Mathematics of 2-dimensional lattices

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 only partially resolved, but standard reductions remained discontinuous under perturbations modelling crystal vibrations. 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 a lattice deviation from a higher-symmetry neighbour.

Left: a 2-dimensional layer of graphene is formed by carbon atoms. Right: one can generate a hexagonal lattice (as any other) by infintiely many bases and continuously deform into a rectangular lattice (or any other) whose bases {v 1 , v 2 } 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 ∈ R 2 that are non-strictly closer to p than to other points of Λ − p.
lattices Λ ∼ = Λ are in different isometry classes, this pair Λ, Λ 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 [19], see Definition 2.3 in section 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 [3], highlighted in [17, section 1] and formally proved in [32,Theorem 15].
A more practically important goal is to find a complete invariant that is continuous under any perturbations of lattices. Such a continuous and complete invariant will unambiguously parameterise the Lattice Isometry Space (LIS) consisting of infinitely many isometry classes of lattices. For example, the latitude and longitude also continuously parameterise the surface of Earth.
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, whose full geometry remained unknown even for 2-dimensional lattices.
The main contribution is a full solution to the mapping problem below.  The space Inv will be the root invariant space RIS(R 2 ) of ordered triples with continuous metrics. Similar invariants will solve Problem 1.1 up to three other equivalence relations: rigid motion, similarity and orientation-preserving similarity. 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 [8], later Conway and Sloane [13] reduced ambiguity of lattice representations by using obtuse superbases. Hence new continuous metrics and other structures on lattice spaces below are the next natural step.
The inverse design in (1.1e) will raise Problem 1.1 above metric geometry to define a richer structure of a vector space on LIS. It is easy to multiply any lattice by a fixed scalar, but a sum of any two lattices is harder to define in a meaningful way independent of lattice bases. We will overcome this obstacle due to a linear structure on the root invariant space (RIS) completely solving Problem 1.1.

Main definitions and an overview of past work and new results
This section defines the main concepts and reviews past work on lattice comparisons, see the definition of an isometry and orientation in the appendix. Any point p in Euclidean space R n can be represented by the vector from the origin 0 ∈ R n to p. This vector is also denoted by p, An equal vector p can be drawn at any initial point. The Euclidean distance between points p, q ∈ R n is |p − q|. The conditions 0 ≤ c i < 1 on the coefficients c i above guarantee that the copies of unit cells U (v 1 , . . . , v n ) translated by all v ∈ Λ are disjoint and cover R n . Definition 2.2 (orientation, isometry, rigid motion, similarity) For a basis v 1 , . . . , v n of R n , the signed volume of U (v 1 , . . . , v n ) is the determinant of the n × n matrix with columns v 1 , . . . , v n . The sign of this det(v 1 , . . . , v n ) can be called an orientation of the basis v 1 , . . . , v n . An isometry is any map f : R n → R n such that |f (p) − f (q)| = |p − q| for any p, q ∈ R n . The unit cells U (v 1 , . . . , v n ) and U (f (v 1 ), . . . , f (v n )) have non-zero volumes with equal absolute values. If these volumes have equal signs, f is orientation-preserving, otherwise f is orientationreversing. Any orientation-preserving isometry f is a composition of translations and rotations, and can be included into a continuous family of isometries f t (a rigid motion), where t ∈ [0, 1], f 0 is the identity map and f 1 = f . A similarity is a composition of isometry and uniform scaling v → sv for a fixed scalar s > 0.
Any orientation-reversing isometry is a composition of a rigid motion and one reflection in a linear subspace of dimension n − 1 (a line in R 2 ).
Any lattice Λ can be generated by infinitely many bases or unit cells, see Fig. 1. A standard approach to resolve this ambiguity is to consider a reduced basis below. In R 3 , there are several ways to define a reduced basis [19]. The most commonly used is Niggli's reduced cell [26], whose 2-dimensional version is defined below.
All bases marked in Fig. 1 are reduced. The condition |v 1 · v 2 | ≤ 1 2 v 2 1 in Definition 2.3 geometrically means that v 1 , v 2 are close to being orthogonal: the projection of v 1 to v 2 is between ± 1 2 |v 2 |. The isometry conditions |v 1 | ≤ |v 2 | and − 1 2 v 2 1 ≤ v 1 · v 2 ≤ 0 in Definition 2.3 coincide with the conventional definition from [7, section 9.2.2] for type II (obtuse) cells in R 3 if we choose v 3 to be very long and orthogonal to v 1 , v 2 . If a basis v 1 , v 2 satisfies the isometry conditions above, then so do three more bases obtained from v 1 , v 2 by reflections in the lines parallel and orthogonal to v 1 . Up to rigid motion, we have maximum two mirror images of the same lattice Λ, so the two extra bases are excluded due to det(v 1 , v 2 ) > 0. If Λ is mirror-symmetric to itself due to |v 1 | = |v 2 |, one more basis is excluded by v 1 · v 2 ≥ 0. Proposition 3.10(a) proves uniqueness of a reduced basis in all cases.   Fig. 3, where the previously reduced basis v 1 , v + 2 (t) = (t, 1) becomes non-reduced for t ∈ ( 1 2 , 1] and substantially differs from the new reduced basis v 1 , v − 2 (t) = (t − 1, 1) at t = 1 2 . In general, [32,Theorem 15] proved that there is no continuous reduction of bases if we compare them coordinate-wise. Theorems 7.5, 7.7 and Corollary 7.9 will settle all (dis)continuity cases in R 2 .
The book [18] considered actions on lattices by groups with reflections. In R 3 , crystallography classifies symmetry groups into 219 classes up to affine transformations including orientation-reversing maps, more often into 230 classes when orientation is preserved as by rigid motion of real crystals. So the classification of lattices up to rigid motion is more practically important than up to isometry.
Another well-known cell of a lattice is the Voronoi domain [29], also called the Wigner-Seitz cell, Brillouin zone or Dirichlet cell. We use the word domain to avoid a confusion with a unit cell in Definition 2.1. Though the Voronoi domain can be defined for any point of a lattice, it suffices to consider only the origin 0.
The Voronoi domain of a lattice Λ is the neighbourhood V (Λ) = {p ∈ R n : |p| ≤ |p − v| for any v ∈ Λ} of the origin 0 ∈ Λ consisting of all points p that are non-strictly closer to 0 than to other then v is called a strict Voronoi vector. Fig. 4 shows how the Voronoi domain V (Λ) can be obtained as the intersection of the closed half-spaces S(0, v) = {p ∈ R n : p · v ≤ 1 2 v 2 } whose boundaries H(0, v) are bisectors between 0 and all strict Voronoi vectors v ∈ Λ. A generic lattice Λ ⊂ R 2 has a hexagonal Vronoi domain V (Λ) with six Voronoi vectors.
Any lattice is determined by its Voronoi domain by Lemma A.1 in the appendix. However, the combinatorial structure of V (Λ) is discontinuous under perturbations. Almost any perturbation of a rectangular basis in R 2 gives a nonrectangular basis generating a lattice whose Voronoi domain V (Λ) is hexagonal, not rectangular. Hence any integer-valued descriptors of V (Λ) such as the numbers of vertices or edges are always discontinuous and unsuitable for continuous quantification of similarities between arbitrary crystals or periodic point sets.
which is unique up to permutations and central symmetry. Other pictures: two pairs of obtuse superbases (related by reflection) for a rectangular lattice.
