On Maximal Homogeneous 3-Geometries and Their Visualization

The motivation for this talk and paper is related to the classification of the homogeneous simply connected maximal 3-geometries (the so-called Thurston geometries: E 3 , S 3 , H 3 , S 2 × R , H 2 × R , S L 2 R ˜ , Nil , and Sol ) and their applications in crystallography. The first author found in (Molnár 1997) (see also the more popular (Molnár et al. 2010; 2015) with co-author colleagues, together with more details) a unified projective interpretation for them in the sense of Felix Klein’s Erlangen Program: namely, each S of the above space geometries and its isometry group Isom ( S ) can be considered as a subspace of the projective 3-sphere: S ⊂ P S 3 , where a special maximal group G = Isom ( S ) ⊆ Coll ( P S 3 ) of collineations acts, leaving the above subspace S invariant. Vice-versa, we can start with the projective geometry, namely with the classification of Coll ( P S 3 ) through linear transforms of dual pairs of real 4-vector spaces ( V 4 , V 4 , R , ∼ ) = P S 3 (up to positive real multiplicative equivalence ∼) via Jordan normal forms. Then, we look for projective groups with 3 parameters, and with appropriate properties for convenient geometries described above and in this paper.