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            <td style="vertical-align: top;"><a href="bieberbach.html"><small
style="color: rgb(0, 0, 102);">Previous</small></a><br>
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            <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span
style="font-weight: bold;">About HAPcryst:  Three-dimensional
flat manifolds <br>
described as quotients of polytopes<br>
            </span></big></td>
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href="aboutLie.html"><small style="color: rgb(0, 0, 102);">next</small></a><br>
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style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A
      <span style="font-style: italic;">euclidean crystallographic group</span>
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 <span style="font-style: italic;">Bieberbach</span>
if each non-trivial transformation has no fixed point. If G is
Bieberbach then the quotient M=R<sup>n</sup>/G is a flat manifold. <br>
      <br>
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.   </td>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
3dBieberbach:=[];<br>
      <br>
gap> for n in [1..219] do<br>
gap> if IsAlmostBieberbachGroup(Range(IsomorphismPcpGroup(
   SpaceGroup(3,n) ))) then<br>
gap> Add(3dBieberbach,n);<br>
gap> od;<br>
      <br>
gap> 3dBieberbach;<br>
[ 1, 4, 7, 9, 19, 33, 34, 76, 142, 165 ]<br>
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style="vertical-align: top; background-color: rgb(255, 255, 255);">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.<br>
      <br>
For example, the Bieberbach group G=SpaceGroup(3,9) admits a
permutaheral fundamental domain:<br>
      <br>
      <div style="text-align: center;"><img alt=""
src="spacegroup39.gif" style="width: 642px; height: 511px;"> <br>
      <br>
      <div style="text-align: left;">Of course, a given Bieberbach
group can admit several combinatorially different convex fundamental
domains.<br>
      </div>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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 <a
href="http://alberti.vlan.nuigalway.ie/%7Eroeder/CHA/HAPcryst/flatManifolds3d/index.shtml">here</a>.
(If you don't have Javaview installed then an html example is
given  <a href="snow2.gif">here</a>.)<br>
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href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
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href="aboutLie.html"><span style="color: rgb(0, 0, 102);">Next
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