Optimal geometric matching of Voronoi domains with a shared centre led [24] to two continuous metrics (up to orientation-preserving isometry and similarity) on lattices. The minimisation over infinitely many rotations was implemented in [24] by sampling and gave approximate algorithms for these metrics. The complete invariant isoset [5] for periodic point sets in R n has a continuous metric that can be approximated [4] with a factor O(n). The metric on invariant density functions [17] required a minimisation over R, so far without approximation guarantees. Lemma 2.5 shows how to find all Voronoi vectors of any lattice Λ ⊂ R n . The doubled lattice is 2Λ = {2v : v ∈ Λ}. Vectors u, v ∈ Λ are called 2Λ-equivalent if u−v ∈ 2Λ. Then any vector v ∈ Λ generates its 2Λ-class v+2Λ = {v+2u : u ∈ Λ}, which is 2Λ translated by v and containing −v. All classes of 2Λ-equivalent vectors form the quotient space Λ/2Λ. Any 1-dimensional lattice Λ generated by a vector v has the quotient Λ/2Λ consisting of only two classes Λ and v + Λ. Appendix A includes detailed proofs of key past results such as Lemma 2.5. Fig. 4 (left), then Λ has three pairs of strict Voronoi vectors ±v 1 , ±v 2 , ±(v 1 + v 2 ). If v 1 ± v 2 have the same length, the unit cell spanned by v 1 , v 2 degenerates to a rectangle, Λ has four non-strict Voronoi vectors ±v 1 ± v 2 .
The triple of vector pairs ±v 1 , ±v 2 , ∓(v 1 + v 2 ) in Fig. 4 motivates the concept of a superbase with the extra vector v 0 = −v 1 −v 2 , which extends to any dimension n by setting v 0 = − n i=1 v n . For dimensions 2 and 3, Theorem 2.9 will prove that any lattice has an obtuse superbase of vectors whose pairwise scalar products are non-positive and are called Selling parameters [28]. For any superbase in R n , the negated parameters p ij = −v i · v j can be interpreted as conorms of lattice characters, functions χ : Λ → {±1} satisfying χ(u + v) = χ(u)χ(v)), see [13,Theorem 6]. So p ij will be defined as conorms only for an obtuse superbase below. Definition 2.6 (obtuse superbase, conorms p ij ) For any basis v 1 , . . . , v n in are the negative scalar products of the vectors above. The superbase is obtuse if all conorms p ij ≥ 0, so all angles between vectors v i , v j are non-acute for distinct indices i, j ∈ {0, 1, . . . , n}. The superbase is strict if all p ij > 0.
[13, formula (1)] has a typo initially defining p ij as exact Selling parameters, but later Theorems 3, 7, 8 use the non-negative conorms The indices of a conorm p ij are distinct and unordered. We set p ij = p ji for all i, j. For n = 1, the 1-dimensional lattice generated by a vector v 1 has the obtuse superbase consisting of the two vectors v 0 = −v 1 and v 1 , so the only conorm is the squared length of v 1 . Any superbase of R n has n(n + 1) 2 conorms p ij , for example, three conorms p 01 , p 02 , p 12 in dimension 2.
The above formulae allow us to express the conorms via vonorms as follows Lemma 2.8 will later help to prove that a lattice is uniquely determined up to isometry by an obtuse superbase, hence by its vonorms or, equivalently, conorms. By Conway and Sloane [13, section 2], any lattice Λ ⊂ R n that has an obtuse superbase is called a lattice of Voronoi's first kind. It turns out that any lattice in dimensions 2 and 3 is of Voronoi's first kind by Theorem 2.9. Theorem 2.9 (reduction to an obtuse superbase) Any lattice Λ in dimensions 2 and 3 has an obtuse superbase Conway and Sloane in [13, section 7] attempted to prove Theorem 2.9 for n = 3 by example, which is corrected in [22]. Appendix A proves Theorem 2.9 for n = 2. Finding an obtuse superbase is related to solving the shortest vector problem in a lattice. The latter problem is NP-hard [2] in R n , see the great review in [25].
The following result implies that the space of all lattices and general periodic point sets in any R n is continuous, path-connected in the language of topology. Due to Proposition 2.10, if we call any periodic structures equivalent (or similar) when they differ up to any small perturbation of points, then any two point sets become equivalent by the transitivity axiom: if Proposition 2.10 Any periodic point sets Λ+M = {v +p | v ∈ Λ, p ∈ M }, where Λ ⊂ R n is a lattice, M is a finite set (motif ) of points in a unit cell of Λ, can be deformed into each other so that coordinates of all points change continuously.
Proof Starting from any basis v 1 , . . . , v n , continuously rotate v 2 , . . . , v n to make all basis vectors pairwise orthogonal. If given periodic sets have different numbers m 1 = m 2 of points in their unit cells, we can enlarge their cells in the direction of v 1 by factors m 2 , m 1 so that both sets have the same number of m = m 1 m 2 points in their cells. If the coordinates of points p ∈ M remain constant in the moving basis v 1 , . . . , v n , they change continuously in a fixed basis of R n . Then we can continuously elongate v 1 , . . . , v n to a get a sufficiently large unit cubic cell. After two periodic point sets are put in this 'gas state' in a common large cube, continuously move all points from one motif into any other configuration without collisions. A composition of the above movements connects any periodic sets.
Any lattice Λ ⊂ R 2 with a basis v 1 , v 2 defines the positive quadratic form Q(x, y) = (xv 1 + yv 2 ) 2 = q 11 x 2 + 2q 12 xy + q 22 y 2 ≥ 0 for all x, y ∈ R, Changing the basis v 1 , v 2 is equivalent to replacing x, y by the linear combinations of the coordinates of the vector xv 1 + yv 2 in a new basis. Conversely, any positive quadratic form Q(x, y) can be written as a sum of two squares (a 1 x + b 1 y) 2 + (a 2 x + b 2 y) 2 , see [15, Theorem 2 on p. 116], and defines the lattice with the basis v 1 = (a 1 , a 2 ), v 2 = (b 1 , b 2 ).
In 1773 Lagrange [14] proved that any positive quadratic form can be rewritten so that 0 < q 11 ≤ q 22 and −q 11 ≤ 2q 12 ≤ 0. The resulting reduced basis v 1 , v 2 satisfies 0 < v 2 1 ≤ v 2 2 and −v 2 1 ≤ 2v 1 · v 2 ≤ 0 without the new special conditions in Definition 2.3. Then the lattice generated by To resolve the basis ambiguity, Delone defined the parameters [16, section 29] equal to the conorms from Definition 2.6. We use the notations p ij from [13]: The quadratic form becomes a sum of squares: The inequalities for q ij are equivalent to the simple ordering 0 ≤ p 12 ≤ p 01 ≤ p 02 , which Definition 3.1 will use to introduce a more convenient root invariant.
