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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
<div class="pcenter">
<h1><strong class="pkg">HAPcryst</strong> – An extension of the <strong class="pkg">GAP</strong> package <strong class="pkg">HAP</strong> for crystallographic groups</h1>
<p>( Version
0.1.8
)</p>
</div>
<p><b>Marc Röder
</b>
<br />e-mail: <span class="URL"><a href="mailto:marc_roeder(at)web.de">marc_roeder(at)web.de</a></span>
</p>
<p><b>Address:</b><br />
Marc Röder, Department of Mathematics, NUI Galway, Irleland</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2007 Marc Röder.</p>
<p>This package is distributed under the terms of the GNU General Public License version 2 or later (at your convenience). See the file <code class="file">LICENSE.txt</code> or <span class="URL"><a href="http://www.gnu.org/copyleft/gpl.html">http://www.gnu.org/copyleft/gpl.html</a></span></p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This work was supported by Marie Curie Grant No. MTKD-CT-2006-042685</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1. <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X813275957BA5B5E0">1.1 <span class="Heading">Abstract and Notation</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F4A00F481A1FB39">1.1-1 <span class="Heading">The natural action of crystallographic groups</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X85A08CF187A6D986">1.2 <span class="Heading">Requirements</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7990986A8114E0DB">1.2-1 <span class="Heading">Recommendation concerning polymake</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X7D9044767BEB1523">1.3 <span class="Heading">Global Variables</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78B0A21E7FD0F3BB">1.3-1 InfoHAPcryst</a></span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X86FBE5B77C2F9442">2. <span class="Heading">Bits and Pieces</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8019925B8294F5B4">2.1 <span class="Heading">Matrices and Vectors</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D58A1848182EC26">2.1-1 SignRat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C0552BA873515B9">2.1-2 VectorModOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7BB083A57C474F45">2.1-3 IsSquareMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78C932A48515EF10">2.1-4 DimensionSquareMat</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X86BD4FE4871379AD">2.2 <span class="Heading">Affine Matrices OnRight</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X838946957FC75C17">2.2-1 LinearPartOfAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80DD4F2286D49F8D">2.2-2 BasisChangeAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81F3C49580958FB6">2.2-3 TranslationOnRightFromVector</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X84A0B0637F269E37">2.3 <span class="Heading">Geometry</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A94DAE679AD73E3">2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X866942167802E036">2.3-2 <span class="Heading">Inequalities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80C365AA87BDDAFA">2.3-3 BisectorInequalityFromPointPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8790464D86D189F4">2.3-4 WhichSideOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83392C417B311B6B">2.3-5 RelativePositionPointAndPolygon</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7B14774981F80108">2.4 <span class="Heading">Space Groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E2F20607F278B70">2.4-1 PointGroupRepresentatives</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X7F6789767FB36E74">3. <span class="Heading">Algorithms of Orbit-Stabilizer Type</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X85CAB2EB85A6E17A">3.1 <span class="Heading">Orbit Stabilizer for Crystallographic Groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BED233684B2F811">3.1-1 OrbitStabilizerInUnitCubeOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82BD20307A67C119">3.1-2 OrbitStabilizerInUnitCubeOnRightOnSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82CCCD597C86A08A">3.1-3 OrbitPartInVertexSetsStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8022CD75819DE536">3.1-4 OrbitPartInFacesStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86A80FA17A4D6664">3.1-5 OrbitPartAndRepresentativesInFacesStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X81DE08687F0DBCC6">3.1-6 StabilizerOnSetsStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83C6448679A54F9D">3.1-7 RepresentativeActionOnRightOnSets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8002407080DB3EA2">3.1-8 <span class="Heading">Getting other orbit parts</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8751429287034F4A">3.1-9 ShiftedOrbitPart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82656DE584E8DC5D">3.1-10 TranslationsToOneCubeAroundCenter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X858C787E83A902AB">3.1-11 TranslationsToBox</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X852C41A77C759D82">4. <span class="Heading">Resolutions of Crystallographic Groups</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7F48638F817A14B0">4.1 <span class="Heading">Fundamental Domains</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79F4F7938116201E">4.1-1 FundamentalDomainStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A07AB55831F212F">4.1-2 FundamentalDomainBieberbachGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86A09AC4842827C9">4.1-3 FundamentalDomainFromGeneralPointAndOrbitPartGeometric</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X824BC99584F5F865">4.1-4 IsFundamentalDomainStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D79DD6E87BCC1DC">4.1-5 IsFundamentalDomainBieberbachGroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X78D68F6087238F97">4.2 <span class="Heading">Face Lattice and Resolution</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CA87AA478007468">4.2-1 ResolutionBieberbachGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E854FC47F9E479E">4.2-2 FaceLatticeAndBoundaryBieberbachGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79A0B28A7FAE31B9">4.2-3 ResolutionFromFLandBoundary</a></span>
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