Silicon is the most important material for solar cells. However, its low conversion rate/quantum efficiency requires a rather high material consumption. Thus, researchers undertake enormous experimental and theoretical efforts to find alternative Si allotropes with a better efficiency and in the ideal case a direct electronic band gap. In recent years and months many new allotropic Si structures have been reported; however, they are often described incoherently and without context to existing structures. Our approach allows a classification of many of these allotropes and a relation of their structures to substructures of, for example, known Zintl phases. For this so-called "chemi-inspired" search for promising Si structures we present a "construction kit" as a guide to introducing a large number of tetrahedral Si allotropes, all derived by a modification of the pristine cubic diamond structure. In addition, this approach allows us to realize structural (topological) relationships between experimentally accessible Zintl phases of different composition, such as open tetrahedral frameworks, to a yet unknown extent, including even known phase transitions of specific Zintl phases. In topological, structural, and computational analyses (on a DFT-PBE0/SVP level of theory) we show the close relationship of ten low-energy Si structures derived from the cubic diamond modification; five of them are new, and some show quasi-direct band gaps. A simple deviation of this approach enables the construction of many more tetrahedral structures.