Add(3dBieberbach,n); gap> od; gap> 3dBieberbach; [ 1, 4, 7, 9, 19, 33, 34, 76, 142, 165 ] 

A
convex polytopal fundamental domain for the action of a Bieberbach
group can be computed using the HAPcryst package (written by Marc
Röder)
and Polymake software. For the 3dimensional case these fundamental
domains can be visualized using Javaview. The corresponding flat
manifold is obtained by appropriately identifying facets of the
fundamental domain: identified faces are given identical colours. For example, the Bieberbach group G=SpaceGroup(3,9) admits a permutaheral fundamental domain: Of course, a given Bieberbach
group can admit several combinatorially different convex fundamental
domains.


Javaview
images (which can be rotated etc.) of fundamental domains and
tesselations for the 10
threedimensional Bieberbach groups have been produced by Marc
Röder and can be viewed here.
(If you don't have Javaview installed then an html example is
given here.) 

