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Symmetry in one two and three dimensions
Symmetry in four dimensions
A notation for the fourdimensional crystal classes
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addition angles application appropriate arise arrow axes axial axis centre Chapter circle classes coincident combination complete component coordinates corresponding crypto crypto-rotation plane crystal classes defined degrees of freedom derived dimensional direction effect enantiomorphic equal equator equivalent example explicit Family family contains figure follows four-dimensional four-fold geometrical given hand hexagonal hyperplane hyperstereogram identity indicated inside intersects involved leads m-hyperplanes marked matrix mirror mirror plane monoclinic moving multiplicity necessary negative points noted occur opposite orientation orthogonal pairs parameters particular perpendicular places position possible present primitive projection reflection regarded relationships repeated representation represented respectively result rotation planes rotation-inversion round scale seen sequence shown in Fig space specified sphere stereogram symbol symmetrically equivalent symmetry elements symmetry operation three dimensions three-dimensional transformed two-dimensional two-fold rotation planes unique vector VIII operation