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Algebraic and Combinatoric Preliminaries
Isometries and Similarities An Intuitive Approach
An Introduction to Crystallography
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abelian affine space affine subspace affine transformation algebraic Assume axis basis central dilatation Chapter characteristic choose collineation commutative completes the proof conjugacy classes conjugate contains coordinate map coordinate system corollary correlation correspondence crystallographic point group cube defined Definition dimension direct isometries elements equation euclidean space example exercise field F fixed points function G-orbit geometry half-turn hence homogeneous homogeneous coordinates homomorphism hyperplane implies induced inner automorphism integers intersection invariant involutions isomorphic kernel lattice group leaves left fixed lemma Let G linear transformation matrix midpoint n-dimensional n-space nonsingular nontrivial nonzero normal subgroup notation permutation perpendicular perspectivity pointwise fixed polynomial problem projective plane projective space proper quadric prove quadric real affine real numbers reflection representation rotation scalar sending similarity space groups space over F subgroup of G subset symmetry group tion translation unique vector space vertex vertices