Mathematics of 2-Dimensional Lattices

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-022-09601-8 Mathematics of 2-Dimensional Lattices Vitaliy Kurlin1 Received: 22 March 2022 / Revised: 19 September 2022 / Accepted: 19 October 2022 © The Author(s) 2022 Abstract A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane. Keywords Lattice · Rigid motion · Isometry · Invariant · Metric · Continuity Mathematics Subject Classification 52C05 · 52C25 · 54H12 · 11H06 1 Motivations for a Continuous Classification of Lattices and Crystals A lattice Λ ⊂ Rn consists of all integer linear combinations of basis vectors v1 , . . . , vn . This basis spans a parallelepiped called a unit cell U ⊂ Rn . A periodic point set is obtained as a union of translated copes Λ + pi for finitely many p1 , . . . , pm ∈ U . Any periodic crystal can be modelled as a periodic set whose points represent atomic centres. For example, graphene is a hexagonal periodic set of carbon atoms, see Fig. 1. The book [32] reviews non-periodic quasicrystals. Communicated by Peter Bubenik. B Vitaliy Kurlin 1 Computer Science, University of Liverpool, Liverpool, UK 123 Foundations of Computational Mathematics Since crystal structures are determined in a rigid form, the most fundamental equivalence of their underlying lattices is a rigid motion. Any rigid motion in R2 is a composition of translations and rotations. A more general isometry includes mirror reflections and is sometimes called a congruence in Euclidean geometry. In the language of Computer Science, the classification of lattices up to isometry is a binary classification problem deciding if lattices Λ, Λ are isometric, which can be denoted as Λ ∼ = Λ . If a descriptor takes different values on distinct representations of isometric lattices Λ ∼ = Λ , this pair of representations is called a false negative. Many descriptors of crystals and their lattices allow false negatives by a simple comparison of lattice bases. Any lattice can be represented by a reduced cell [18], see Definition 2.3 in Sect. 2, which is unique up to isometry, but this cell still has different bases as in Fig. 1. A descriptor without false negatives takes the same value on all isometric lattices and can be called an isometry invariant. For example, the area of the unit cell U spanned by any basis of a lattice Λ is an isometry invariant because a change of basis is realised by a 2 × 2 matrix with determinant ±1, which preserves the absolute value of the area. Such an invariant I may allow false positives Λ  Λ with I (Λ) = I (Λ ). All lattices in Fig. 1 have unit cells of the same area. The area and many other invariants allow infinitely many false positives. An invariant I without false positives is called complete and distinguishes all non-isometric lattices so that if I (Λ) = I (Λ ) then Λ ∼ = Λ . The traditional approach to deciding if lattices are isometric is to compare their conventional or reduced cells up to isometry. Though this comparison theoretically gives a complete invariant, in practice all real crystal lattices are non-isometric because of inevitable noise in measurements. All atoms vibrate above the absolute zero temperature; hence, any real lattice basis is always slightly perturbed. The discontinuity of reduced bases under perturbations was experimentally known since 1980 [2], highlighted in [16, Sect. 1] and formally proved in [41, Theorem 15]. A more practical goal is to find a complete invariant that is continuous under any perturbations of (bases of) lattices. Such a continuous and complete invariant will unambiguously parameterise the Lattice Isometry Space LIS(Rn ) consisting of infinitely many isometry classes of lattices in Rn . For example, the latitude and longitude continuously parameterise the surface of Earth. Fig. 1 Left a 2-dimensional layer of graphene is formed by carbon atoms. Right one can generate a hexagonal lattice (as any other) by infinitely many bases and continuously deform into a rectangular lattice (far right) whose bases {v1 , v2 } and {u 1 , u 2 } are related by an orientation-reversing map. The yellow Voronoi domain V (Λ) of any point p in a lattice Λ consists of all points q ∈ R2 that are non-strictly closer to p than to other points of Λ − p (Color figure online) 123 Foundations of Computational Mathematics The space LIS of isometry classes is continuous and connected because any two lattices can be joined by a continuous deformation of their bases as in Fig. 1. Such deformation can be always visualised as a continuous path in the space LIS(Rn ), whose full geometry remained unknown even for 2-dimensional lattices. Lattices were previously represented by ambiguous or reduced bases, which are discontinuous under perturbations. Most discrete invariants such as symmetry groups are also discontinuous and cut the Lattice Isometry Space (LIS) into finitely many disjoint strata. Delone [13], later Conway and Sloane [11] reduced ambiguity of lattice representations by using obtuse superbases. Hence new continuous metrics and parameterisations on lattice spaces in Problem 1.1 are the next natural step. The main contribution is a full solution to the mapping problem below. Problem 1.1 (Lattice mapping) Find a bijective and continuous invariant I : LIS(R2 ) → Inv mapping the Lattice Isometry Space to a simpler space of complete invariants. In detail, an invariant I should satisfy the following conditions. (1.1a) Invariance : if Λ ∼ = Λ then I (Λ) = I (Λ ), so I is preserved by isometry. (1.1b) Completeness (or injectivity) : if I (Λ) = I (Λ ), then Λ ∼ = Λ are isometric. (1.1c) Continuity : the invariant map I is continuous in a suitable metric d satisfying all axioms: (1) d(I (Λ), I (Λ )) = 0 if and only if Λ ∼ = Λ , (2) symmetry d(I1 , I2 ) = d(I2 , I1 ), (3) triangle inequality d(I1 , I2 ) + d(I2 , I3 ) ≥ d(I1 , I3 ). (1.1d) Computability : the metric d(I (Λ), I (Λ )) can be exactly computed in a constant time from reduced bases of Λ, Λ , introduced Definition 2.3 in Sect. 2. (1.1e) Inverse design : a basis of Λ can be explicitly reconstructed fro (...truncated)