Table 1 Aspects of the Bravais lattice types of the Euclidean plane in their crystallographic standard [25] settings

Bravais lattice types’ names and standard abbreviation letters [25] Parameters of the conventional unit cells Limiting cases of lattice parameters that lead from a lower symmetric Bravais lattice type to its higher symmetric counterpart(s), also known as metric specializations in 2D Conventional basis vectors (a′, b′) for limiting cases [29] Number of geometric or symmetry constraints that enter the G-AIC for the assignment of Bravais lattice types to sets of lattice parameters [17]
Hexagonal, hp a = b, γ = 120° None as top of a hierarchy branch is reached None as there is no such case Four, e.g., opposite sides are parallel, diagonals are orthogonal and posses ratios of $$\sqrt 3$$ or $$\sqrt 3^{ - 1}$$ (equivalent to lattice defining angle 120°)
Square, tp a = b, γ = 90° None as top of a hierarchy branch is reached None as there is no such case Four, e.g., opposite sides are parallel, adjacent sides are orthogonal, and diagonals are both orthogonal and of equal length (equivalent to lattice defining angle 90°)
Rectangular centered, oc a < b ≠ a$$\sqrt 3$$,
γ = 90°
For primitive sub-unit cell:
a = b, γ ≠ 60° or 90°
a = b → tp
a$$\sqrt 3$$ = b → hp
For primitive sub-unit cell:
γ = 60° → hp
γ = 90° → tp
(a + b)/2, (b − a)/2 in both cases
For primitive sub-unit cell:
a, b in both cases
Three, e.g., opposite sides are parallel and adjacent sides are orthogonal while conventional unit cells encompass 2 lattice points
For primitive sub-unit cell: opposite sides are parallel and diagonals are orthogonal but of different lengths so that their ratio is never 1, $$\sqrt 3$$, or $$\sqrt 3^{ - 1}$$
Rectangular (primitive), op a < b, γ = 90° a = b → tp a, b Three, e.g., opposite sides are parallel and adjacent sides are orthogonal
Oblique, mp − 2b cos γ < a < b, 90° < γ γ = 90° → op
− 2b cos γ = a → oc
a = b → oc
a, b
a, 2b + a (a′$$\sqrt 3$$ < b′)
a + b, b  a (a′$$\sqrt 3$$ > b′)
Two, e.g., (two) opposite sides are parallel
1. This setting arises from a projection through a crystal along its third dimension and is employing a right-handed coordinate system so that the c (or z) axis vector points into the page away from the reader. Vectors are given in bold face font and their magnitudes are in an ordinary font. The third column of this table lists the limiting cases (physical degeneracies) that are pertinent to the metric hierarchy of the Bravais lattice types as shown graphically in the middle-left sketch of Fig. 1. The sign refers to the primitive sublattice of the rectangular centered Bravais lattice