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: $E^3$, $S^3$, $H^3$, $S^2\times R$, $H^2\times R$, $\widetilde{S{L}_{2}R}$, $\mathrm{Nil}$, and $\mathrm{Sol}$) 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 html-italic^>S of the above space geometries and its isometry group $\mathrm{Isom}\left(S\right)$ can be considered as a subspace of the projective 3-sphere: $S\subset P{S}^3$, where a special maximal group $G=\mathrm{Isom}\left(S\right)\subseteq \mathrm{Coll}\left(P{S}^3\right)$ of collineations acts, leaving the above subspace html-italic^>S invariant. Vice-versa, we can start with the projective geometry, namely with the classification of $\mathrm{Coll}\left(P{S}^3\right)$ through linear transforms of dual pairs of real 4-vector spaces $(V^4,V_4,R,\sim )$ = $P{S}^3$ (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.