In this paper, all the real crystal rhombododecahedra (625 in total) are enumerated and characterized by facet symbols and symmetry point groups. All the dodecahedra (34 in total) are drawn. Nine combinatorially different types are found among the garnets from Mt Makzapakhk, Kola Peninsula.

Eudialyte-group minerals (EGMs) are Na-Ca zirconosilicates typical for peralkaline plutonic rocks. In the zeolite-like crystal structure of these minerals, there are many sites of different volumes and configurations, and therefore EGMs can include up to one-third of the periodic table. Although there are preferred sites for many elements in the crystal structure of eudialyte-group minerals, the same element can appear in several sites. In addition, many sites may be partially or fully vacant. Currently, 30 mineral species are established in the eudialyte group. However, this diversity is, in fact, limited to holotype specimens. To name any mineral from the eudialyte group, you need to solve its crystal structure and compare it with holotypes. Meanwhile, the composition (and, therefore, the name) of any mineral of the eudialyte group is an excellent indicator of the composition of the mineral-forming media, which is very important to petrological and mineralogical studies. In this article, we propose a diagnostic scheme for minerals of the eudialyte group, based only on the chemical composition. The scheme includes five consecutive steps, each of which evaluates the content of a species-forming element (or the sum of such elements). This scheme can be supplemented by new members without changing its hierarchical structure.

The symmetry point groups for all combinatorially non-isomorphic 9-and 10-hedra (2606 and 32 300, respectively) are contributed in the paper for the ®rst time. The most symmetrical polyhedra of 3 to 32 automorphism group orders (50 and 187, respectively) are drawn in the Schlegel projections and characterized by the facet symbols and symmetry point groups.