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Trigonal planar and tetrahedrral geometries tend to be present in polar molecules.
Five and six coordinate geometries are special because of the number of valence electrons. Five coordinate geometries have ten valence electrons while six coordinate geometries have six.
There are two non-Euclidean geometries: hyperbolic geometry and ellptic geometry.
No, both spherical and hyperbolic geometries are noneuclidian.
Yes - except in finite geometries (projective, for example).