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big><
span
style=
"font-weight: bold;">About HAPcryst: Three-dimensional
flat manifolds <
br>
described as quotients of polytopes<
br>
</
span></
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<
td
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. </
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
3dBieberbach:=[];<
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<
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gap> for n in [1..219] do<
br>
gap> if IsAlmostBieberbachGroup(Range(IsomorphismPcpGroup(
SpaceGroup(3,n) ))) then<
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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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<
td
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>
</
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>.)<
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