Axonometric projections together with corresponding graphs for the first branch of the family of tetra-hexa-cell equator fullerenes including some isomers are constructed in the range from 20 to 60. Some of graphs were obtained earlier but the majority is given for the first time. A unified approach for drawing axonometric projections of these fullerenes is used; namely, it is the dimetric representation which symmetry coincides with that of a corresponding graph. The growth of fullerenes is studied on the basis of two mechanisms, according to which a carbon dimer embeds either in a hexagon or a pentagon of an initial fullerene. This leads to stretching and breaking the covalent bonds which are parallel to arising tensile forces. In both cases there arises a new atomic configuration and there is mass increase of two carbon atoms. We considered direct descendents of the first branch of the tetra-hexa-cell equator family beginning with C20; namely C22, C24, C26, C28, C30, C32, C34, C36, C38, C40, C42, C44, C46, C48, C50, C52, C54, C56, C58, and C60.