The paper starts with examples which illustrate how to get the \(D\)-symbols \(({\mathcal D},{\mathcal M})\) of a tiling \({\mathcal T}\) in the Euclidean 3-space and its corresponding isometry group \(\Gamma\) (\(\Gamma\leq \text{Aut} {\mathcal T}\)). After this introduction the inverse problem is formulated: How to describe the possible matrix functions such that each \(D\)-symbol \(({\mathcal D},{\mathcal M})\) shall be realizable by a topological tiling \(({\mathcal T}, \Gamma)\) in a simply connected homogeneous Riemann 3-space \({\mathcal S}^3\). How to give metric realizations of \(D\)-symbols and tilings \(({\mathcal T}, \Gamma)\) (if exists). The problem is solved for 3-dimensional \(D\)-symbols up to cardinality \(|{\mathcal D}|=3\) of the vertices of \(D\)-diagram \({\mathcal D}\). The metric realizations are given in Thurston geometries. If a metric realization does not exist then the author gives the corresponding spherical or Euclidean suborbifolds and the Thurston splitting along them. The results are summarized in tables and figures. The paper consists of algorithms for the classification of \(D\)-diagrams and \(D\)-symbols in \({\mathcal S}^d\) in general case. According to the author's abstract: ''After having implemented our algorithms to computer we can proceed by the dimension \(d\) of space, by the increasing vertex numbers of \(D\)-diagrams.