The isometry classification in (1.1ab) can be interpreted via group actions, see [18] and [33]. Let B n be the space of all linear bases in R n . Up to a change of basis, all lattices in R n form the n 2 -dimensional orbit space L n = B n /GL n (R), see [18, formula (1.37) on p. 34]. Up to orthogonal maps from the group O n (R), the orbit space of lattices can be identified with the cone C + (Q n ) = B n /O n (R) of positive quadratic forms, where Q n denotes the space of real symmetric n × n matrices, see The past approach to uniquely identify an intrinsic lattice (isometry class), say for n = 2, was to choose a fundamental domain of the action of GL 2 (Z) on the cone C + (Q 2 ). This choice is equivalent to a choice of a reduced basis, which can be discontinuous. Mirror reflections of any lattice Λ correspond to quadratic forms q 11 x 2 ± 2q 12 xy + q 22 y 2 that differ by a sign of q 12 . To distinguish mirror images of lattices, Definition 3.4 will introduce sign(Λ). Then continuous deformations of lattices become continuous paths in a space of invariants, see Remark 4.8.  3.10 establishes a 1-1 correspondence between obtuse superbases and reduced bases. The latter bases are common in crystallography and implemented by many fast algorithms [6]. So our lattice input will be any obtuse superbase. Fig. 5 shows logical flows from key concepts to new contributions.
The main results are complete classifications in Theorem 4.2, Corollary 4.6, and metrics on lattice invariants in Definitions 5.1, 5.4. Continuity of invariants in Theorems 7.5, 7.7 convert the Lattice Isometry Space LIS(R 2 ) into a continuously parameterised map solving Problem 1.1. Definition 6.1 extends the binary chirality to a real-valued deviation of a lattice from a higher-symmetry neighbour.
Petitjean [27] comprehensively described past approaches to quantify chirality of bounded objects such as a rigid molecule. The most rigorous approach is to use a metric between such rigid objects. However, even for the simplest case of a finite set of points, the Hausdorff-like distances between finite sets required approximate minimisations over infinitely many rotations. Definition 6.1 will introduce chiral distances for 2D lattices, which are easily computable by Propositions 6.5, 6.6.
3 Isometry invariants of an obtuse superbase of a 2-dimensional lattice Definition 3.1 introduces voforms VF and coforms CF, which are triangular cycles whose three nodes are marked by vonorms and conorms, respectively. We start from any obtuse superbase B of a lattice Λ ⊂ R 2 to define VF, CF, and a root invariant RI. Lemma 3.8(a) will justify that RI depends only on Λ, not on B.   (b) For any lattice Λ ⊂ R 2 whose Voronoi domain V (Λ) is a mirror-symmetric hexagon, assume that the x-axis is its line of symmetry. Since V (Λ) is centrally symmetric with respect to the origin 0, the y-axis is also its line of symmetry, see Fig. 7. Then Λ has the centred rectangular (non-primitive) cell with sides to get an ordered root invariant RI(B).
A lattice Λ ⊂ R n that can be mapped to itself by a mirror reflection with respect to a (n − 1)-dimensional hyperspace can be called mirror-symmetric or achiral. Since a mirror reflection of any lattice Λ ⊂ R 2 with respect to a line L ⊂ R 2 can be realised by a rotation in R 3 around L through 180 • , the term achiral sometimes applies to all 2D lattices and becomes non-trivial only for 3D lattices. This paper for 2D lattices uses the clearer adjective mirror-symmetric.  Proof The part if ⇐. Let RI(B) include a zero, which should be the first root product, say 0 = r 12 = √ −v 1 · v 2 . The vectors v 1 , v 2 of the superbase B are orthogonal and generate a rectangular lattice, which is mirror-symmetric. If RI(B) has two equal root products, say r 01 = r 02 , the conorms are also equal: p 01 = p 02 . Formulae (2.7a) imply that v 2 1 = p 01 + p 12 = p 02 + p 12 = v 2 2 . The vectors v 1 , v 2 have equal lengths and can be swapped (v 1 ↔ v 2 ) by the reflection in the bisector The part only if ⇒. If Λ(B) is mirror-symmetric, then so is its Voronoi domain V (Λ). If V (Λ) is a rectangle or a mirror-symmetric hexagon as in Fig. 7, RI(B) computed in Example 3.2 contains either a zero or two equal root products.    Proof Any isometry of an ordered obtuse superbase B preserves the lengths and scalar products of the ordered vectors, so RI(B) is unchanged. Any re-ordering of vectors of B permutes conorms. RI(B) is unique due to ordered root products.
If a lattice is mirror-symmetric, then so is its image under any rigid motion in R 2 , hence sign(B) = 0 is preserved. If B generates a non-mirror symmetric lattice, B has unique shortest vectors v 1 , v 2 . A rigid motion acts on v 1 , v 2 as a special orthogonal matrix with determinant 1, hence preserving det(v 1 , v 2 ), sign(B). Theorem 3.7 below is crucial for a complete classification of 2D lattices in Theorem 4.2 and Corollary 4.6. Theorem 3.7 highlights that mirror-symmetric lattices have more options for obtuse superbases up to rigid motion. The same rectangular lattice can have two obtuse bases with v 1 = (1, 0), v 2 = (0, ±2), which are related by reflection in the x-axis, not by rigid motion. This symmetry-related ambiguity is much harder to resolve for 3D lattices even up to isometry, see [22]. Theorem 3.7 (isometric obtuse superbases) Any lattices Λ, Λ ⊂ R 2 are isometric if and only if any obtuse superbases of Λ, Λ are isometric. If Λ, Λ are not rectangular, the same conclusion holds for rigid motion instead of isometry. Any rectangular (non-square) lattice has two obtuse superbases related by reflection.
Proof Part if (⇐): any isometry between obtuse superbases of Λ, Λ linearly extends to an isometry Λ → Λ . Part only if (⇒) means that any obtuse superbase of Λ is unique up to isometry. By Lemma 2.8 for n = 2, if a lattice Λ has a strict obtuse superbase B = {v 0 , v 1 , v 2 }, the Voronoi vectors of Λ are the pairs of opposite partial sums ±v 0 , ±v 1 , ±v 2 , see Fig. 4 (left). Hence B is uniquely determined by the strict Voronoi vectors up to a sign. So B is one of only two obtuse superbases ±{v 0 , v 1 , v 2 } related by central symmetry or rotation through 180 • around 0. Hence Λ has a unique obtuse superbase up to rigid motion.
If a superbase of Λ is non-strict, one conorm vanishes, say p 12 = 0. Then v 1 , v 2 span a rectangular unit cell and Λ has four non-strict Voronoi vectors ±v 1 ± v 2 with all possible combinations of signs. Hence Λ has four obtuse superbases  Proof (a) An obtuse superbase B of any lattice Λ is unique up to isometry by Theorem 3.7. Lemma 3.6 implies that the root invariant RI is an isometry invariant of Λ, independent of any obtuse superbase B, hence can be denoted by RI(Λ).
Since an obtuse superbase B of any non-mirror-symmetric lattice Λ is unique up to rigid motion by part (a), Lemma 3.6 implies that sign(B) and RI o (B) are invariant up to rigid motion, hence can be denoted by sign(Λ) and RI o (Λ), respectively. If Λ is mirror-symmetric, then any rigid motion preserves sign(Λ) = 0 as well as RI(Λ). So RI o (Λ) is invariant up to rigid motion for all Λ ⊂ R 2 .
Any orientation-preserving similarity is a composition of a rigid motion and a uniform scaling (or a dilation) of all vectors by a factor s > 0. This similarity preserves any symmetries of the lattice Λ and multiplies the determinant det(v 1 , v 2 ) from Definition 3.4 by s 2 > 0, hence preserving sign(Λ).
and Table 1. Right: re-ordering and re-scaling vectors of an obtuse superbase is realised by the symmetries acting on v 2 = (x, y) within the yellow region Obt, see Lemma 3.9(b).
