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.
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There are two non-Euclidean geometries: hyperbolic geometry and ellptic geometry.
Trigonal planar and tetrahedrral geometries tend to be present in polar molecules.
No, both spherical and hyperbolic geometries are noneuclidian.
Yes - except in finite geometries (projective, for example).
Hyperbolic, elliptic, projective are three possible answers.