How are the trigonal and hexagonal crystal systems different? Why are they "symmetrically" equivalent?
Gervais Chapuis, Author of more than 300 peer reviewed publications in crystallography
Indeed, the hexagonal and trigonal are two of the seven crystal systems. It is not correct to claim that the two systems are “symmetrically” equivalent.
You can always find six-, three- and two-fold rotational symmetry operations in the hexagonal system. The three- and two-fold rotational symmetry are parallel to the hexagonal axis. There is however one exception when 6bar is present. Here only 6bar axis are present.
In the trigonal system you can only find three fold rotation symmetry. Neither hexagonal nor two-fold axis are present parallel to the trigonal axis.
Below you will find a diagram indicating different types of classifications of space groups depending on the criteria.
You can notice that the crystal systems is a classification of space groups AND crystallographic point groups but NOT lattice types. Lattice systems only classify space groups AND lattice types but NOT crystallographic point groups.
The following table taken from the 2016 edition of vol. A of the International Tables of Crystallography illustrates the possible classifications which can be adopted for space groups with 3-and 6-fold symmetries
It should be noted that the classification into crystal systems, lattice systems or crystal family is the result of a strict mathematical operation based on the concept of equivalence class.
If the crystal system classification is selected, we obtain the hexagonal and trigonal cases (horizontal separation). In this classification, e.g. P6bar and P3bar are in different crystal systems.
If the lattice system is selected, we obtain the hexagonal and rhombohedral cases (vertical separation). In this classification, both P6bar and P3bar are in the same hexagonal lattice
In the family classification, all the space groups indicated in the table belongs to the same hexagonal crystal family.
It should be mentioned that the International Tables are based on the crystal system classification to classify space groups.
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