[1X[5XHAPcryst[0X â An extension of the [5XGAP[0X package [5XHAP[0X for crystallographic groups[0X ( Version 0.1.8 ) Marc Röder Marc Röder Email: [7Xmailto:marc_roeder(at)web.de[0X Address: Marc Röder, Department of Mathematics, NUI Galway, Irleland ------------------------------------------------------- [1XAbstract[0X ------------------------------------------------------- [1XCopyright[0X © 2007 Marc Röder. This package is distributed under the terms of the GNU General Public License version 2 or later (at your convenience). See the file [11XLICENSE.txt[0X or [7Xhttp://www.gnu.org/copyleft/gpl.html[0X ------------------------------------------------------- [1XAcknowledgements[0X This work was supported by Marie Curie Grant No. MTKD-CT-2006-042685 ------------------------------------------------------- [1XContent (HAPcryst)[0X 1. Introduction 1.1 Abstract and Notation 1.1-1 The natural action of crystallographic groups 1.2 Requirements 1.2-1 Recommendation concerning polymake 1.3 Global Variables 1.3-1 InfoHAPcryst 2. Bits and Pieces 2.1 Matrices and Vectors 2.1-1 SignRat 2.1-2 VectorModOne 2.1-3 IsSquareMat 2.1-4 DimensionSquareMat 2.2 Affine Matrices OnRight 2.2-1 LinearPartOfAffineMatOnRight 2.2-2 BasisChangeAffineMatOnRight 2.2-3 TranslationOnRightFromVector 2.3 Geometry 2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup 2.3-2 Inequalities 2.3-3 BisectorInequalityFromPointPair 2.3-4 WhichSideOfHyperplane 2.3-5 RelativePositionPointAndPolygon 2.4 Space Groups 2.4-1 PointGroupRepresentatives 3. Algorithms of Orbit-Stabilizer Type 3.1 Orbit Stabilizer for Crystallographic Groups 3.1-1 OrbitStabilizerInUnitCubeOnRight 3.1-2 OrbitStabilizerInUnitCubeOnRightOnSets 3.1-3 OrbitPartInVertexSetsStandardSpaceGroup 3.1-4 OrbitPartInFacesStandardSpaceGroup 3.1-5 OrbitPartAndRepresentativesInFacesStandardSpaceGroup 3.1-6 StabilizerOnSetsStandardSpaceGroup 3.1-7 RepresentativeActionOnRightOnSets 3.1-8 Getting other orbit parts 3.1-9 ShiftedOrbitPart 3.1-10 TranslationsToOneCubeAroundCenter 3.1-11 TranslationsToBox 4. Resolutions of Crystallographic Groups 4.1 Fundamental Domains 4.1-1 FundamentalDomainStandardSpaceGroup 4.1-2 FundamentalDomainBieberbachGroup 4.1-3 FundamentalDomainFromGeneralPointAndOrbitPartGeometric 4.1-4 IsFundamentalDomainStandardSpaceGroup 4.1-5 IsFundamentalDomainBieberbachGroup 4.2 Face Lattice and Resolution 4.2-1 ResolutionBieberbachGroup 4.2-2 FaceLatticeAndBoundaryBieberbachGroup 4.2-3 ResolutionFromFLandBoundary -------------------------------------------------------