(b) Up to similarity, any lattice Λ ⊂ R 2 with an obtuse superbase v 0 , v 1 , v 2 can be represented by up to six points (x, y) in the subregions of Obt. Swapping v 0 ↔ v 2 is realised by the reflection in the line x = − 1 2 , so v 2 = (x, y) → (−1−x, y). Swapping and re-scaling the vectors v 1 ↔ v 2 is realised by the inversion with respect to the circle Proof (a) Let B = {v 0 , v 1 , v 2 } be an obtuse superbase of Λ. Any point p ∈ Λ can be translated to the origin. Then a suitable rotation puts the basis vector v 1 along the positive x-axis so that v 1 = (s, 0) for s > 0. The uniform scaling by the factor s, maps v 1 to (1, 0). Since both vectors v 0 , v 2 have non-acute angles with v 1 , they should have non-positive x-coordinates. Since the vectors v 0 , v 2 have a non-acute angle, one of them should be in the second quadrant {x ≤ 0 < y}. Since we can swap v 0 , v 2 without affecting Λ, we can assume that v 2 = (x, y) for x ≤ 0 < y. Table 1 The sign of a lattice Λ ⊂ R 2 can be found from an obtuse superbase with v 1 = (1, 0), v 2 = (x, y), see Lemma 3.9(a), Fig. 9. If any inequality becomes equality, then sign(Λ) = 0.
k sign(Λ) conditions on v 2 = (x, y) in the k-th subregion in Fig. 9 p ij inequalities Table 2 Inequalities between conorms are interpreted in terms of endpoints ( p ij inequality condition on (x, y) subregion within the yellow region Obt in Fig. 9 p 02 ≥ 0 the right hand side vertical strip of the region Obt p 01 < p 02 x 2 + y 2 > 1 the subregion in Obt above the circle C(0; 1) Since all conorms should be non-negative, we need that 0 non-strictly above the green circle C(− 1 2 ; 1 2 ) with the centre (− 1 2 , 0) and radius 1 2 in Fig. 9. The yellow region Obt of allowed endpoints (x, y) of v 2 in Fig. 9 is bounded by the vertical lines x = 0, x = −1 and the green circle C(− 1 2 ; 1 2 ). All boundary points represent all rectangular lattices. For example, the points (x, y) = (0, 1) and (x, y) = (−1, 1) in the vertical boundaries represent the same square lattice. For (x, y) = (− 1 2 , 1 2 ) in the green circle C(− 1 2 ; 1 2 ), the vectors v 0 = (− 1 2 , − 1 2 ) and v 2 = (− 1 2 , 1 2 ) span a square unit cell with edge-length 1 √ 2 . Now we split the yellow region into three pairs of symmetric subregions according to inequalities between three conorms.
The inequalities on p ij from Table 2 justify that the region Obt splits into six subregions split by the vertical line x = − 1 2 and two circles C(−1; 0) and C(0; 1). Each subregion is defined by one of six possible orderings of the conorms p 12 , p 01 , p 02 , see the last column of Table 1. To check the signs in the second column of Table 1, notice that if p ij is a minimal conorm, the formula For example, Table 1 says that v 1 = (1, 0) and v 2 are the two shortest vectors in the cases of the first and last rows. In the first row, v 2 = (x, y) has the length |v 2 | = x 2 + y 2 > 1 = |v 1 |, hence by Definition 3.4 sign(Λ) equals the sign of det(v 1 , v 2 ) = y > 0. In the last row, v 2 = (x, y) has the length |v 2 | = x 2 + y 2 < 1 = |v 1 |, hence by Definition 3.4 sign(Λ) equals the sign of det(v 2 , v 1 ) = −y < 0. The signs in the remaining four rows of Table 1 are similarly checked.
If any of the strict inequalities above becomes equality, we get a point either on the boundary of Obt (representing all rectangular lattices in |R 2 ) or in one of the lines x = − 1 2 or the circles C(0; 1) and C(−1; 1). These internal curves contain points (x, y) representing centred rectangular lattices. For instance, the triple intersection of the internal curves at (x, y) = (− 1 2 , 2 ) represents all hexagonal lattices. All these lattices are mirror-symmetric and have sign(Λ) = 0.
(b) Any two vectors of an obtuse superbase B = {v 0 , v 1 , v 2 } can be mapped by similarity to (1,0) and (x, y). Each of the resulting six pairs (x, y) belongs to one of the six subregions marked by k = 1, 2, 3, 4, 5, 6 in the middle picture of Fig. 9. It suffices to understand the action of two plays the role of v 2 in the reflected lattice and is symmetric to v 2 = (x, y) in the vertical line x = − 1 2 within the yellow region Obt in Fig. 9 (right). When we swap v 1 = (1, 0) and v 2 = (x, y), the second vector is divided by its length |v 2 | = x 2 + y 2 . Hence the first vector v 1 maps to the vector that is parallel to v 2 = (x, y) and has the length 1/ x 2 + y 2 . This new vector v 2 = v 2 /(x 2 + y 2 ) plays the role of v 2 and is obtained from v 2 = (x, y) by the inversion with respect to the circle x 2 + y 2 = 1. The inversion keeps all points on x 2 + y 2 = 1 fixed, maps the y-axis x = 0 to itself, swaps the half-line {x = 0, y > 0} with the upper half-circle {x 2 + x + y 2 = 0, y > 0}. Compositions of the symmetry in x = − 1 2 and this inversion generate up to six images of (x, y) in the six subregions of Obt, though the point (x, y) = (− 1 2 , 2 ) representing all hexagonal lattices is fixed.
Up to isometry, any lattice Λ ⊂ R 2 has a unique reduced basis specified by the conditions of Definition 2.3.
(b) Up to rigid motion, any lattice has a unique reduced basis in Definition 2.3.
Proof (a) Up to similarity by Lemma 3.9(b), any lattice Λ has an obtuse superbase where the point (x, y) belongs to the yellow region Obt in Fig. 9. By Lemma 3.9(b) the six permutations of v 0 , v 1 , v 2 are realised by internal symmetries of Obt, so we may assume that |v 1 | ≤ |v 2 | ≤ |v 0 |. The equivalent inequalities in conorms p 12 ≤ p 01 ≤ p 02 define the 1st subregion of Obt in Fig. 9(middle), which coincides with the closure of the region Fig. 10 (left). Due to uniqueness of B up to isometry by Theorem 3.7, the position of v 2 = (x, y) ∈ Red + is unique for Λ. Up to uniform scaling and reflection y ↔ −y, the closure Red + is defined by the same conditions v 2 x ≤ 0 as a reduced basis whose uniqueness up to isometry follows now.
(b) If orientation should be preserved, Theorem 3.7 proves the uniqueness of the obtuse superbase B = {v 0 , v 1 , v 2 } from part (a) up to rigid motion for any nonrectangular lattice Λ. Cyclic permutations of v 0 , v 1 , v 2 allow us to assume that v 1 is the shortest vector. The equivalent condition on conorms says that p 02 is the largest, hence v 2 = (x, y) belongs to the first two subregions of Obt in Fig. 9.
For the first open subregion with sign(B) = +1, the conditions v 2 belongs to the interior of the right-half region in Fig. 10 It remains to consider singular cases. We include the common boundary line and the boundary round arcs of both subregions represent mirror-symmetric lattices with a unique (up to rigid motion) obtuse superbase. We exclude these boundaries from the first region, include them into the second region and shift by x → x + 1 so that the unique reduced basis v The above boundaries in Fig. 10 (left) include the blue and red points representing the basis vectors v 2 = (x, y) of the lattices Λ, Λ , respectively.
