Axonometric projections together with corresponding graphs for tetra-hexa-cellequator fullerenes are constructed in the range from 48 to 72. A unified approach for drawing axonometric projections of the fullerenes was used. The process of growth of fullerenes is studied on the basis of the mechanism, according to which a carbon dimer embeds in a hexagon of an initial fullerene. As a result, there arises a new atomic configuration and there is mass increase of two carbon atoms. We obtained direct descendents of the fifth branch of tetra-hexa-cell-equator family which begins with C48; namely C50, C52, C54, C56, C58, C60, C62, C64, C66, C68, C70, and C72. Among them only three fullerenes, C48, C60 and C72 are highly symmetric. The symmetry can be easily discovered looking at their graphs. On the basis of this study, as well as previous investigations the periodic system of fullerenes is suggested.