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# HAPcryst – An extension of the GAP package HAP for crystallographic groups

( Version 0.1.11 )

Marc Röder
Email: marc_roeder(at)web.de

Address:
Marc Röder, Department of Mathematics, NUI Galway, Irleland

### Abstract

### Copyright

© 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 `LICENSE.txt` or http://www.gnu.org/copyleft/gpl.html

### Acknowledgements

This work was supported by Marie Curie Grant No. MTKD-CT-2006-042685

### Contents

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
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