About HAPcryst:  Three-dimensional flat manifolds
described as quotients of polytopes

A euclidean crystallographic group G is, by definition, a group of affine transformations of n-dimensional euclidean space whose subgroup of translations is free abelian of rank n. One says that G is Bieberbach if each non-trivial transformation has no fixed point. If G is Bieberbach then the quotient M=Rn/G is a flat manifold.

Tha GAP package Cryst contains the list of 219 three-dimensional 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 3-dimensional 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 three-dimensional 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.)
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