Inversion symmetry?

1. P. Marder ever said that there is no special symmetry results in two dimensional oblique lattice. But it still possesses inversion symmetry. r-r
How to understand r-r?

2. Many book ever states that space symmetry broken by atomic displacment can bring ferroelectricity. But why this kind of displacement breaks the space symmetry?

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The definition of the inversion symmetry operator I that it transforms a vector into a different vector of same magnitude but antiparallel orentation. This can be written in many ways, e.g. I(r) = -r, or r --> -r, where r is a vector. All "naked" Bravais lattices have inversion symmetry (=they are invariant under inversion symmetry).

Special symmetry elements in 2D are mirror axes and 60, 90 or 180deg rotation symmetry.

http://en.wikipedia.org/wiki/Bravais_lattice

Electric polarization is a vector. Therefore I(P) = -P. If the crystal is invariant under inversion symmetry, then P=0 and the crystal cannot be ferroelectric.

The definition of the inversion symmetry operator I that it transforms a vector into a different vector of same magnitude but antiparallel orentation. This can be written in many ways, e.g. I(r) = -r, or r --> -r, where r is a vector. All "naked" Bravais lattices have inversion symmetry (=they are invariant under inversion symmetry).

Special symmetry elements in 2D are mirror axes and 60, 90 or 180deg rotation symmetry.

http://en.wikipedia.org/wiki/Bravais_lattice

Electric polarization is a vector. Therefore I(P) = -P. If the crystal is invariant under inversion symmetry, then P=0 and the crystal cannot be ferroelectric.
Dear Quack,