The final boundary lines x = −1 and x = 0 represent rectangular lattices with a unit cell a × b for 0 < a < b and two obtuse superbases related by reflection, not by rigid motion, for example v 1 = (a, 0), v 2 = (0, ±b). In this case there is no 1-1 correspondence between obtuse superbases and reduced bases.
The dotted arc A in Fig. 10 Indeed, all basis vectors have length 1 and the second basis can be rotated to The second basis (u 1 , u 2 ) of the same lattice Λ is related to (v 1 , v 2 ) by a reflection, but not by rigid motion. So the region Red with the excluded left boundary for x < 0 contains a unique vector v 2 = (x, y) of a reduced basis up to orientation-preserving similarity. Forgetting about the uniform scaling, we get uniqueness of a reduced basis up to rigid motion.
The region Red in Fig. 10 (left) is a fundamental domain of all bases by the action of SO(R 2 ) × R + and GL 2 (Z) in the sense that any lattice up to orientationpreserving similarity can be represented by a unique point (x, y) ∈ Red. Red or any other half-open fundamental domain of a group action suffers from discontinuity on boundary when close lattices are represented by distant bases. For each of the lattices Λ, Λ in Fig. 10, a slight perturbation of the non-reduced basis makes it reduced but distant from the initial reduced basis up to rigid motion. The discontinuity above can be resolved by identifying boundary points of of Red by the reflection x ↔ −x. Section 4 will describe a simpler way to continuously parameterise lattices up to orientation-preserving similarity in Corollary 4.6.
4 Complete classifications of 2D lattices up to isometry and similarity  Proof Assuming that a root invariant RI(Λ) is ordered as r 12 ≤ r 01 ≤ r 02 , we will build an obtuse superbase Up to rigid motion, the length |v 1 | is enough to fix the vector v 1 along the positive x-axis. The length |v 2 | and cos ∠(v 1 , v 2 ) determine the position of v 2 relative to the fixed vector v 1 up to reflection in the x-axis. Up to isometry or if sign(Λ) = 0 (when Λ is mirror-symmetric), the above options for v 2 are not important. If sign(Λ) = +1, then we choose v 2 in the upper half-plane above the x-axis so that Finally, v 0 = −v 1 − v 2 and the reconstructed ordered obtuse superbase B = {v 0 , v 1 , v 2 } is unique up to isometry and up to rigid motion by Theorem 3.7. The above classification helps prove that some other isometry invariants of lattices are also complete and continuous. By (2.7ab) the voform VF = (v 2 0 , v 2 1 , v 2 2 ) and coform CF = (p 12 , p 01 , p 02 ) are both complete if considered up to 3! permutations. The root invariant RI is a uniquely ordered version of CF and deserves its own name. The square roots r ij = √ p ij have original units of vector coordinates.
The oriented part of Theorem 4.2 didn't appear in the past to the best of our knowledge. Conway and Sloane studied 2D lattices in [13, section 6] only up to general isometry including reflections. Here is the closest formal claim from [13]. Theorem 4.2 and Lemma 4.3 imply that, after taking square roots of vonorms, the ordered lengths, say |v 1 | ≤ |v 2 | ≤ |v 0 |, form a complete invariant that should satisfy the triangle inequality |v 1 | + |v 2 | ≥ |v 0 |. This inequality is the only disadvantage of the complete invariant |v 1 | ≤ |v 2 | ≤ |v 0 | in comparison with ordered root products r 12 ≤ r 01 ≤ r 02 , which are easier to visualise in Fig. 11, 12.
Classification Theorem 4.2 says that all isometry classes of lattices Λ ⊂ R 2 are in a 1-1 correspondence with all ordered triples 0 ≤ r 12 ≤ r 01 ≤ r 02 of root products in RI(Λ). Only the smallest root product r 12 can be zero, two others r 01 ≤ r 02 should be positive, otherwise v 2 1 = r 2 12 + r 2 01 = 0 by formulae (2.7a).
We explicitly describe the set of all possible root invariants, which will be later converted into metric spaces with continuous metrics in Definitions 5.1 and 5.4.  To classify lattices up to similarity, it is convenient to scale them by the size σ(Λ) = r 12 + r 01 + r 02 . This sum is a simpler uniform measure of size than (say) the unit cell area A(Λ) from Lemma 4.1, which can be small even for long cells.
All oriented root invariants RI o (Λ) live in the doubled cone DC that is the union of two triangular cones TC ± , where we identify any two boundary points representing the same root invariant RI(Λ) with sign(Λ) = 0. The oriented projected invariant PI o (Λ) = (x, y) ± is PI(Λ) with the superscript from sign(Λ).
The set of oriented projected invariants PI o is visualised in Fig. 12 (right) as the quotient square QS obtained by gluing the quotient triangle QT + with its mirror image QT − . The boundaries of both triangles excluding the vertex (x, y) = (1, 0) are glued by the diagonal reflection (x, y) ↔ (1 − y, 1 − x). Any pair of points (x, y) ∈ QT + and (1 − y, 1 − x) ∈ QT − in Fig. 12 (right) represent mirror images of a lattice up to similarity, see Corollary 4.6. So QS is a topological sphere without a single point and will be parameterised by geographic-style coordinates in [10].
Following Fig. 6, any square lattice has a root invariant RI = (0, a, a), so its projected invariant PI = (0, 0) is at the bottom left vertex of QT in Fig. 12 (left), identified with top right vertex of QS in Fig. 12 (right). Any hexagonal lattice has a root invariant RI = (a, a, a), so its projected invariant PI = (0, 1) is at the top left vertex of QT in Fig. 12 (left), identified with bottom right vertex of QS.
By Example 3.2(a) any rectangular lattice has RI = (0, a, b) for a < b, hence its projected invariant PI = ( b−a a+b , 0) belongs to the bottom edge of QT identified with the top edge of QS. By Example 3.2(b) any lattice with a mirror-symmetric Voronoi domain has RI with 0 or two equal root products. Such lattices have a rhombic unit cell and form the centred rectangular Bravais class. Their projected invariants belong to the vertical edges and diagonal of QS in Fig. 12 (right). The companion paper [10] discusses Bravais classes of 2-dimensional lattices in detail.
In the theory of complex functions, any lattice Λ ⊂ R 2 can be considered as a subgroup of the complex plane C whose quotient C/Λ is a torus. By the Riemann mapping theorem any compact Riemann surface of genus 1 is conformally equivalent (holomorphically homeomorphic) to the quotient C/Λ for some lattice Λ, see [   or with two equal root products (centred rectangular lattices). The last conditions on RI define the boundary ∂TC of the triangular cone in Fig. 11 or, equivalently, the projected invariant PI(Λ) belongs to the boundary of QT in Fig. 12 (left).

Remark 4.8 (lattices via group actions)
Another parameterisation of the Lattice Similarity Space LSS(R 2 ) can be obtained from a fundamental domain of the action of GL 2 (Z) × R × + on the cone C + (Q 2 ) of positive quadratic forms. Recall that any lattice Λ ⊂ R 2 with a basis v 1 , v 2 defines the positive quadratic form whose positivity for all (x, y) ∈ R 2 − 0 means that q 2 12 < q 11 q 22 . The cone C + (Q 2 ) of all positive quadratic forms projects to the unit disk ξ 2 + η 2 < 1 parameterised by ξ = q 22 − q 11 q 11 + q 22 and η = −2q 12 q 11 + q 22 . Indeed, the positivity condition q 2 12 < q 11 q 22 for the form Q Λ (x, y) is equivalent to ξ 2 + η 2 < 1 in the coordinates above.
