Fig. 1

Aspects of the intertwined hierarchy of 2D-Bravais lattice types (modified after Refs. [17, 28, 29]). From the bottom to the top of this figure, the number of independent lattice parameters (most to the left, which is also the number of independent components of the metric tensors) decreases while the number of geometry/symmetry constraints (bold large font numbers most to the right) increases. The plane symmetry hierarchy of the Bravais lattice types is illustrated by the middle-left sketch. The type of Bravais lattice at the upper end of a line in this sketch is a special case (metric specialization) of the type at its lower end. Solid lines indicate ordinary subgroups in this sketch, the dashed line stands for a set of three conjugate plane symmetry subgroups. The plane symmetry groups of the Bravais lattice types (also known as the holohedries) are explicitly given by their symbol and number in Ref. [25], e.g., p2 and number 2. The two letter symbols within the nodes of this sketch are the standard abbreviations of the 2D-Bravais lattice types, e.g., mp. The order of the plane symmetry groups of the Bravais lattice types is given to the left of this sketch and corresponds to the multiplicity of the general position within these groups [25]. The middle-right sketch shows the related hierarchy of the primitive unit and sub-unit cell shapes. Short lines that are perpendicular to the basis vectors mark congruence (equal length) in this sketch. The ^♦ sign signifies the parameters of the primitive sub-unit of the conventional rectangular centered unit cell. All four primitive unit cells and the primitive sub-unit of the oc Bravais lattice possess the same area in this sketch. The number of geometric (metric and symmetry) constraints on the unit and sub-unit cells has been taken from Refs. [17, 18] and is further elaborated on in Table 1