Trigonal crystal system

In crystallography, the trigonal crystal system is one of the seven crystal systems. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are associated with a lattice system. The trigonal crystal system consists of those five point groups that have a single three-fold rotation axis (see table in Crystal_system#Crystal_classes). Sometimes the term rhombohedral lattice system is used as an exact synonym, whereas it is more akin to a subset. Crystals in the rhombohedral lattice system are always in the trigonal crystal system, but some crystals such as alpha-quartz are in the trigonal crystal system but not in the rhombohedral lattice system (alpha-quartz is in the hexagonal lattice system). There are 25 space groups (143-167) whose corresponding point groups are one of the five in the trigonal crystal system, consisting of the seven space groups associated with the rhombohedral lattice system together with 18 associated with the hexagonal lattice system. The crystal structures of alpha-quartz in the previous example are described by two of those 18 space groups (152 and 154) associated with the hexagonal lattice system. To distinguish: The rhombohedral lattice system consists of the rhombohedral lattice, while the trigonal crystal system consists of the five point groups that have seven corresponding space groups associated with the rhombohedral lattice system (and 18 corresponding space groups associated with the hexagonal lattice system). An additional source of confusion is that all members of the trigonal crystal system with assigned rhombohedral lattice system (space groups 146, 148, 155, 160, 161, 166, and 167), can be represented with an equivalent hexagonal lattice with so called R-centering (rhombohedral-centering); there is a choice of using a R-centered hexagonal or a primitive rhombohedral setting for the lattice.