Another complete invariant is an ordered Voronoi form v 2 1 ≤ v 2 2 ≤ v 2 0 or the lengths |v 1 | ≤ |v 2 | ≤ |v 0 | of three shortest Voronoi vectors from Lemma 4.3. However, this invariant doesn't extend even to dimension n = 3 due to a 6-parameter family of pairs of non-isometric lattices Λ 1 ∼ = Λ 2 that have the same lengths of seven shortest Voronoi vectors in R 3 , see [22]. The above reasons justify the choice of homogeneous coordinates r ij , which easily extend to higher dimensions.
The projected invariant PI = (x, y) obtained from RI is preferable to the coordinates (ξ, η), which define a non-isosceles triangle, while the isosceles quotient triangle QT will lead to easier formulae for metrics in the next section. Since the metric tensor (v 1 · v 2 , v 2 1 , v 2 2 ) = (−q 12 , q 11 , q 22 ) and its 3-dimensional analogue are more familiar to crystallographers, we will rephrase key results from sections 5-6 by using these non-homogeneous cooordinates in the companion paper [10].  If (x, y) is in the interior of QT, the invariant RI defines a pair of lattices Λ ± that have opposite signs and unique (up to isometry) reduced basis vectors v 1 , v 2 with the lengths |v 1 | = r 2 12 + r 2 01 , |v 2 | = r 2 12 + r 2 02 and the anticlockwise angle Proof In Definition 4.5 the projected invariant PI(Λ) = (x, y) is obtained from the coordinates (r 12 ,r 01 ,r 02 ) of RI(Λ) satisfying the equations x =r 02 −r 01 , y = 3r 12 , r 12 +r 01 +r 02 = 1.
, see all forms in Table 3.

Metrics on spaces of lattices up to isometry, rigid motion, similarity
All lattices Λ ⊂ R 2 are uniquely represented up to isometry and similarity by their invariants RI ∈ TC and PI ∈ QT, respectively. Then any metric d on the Table 3 Various forms of the lattices computed in Example 4.10 and shown Fig. 13 and 14. triangular cone TC ⊂ R 3 or the quotient triangle QT ⊂ R 2 gives rise to a metric in Definition 5.1 on the spaces LIS and LSS, respectively. The oriented case in Definition 5.4 will be harder because of identifications on the boundary ∂TC.   Table 4 summarises metric computations for the lattices Λ 4 , Λ 6 , L 0 , L ± ∞ , which were inversely designed in Example 4.10. Proof The metric axioms for RM, PM from Definition 5.1 follow from the same axioms for an underlying metric d. Only the first axiom is non-trivial: by the first axiom for d we know that RM(Λ 1 , Λ 2 ) = d(RI(Λ 1 ), RI(Λ 2 )) = 0 if and only if RI(Λ 1 ) = RI(Λ 2 ). Now Theorem 4.2 says that RI(Λ 1 ) = RI(Λ 2 ) is equivalent to Λ 1 , Λ 2 being isometric. Corollary 4.6 classifying lattices up to similarity by projected invariants similarly justifies the first axiom for PM(Λ 1 , Λ 2 ).
Lemma 5.6 speeds up computations in the oriented case, see Example 6.8.
Lemma 5.6 (reversed signs) If lattices Λ ± 1 , Λ ± 2 ⊂ R 2 have specified signs, then Proof By Definition 5.4, for any base distance d on R 3 , when minimising over mirror-symmetric lattices Λ 3 with sign(Λ 3 ) = 0, the metrics RM o are computed for lattices that have one zero sign and one non-zero sign. Hence RM o (Λ ± 1 , Λ 3 ) can be replaced by the simpler metric RM(Λ 1 , Λ 3 ) = d(RI(Λ 1 ), RI(Λ 3 )) depending only on the unoriented root invariants RI(Λ 1 ) and RI(Λ 3 ) without a sign. After that the metric RM can lifted back to the lattices Λ − 1 , Λ + 3 with reversed signs: The proof for the projected metric PM o is similar to the above arguments. Without loss of generality one can assume that a ≤ b and a ≤ c ≤ d. Then Fig. 16 shows the graphs of y = |a − x| + |x − b|, y = |c − x| + |x − d|, y = S(x) in green, blue, red, respectively.   The triangle inequality for the Euclidean distance with q = 2 implies that Proposition 5.9 (projected metrics for q = 2, +∞) Let Λ 1 , Λ 2 be lattices with opposite signs and projected invariants PI(Λ 1 ) = (x 1 , y 1 ), PI(Λ 2 ) = (x 2 , y 2 ).
Proof (a) By Definition 5.4 PM o (Λ 1 , Λ 2 ) is the minimum value of PM(Λ 1 , Λ 3 ) + PM(Λ 2 , Λ 3 ) achieved for a mirror-symmetric lattices Λ 3 . By Lemma 4.7 the invariant PI(Λ 3 ) belongs to one of the sides of the quotient triangle QT. Let PI be the mirror image of PI(Λ 2 ) with respect to the side of QT containing PI(Λ 3 ).
In the last picture of Fig. 17, any (x 3 , y 3 ) between the triangles with rightangled vertices at (x 1 , y 1 ), (x 2 , y 2 ) gives the minimum values x 2 − x 1 and |y 2 − y 1 |. Hence x 3 = x 2 always gives the minimum value of the distance 6 Real-valued chiral distances measure asymmetry of lattices The classical concept of chirality is a binary property distinguishing mirror images of the same object such as a molecule or a periodic crystal. Continuous classifications in Theorem 4.2 and Corollary 4.6 imply that the binary chirality is discontinuous under almost any perturbations similar to other discrete invariants such as symmetry groups. To avoid arbitrary thresholds, it makes more sense to continuously quantify a deviation of a lattice from a higher-symmetry neighbour.
The term chirality often refers to 3-dimensional molecules or crystal lattices. One reason is the fact that in R 2 a reflection with respect to a line L is realised by the rotation in R 3 around L through 180 • . However, if our ambient space is only R 2 , the concepts of isometry and rigid motion differ. For example, Lemma 3.3 described root invariants of all lattices that are related to their mirror images by rigid motion. Such lattices can be called achiral. We call them mirror-symmetric.
Recall that the crystallographic point group G of a lattice Λ ⊂ R 2 containing the origin 0 consists of all symmetry operations that keep 0 and map Λ to itself. For example, any such group G includes the central symmetry with respect to 0 ∈ Λ ⊂ R 2 . If G has no other non-trivial symmetries, we get G = C 2 in Schonflies notations. All 2D lattices split into four crystal families by their point groups: oblique (C 2 ), orthorhombic (D 2 ), tetragonal or square (D 4 ) and hexagonal (D 6 ). Orthorhombic lattices split into rectangular and centred rectangular, see Fig. 12. For q = 2, +∞, the distances RC q , PC q are computed in Propositions 6.5, 6.6. {RM(Λ, Λ )} = 0 means that RI(Λ) = RI(Λ ) for some mirror-symmetric lattice Λ due to Lemma 4.7. Then Λ is isometric to Λ by Theorem 4.2 and is mirror-symmetric.
||RI(Λ)−RI(Λ )|| ∞ is minimised over mirror-symmetric lattices Λ whose root invariants by Lemma 4.7 belong to one of the three boundary sectors of the triangular cone TC. We consider them below one by one. Proposition 6.6 (chiral distances PC q for q = 2, +∞) Let a lattice Λ have a projected invariant PI(Λ) = (x, y) ∈ QT so that x ∈ [0, 1), y ∈ [0, 1], x + y ≤ 1. Hypotenuse : PI(Λ ) = (s, t) for variables s, t ≥ 0 such that s + t = 1. To compute the M ∞ distance from (x, y) to (s, t), first assume that s ≥ x, t ≥ y. Then One can check that s + t = 1 and s ≥ x, t ≥ y due to x + y ≤ 1 as (x, y) ∈ QT. Then . It remains to show that the minimum M = 1 − x − y 2 of the distance from (x, y) to (s, t) cannot have a smaller value for s ≤ x or t ≤ y.
