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Chapterl MODELING SYMMETRICAL PATTERNS AND GEOMETRIES
A MATHEMATICAL DESCRIPTION OF THE GEOMETRIES OF MOLECULES
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2-fold axes a-quartz Appendix augmented matrix basis matrix basis vectors binary operation Bravais lattice calculate cartesian basis cell dimensions change of basis Chapter coesite column composition coordinates crystal crystallographic point groups crystallographic space group defined Definition denote a basis determine entries equation equivalence classes equivalence relation Example exists Figure G-equivalent group G half-turn Hence identity integer inverse invertible matrix isomorphic kyanite lattice left invariant lattice plane lattice points lattice type lattice vector Let G denote linear combination matrix representation MD(a metrical matrix monaxial normal subgroup Note obtain one-to-one oxygen atoms parallel perpendicular point isometry pole points Problem Proof quarter-turn represents respect right cosets rotation axis rotoinversion RP(TL sanidine Show Soluﬁon Solution space group types structure subgroup of G Suppose symmetry elements Theorem third-turn translation group translational vector turn angle two-fold axes unit cell unit vector vector space