So PC ∞ (Λ) is the minimum of the above three M ∞ distances.
Example 6.7 (distances RC q , PC q ) Table 5 shows the chiral distances computed by Propositions 6.5, 6.6 for the prominent lattices L ± 2 , L ± ∞ in Example 4.10. Table 5 Chiral distances PCq, RCq for the lattices L ± 2 , L ± ∞ in Fig. 13 and 14, see Example 6.7. Table 6 has RM o q , RM o q for q = 2, +∞ and the prominent lattices L ± 2 , L ± ∞ , which were inversely designed in Example 4.10. If lattices have the same sign, then RM o , PM o coincide with their unoriented versions by Definition 5.4. For example, PM o q (L + 2 , L + ∞ ) is the distance M q between the invariants PI(L ∞ ) = ( 1 4 , 1 4 ) and PI(L 2 ) = ( 1 Similarly, RM o q (L + 2 , L + ∞ ) is the M q distance between the root invariants PI(L ∞ ) = (1,4,7) and RI( By Lemma 6.3(b) the distance between mirror images of the same lattice equals the doubled D 2 -chiral distance. For example, Lemma 6.3(b) and Table 5 also Using the above properties, it remains to find four distances. Given  Table 6 Metrics PM o q and RM o q for the lattices given by their forms in Table 5, see Fig. 13. This section studies continuity of the bijection B → Λ(B), where an obtuse superbase B and its lattice Λ(B) are considered up to isometry, rigid motion or two types of similarity. To state continuity results, we need metrics on lattices and superbases. Up to each of the four equivalence relations, a lattice Λ will be identified with its complete invariant with a relevant metric. For example, up to isometry, the space LIS(R 2 ) is represented by root invariants RI with the root metric RM. Now we define natural metrics on obtuse superbases in any R n .  Proof Let B j = {v j0 , . . . , v jn }, j = 1, 2, 3, be any obtuse superbases in R n . The first axiom Since any isometry f preserves Euclidean distance, we get |f To prove the triangle inequality in the third axiom for SIM ∞ , let f, g ∈ O(R n ) be rotations that minimise the distances SIM ∞ (B 1 , B 2 ) = max for other spaces is identical after replacing O(R n ) with the relevant groups. Lemma 7.3 (bounds for root products) Let vectors u 1 , u 2 , v 1 , v 2 ∈ R n have a maximum length l, have non-positive scalar products u 1 · u 2 , v 1 · v 2 ≤ 0, and be δ-close in the Euclidean distance so that |u i − v i | ≤ δ for i = 1, 2. Then , it remains to prove that We estimate the scalar product |u·v| ≤ |u|·|v| by using Euclidean lengths. Then we apply the triangle inequality for scalars and replace vector lengths by l as follows: Lemma 7.4 (a lower bound of the size) If all vectors of an obtuse superbase B = {v 0 , v 1 , v 2 } of a lattice Λ ⊂ R 2 have a maximum length l, the size σ(Λ) = r 12 + r 01 + r 02 of the lattice Λ has the lower bound l ≤ σ(Λ).
Hence the bijection OSI(R 2 ) → LIS(R 2 ) is continuous in the metrics SIM ∞ and RM q .
Proof (a) One can assume that given obtuse superbases B = (v 0 , v 1 , v 2 ) and B = (u 0 , u 1 , u 2 ) satisfy |u i − v i | ≤ δ for i = 0, 1, 2 after applying a suitable isometry to B by Definition 7.1. Lemma 7.3 implies that the root products r ij = √ −v i · v j and √ −u i · u j differ by at most 2lδ for any pair (i, j) of indices. The M q -norm of the vector difference in The coordinates x =r 02 −r 01 and y = 3r 12 have an error bound that is at most three times larger than the error forr ij . In Definition 4.5 eachr ij is obtained by dividing the root product r ij by the sizes with the lower bound l ≤ σ = r 12 + r 01 + r 02 by Lemma 7.4. The above error bound √ 2lδ for r ij gives the error bound 3 2δ/l for x, y. Then PM q (Λ, Λ ) ≤ 2 1/q 3 2δ/l, q ∈ [1, +∞]. Theorem 7.5 is proved for the metrics RM q , PM q only to give explicit upper bounds. A similar argument proves continuity for any metrics RM, PM in Definition 5.1 based on a metric d satisfying d(u, v) → 0 when u → v coordinatewise. All Minkowski norms in R n are topologically equivalent due to the bounds ||v|| q ≤ ||v|| r ≤ n 1 q − 1 r ||v|| q for any 1 ≤ q ≤ r [1], hence continuity for one value of q is enough. Theorem 7.5 implies continuity of OSI o → RIS o , because closeness of superbases up to rigid motion is a stronger condition than up to isometry. Example 7.6 illustrates Theorem 7.5 and shows that the root invariant changes continuously for a deformation when a reduced basis changes discontinuously.
Up to rigid motion in R 2 , one can assume that Λ, Λ share the origin and the first vectors v 0 , u 0 lie in the positive x-axis. Let r ij , s ij be the root products of B, B , respectively. Formulae (2.7a) imply that v 2 i = r 2 ij + r 2 ik and u 2 i = s 2 ij + s 2 ik for distinct indices i, j, k ∈ {0, 1, 2}, for example if i = 0 then j = 1, k = 2.
For any given continuous transformation of root invariants from RI(Λ) to RI(Λ ), all root products have a finite upper bound M , which we use to estimate Since at least two continuously changing conorms are strictly positive to guarantee positive lengths of basis vectors by formula (2.7a), all basis vectors reconstructed by Lemma 4.1 have a minimum length a > 0. Then Since the vectors v 0 , u 0 lie in the positive horizontal axis, the lengths can be replaced by vectors: If the superbases B, B have opposite signs, apply to B the reflection with respect the fixed x-axis. To conclude that SIM o ∞ (B, B ) → 0, we show below that the basis vectors v i , u i from both superbases have close angles α i , β i measured anticlockwise from the positive x-axis for i = 1, 2. To estimate the small difference α i − β i , we first express the angles via the root products by Lemma 4.1: where j = k differ from i = 1, 2. If δ → 0, then s ij → r ij and α i − β i → 0 for all indices because all functions above are continuous for |u j |, |v j | ≥ a, j = 0, 1, 2.
We estimate the squared length of the difference by using the scalar product: where we used that | sin x| ≤ |x| for x ∈ R. The upper bound M of root products guarantees a fixed upper bound for the lengths |u i |, |v i |. If δ → 0, then |v i |−|u i | → 0 and α i − β i → 0 as proved above, so v i − u i → 0 and SIM o ∞ (B, B ) → 0. (b) Since the metric SIM ∞ from Definition 7.1 is minimised over the larger group O(R 2 ) in comparison with SO(R 2 ), we have the inequality SIM ∞ (B, B ) ≤ SIM o ∞ (B, B ), hence SIM ∞ (B, B ) → 0 as δ → 0 by part (a). Up to isometry, the obtuse superbases B, B of Λ, Λ are unique by Theorem 3.7. Since we can start with any obtuse superbase B and can also apply a reflection to B , the above convergence SIM ∞ (B, B ) → 0 for any B, B proves continuity of LIS → OSI.
(c) To extend part (a) to the similarity equivalence, we use the size σ = r 12 + r 01 + r 02 of the given superbase B to reconstruct an obtuse superbase B of Λ from PI(Λ ) with the same size σ by Proposition 4.9. By formula (4.9) the given condition PM q (Λ, Λ ) → 0 implies that RM q (Λ, Λ ) → 0, hence part (a) implies the required conclusions for the smaller metrics SSM ∞ ≤ SIM ∞ and SSM o ∞ ≤ SIM o ∞ . The same argument extends part (b) to the similarity case. Lemma 7.8 proves a non-trivial lower bound needed for Corollary 7.9 later.  One can prove that min-max distance in Lemma 7.8 satisfies metric axioms in R 3 . Corollary 7.9 shows that Theorem 7.7(b) is the strongest possible continuity in the oriented case. In R 3 , a similar discontinuity around high-symmetry lattices will be much harder to resolve for continuous invariants even up to isometry [22]. Proof For any 0 ≤ 3δ < a < b, start from any rectangular lattice with a unit cell a × b and consider the lattices Λ ± (δ) ⊂ R 2 with the obtuse superbases Notice that the vectors in both superbases are ordered anticlockwise around 0. The initial lattice Λ ± (0) has two superbases v 1 = (a, 0), v ± 2 (0) = (0, ±b), v 0 = (−a, ∓b) related by reflection, not by rigid motion, see Fig. 1 (right).
Keeping the anticlockwise order above, write the ordered coforms below.
The above coforms differ by the transposition of the first two conorms. The maximum difference of all corresponding conorms in CF(B ± (δ)) is a 2 − 2δa. If we cyclically shift CF(B − (δ)) to the left, the maximum difference becomes b 2 −a 2 +δ 2 . If we cyclically shift CF(B − (δ)) to the right, the maximum difference becomes b 2 − 2δa + δ 2 . By Lemma 7.8, the cyclic metric between the above coforms is Since the maximum length of vectors from B ± (δ) is l ≤ This lower bound shows that, for any 0 < δ < a 3 , the (unique up to rigid motion) obtuse superbases B ± (δ) of Λ ± (δ) are not close in the metric SIM o ∞ . The lattices Λ ± (δ) have RI(Λ ± (δ)) consisting of √ δa, √ a 2 − δa, √ b 2 − a 2 + δ 2 , which might need to be ordered. Since the lattices Λ ± (δ) are related by reflection, Lemma 6.3(b) computes RM(Λ + (δ), Λ − (δ)) as the double distance 2RC[D 2 ](Λ(δ)) depending only on the root invariant above without signs. For the Minkowski parameter q = +∞, Proposition 6.5(a) computes the required distance as follows: Hence the lattices Λ ± (δ) have close root invariants with RM ∞ (Λ + (δ), Λ − (δ)) → 0 as δ → 0, but their obtuse superbases have a constant lower bound for the metric SIM o ∞ independent of δ. The discontinuity conclusion holds for all q ∈ [1, +∞), because all Minkowski distances M q are topologically equivalent [1]. Corollary 7.9 should be positively interpreted in the sense that we need to study lattices up to rigid motion by their complete oriented root invariants in the continuous space LIS o (R 2 ) rather than in terms of reduced bases (or, equivalently, obtuse superbases due to Proposition 3.10b), which are inevitably discontinuous.  This section first connects the recent invariants of more general periodic point sets with the complete invariants of lattices. Then we discuss linear operations, scalar products, CAT(0) property of LIS(R 2 ) and finally describe the future work.
Below we prove that other continuous isometry invariants AMD (average minimum distances) and PDD (pointwise distance distribution) are complete for lattices, though they make sense for general periodic and finite point sets [31,32]. Other isometry invariants such as persistent homology turned out to be weaker than expected, see infinite families of sets that have identical persistence in [23].
Definition 8.1 (RSD invariant) For any lattice Λ ⊂ R n , both AMD and PDD invariants reduce to the sequence of distances (d 1 , d 1 , d 2 , d 2 , d 3 , d 3 , . . . ) from the origin 0 ∈ Λ to its k-th nearest neighbour in Λ for k ≥ 1. Since any Λ is symmetric with respect to 0, define the Reduced Sequence of Distances RSD(Λ) = (d 1 , d 2 , d 3 , . . . ) containing one distance from each pair of equal distances above.
In 1938 Delone reduced RSD(Λ) even further and considered only distinct increasing distances [16, p. 163]. He proved that the resulting weaker invariant (of only the first four distinct distances) is complete for all lattices Λ ⊂ R 2 except the two lattices Λ 6 , Λ in Fig. 20  Any quadratic form Q(x, y) is uniquely determined (up to a linear change of variables) by the sets of its values with the only exception of Q 6 = x 2 + xy + y 2 and Q = x 2 + 3y 2 corresponding to the lattices in Fig. 20, see references in [30].
For any lattice Λ ⊂ R n , the 'halved' sequence RSD contains the same information as AMD and PDD. We conjecture that PDD is complete for all finite and periodic points sets in R 2 . Proposition 8.2 implies completeness for lattices in R 2 .
In conclusion, Problem 1.1 was resolved by the new invariants RI, RI o , PI, PI o classifying all 2D lattices up to four equivalences, see a summary in Table 7.     Fig. 14 The key contributions are the easily computable metrics in Definitions 5.1,5.4, which led to continuous real-valued deviations of lattices from their higher symmetry neighbours. The chiral distances in Definition 6.1 continuously extend the classical binary chirality and have explicit formulae in Propositions 6.5,6.6.
The discontinuity of reduction in [32,Theorem 15] was proved with a simple metric on bases without isometry. When we consider obtuse superbases up to isometry, continuity holds in Theorem 7.7 without orientation. If orientation should be preserved, Corollary 7.9 proves discontinuity at any rectangular lattice in R 2 .
The structures in Remark 8.3 help treat lattices as vectors in a meaningful way (independent of a basis), for example, as inputs or outputs in machine learning algorithms. Future work [22,11] extends key results to 3D lattices. The author thanks any reviewers for their valuable time and helpful suggestions in advance. is minimal if all even c i are replaced by 0 and all odd c i are replaced by 1. The resulting shortest vectors with coefficients 0, 1 are all 2 n − 1 symmetric pairs of partial sums ±v S for a proper subset S ⊂ {0, 1, . . . , n}. If all conorms p ij > 0, to guarantee a minimum value of N (v), every difference |c i − c j | should be 0 or 1, hence there are no other Voronoi vectors apart from the partial sums above.
To prove that all conorms eventually become non-negative, note that every reduction can make superbase vectors only shorter, but not shorter than a minimum distance between points of Λ. The angle between v i , v j can have only finitely many values when lengths of v i , v j are bounded. Then the scalar product ε = v i · v j > 0 cannot converge to 0. Since every reduction makes one superbase vectors shorter by a positive constant, the reductions will finish in finitely many steps.