Problem 1.1

Definition 2.1

(A lattice \(\varLambda \), a primitive unit cell \(U(v_1,\dots ,v_n)\)) Let vectors \(v_1,\ldots ,v_n\) form a linear basis in \({\mathbb {R}}^n\) so that any vector \(v\in {\mathbb {R}}^n\) can be written as \(v=\sum \nolimits _{i=1}^n c_i v_i\) for some real \(c_i\), and if \(\sum \nolimits _{i=1}^n c_i v_i=0\) then all \(c_i=0\). A lattice \(\varLambda \) in \({\mathbb {R}}^n\) consists of \(\sum \nolimits _{i=1}^n c_i v_i\) with integer coefficients \(c_i\in {\mathbb {Z}}\). The parallelepiped \(U(v_1,\ldots ,v_n)=\left\{ \sum \nolimits _{i=1}^n c_i v_i \,:\,c_i\in [0,1) \right\} \) is called a primitive unit cell of \(\varLambda \). \(\square \)

Definition 2.2

(Orientation, isometry, rigid motion, similarity) For a basis \(v_1,\dots ,v_n\) of \({\mathbb {R}}^n\), the signed volume of \(U(v_1,\dots ,v_n)\) is the determinant of the \(n\times n\) matrix with columns \(v_1,\dots ,v_n\). The sign of this \(\det (v_1,\dots ,v_n)\) can be called an orientation of the basis \(v_1,\dots ,v_n\). An isometry is any map \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) such that \(|f(p)-f(q)|=|p-q|\) for any \(p,q\in {\mathbb {R}}^n\). The unit cells \(U(v_1,\dots ,v_n)\) and \(U(f(v_1),\dots ,f(v_n))\) have nonzero volumes with equal absolute values. If these volumes have equal signs, f is orientation-preserving; otherwise, f is orientation-reversing. 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 \(t\in [0,1]\), \(f_0\) is the identity map and \(f_1=f\). A similarity is a composition of isometry and uniform scaling \(v\mapsto sv\) for a fixed scalar \(s>0\). \(\square \)

Definition 2.3

(Reduced cell) For a lattice \(\varLambda \subset {\mathbb {R}}^2\) up to isometry, a basis and its unit cell \(U(v_1,v_2)\) are reduced (non-acute) if \(|v_1|\le |v_2|\) and \(-\frac{1}{2}v_1^2\le v_1\cdot v_2\le 0\). Up to rigid motion, the conditions are weaker: \(|v_1|\le |v_2|\) and \(|v_1\cdot v_2|\le \frac{1}{2}v_1^2\), \(\det (v_1,v_2)>0\), and the new special condition : if \(|v_1|=|v_2|\) then \(v_1\cdot v_2\ge 0\). \(\square \)

Definition 2.4

(Voronoi domain \(V(\varLambda )\)) The Voronoi domain of a lattice \(\varLambda \) is the neighbourhood \(V(\varLambda )=\{p\in {\mathbb {R}}^n: |p|\le |p-v| \text { for any }v\in \varLambda \}\) of the origin \(0\in \varLambda \) consisting of all points p that are non-strictly closer to 0 than to other points \(v\in \varLambda \). A vector \(v\in \varLambda \) is called a Voronoi vector if the bisector hyperspace \(H(0,v)=\{p\in {\mathbb {R}}^n \,:\,p\cdot v=\frac{1}{2}v^2\}\) between 0 and v intersects \(V(\varLambda )\). If \(V(\varLambda )\cap H(0,v)\) is an \((n-1)\)-dimensional face of \(V(\varLambda )\), then v is called a strict Voronoi vector. \(\square \)

Lemma 2.5

(Criterion for Voronoi vectors [25, 11, Theorem 2]) For any lattice \(\varLambda \subset {\mathbb {R}}^n\), a nonzero vector \(v\in \varLambda \) is a Voronoi vector of \(\varLambda \) if and only if v is a shortest vector in its \(2\varLambda \)-class \(v+2\varLambda \). Also, v is a strict Voronoi vector if and only if \(\pm v\) are the only shortest vectors in the \(2\varLambda \)-class \(v+2\varLambda \). \(\square \)

Definition 2.6

(Obtuse superbase, conorms \(p_{ij}\)) For any basis \(v_1,\ldots ,v_n\) in \({\mathbb {R}}^n\), the superbase \(v_0,v_1,\dots ,v_n\) includes the vector \(v_0=-\sum \nolimits _{i=1}^n v_i\). The conorms \(p_{ij}=-v_i\cdot v_j\) are the negative scalar products of the vectors above. The superbase is obtuse if all conorms \(p_{ij}\ge 0\), so all angles between vectors \(v_i,v_j\) are non-acute for distinct indices \(i,j\in \{0,1,\dots ,n\}\). The superbase is strict if all \(p_{ij}>0\). \(\square \)

Definition 2.7

(Partial sums \(v_S\), vonorms \(v_S^2\)) Let a lattice \(\varLambda \subset {\mathbb {R}}^n\) have a superbase \(B=\{v_0,v_1,\dots ,v_n\}\). For any proper subset \(S\subset \{0,1,\dots ,n\}\) of indices, consider its complement \({\bar{S}}=\{0,1,\ldots ,n\}\setminus S\) and the partial sum \(v_S=\sum \nolimits _{i\in S} v_i\) whose squared lengths \(v_S^2\) are called the vonorms of B and can be expressed as \(v_S^2=(\sum \nolimits _{i\in S} v_i)(-\sum \nolimits _{j\in \bar{S}}v_j)=-\sum \nolimits _{i\in S,j\in {{\bar{S}}}}v_{j}\cdot v_j=\sum \nolimits _{i\in S,j\in {{\bar{S}}}}p_{ij}\). For \(n=2\), we get

The above formulae allow us to express the conorms via vonorms as follows

So \(p_{ij}=\dfrac{1}{2}(v_i^2+v_j^2-v_k^2)\) for distinct \(i,j\in \{0,1,2\}\) and \(k=\{0,1,2\}-\{i,j\}\). \(\square \)

Lemma 2.8

(Voronoi vectors \(v_S\) [11, Theorem 3]) For any obtuse superbase \(v_0,v_1,\dots ,v_n\) of a lattice, all partial sums \(v_S\) from Definition 2.7 split into \(2^n-1\) symmetric pairs \(v_S=-v_{{{\bar{S}}}}\), which are Voronoi vectors representing distinct \(2\varLambda \)-classes in \(\varLambda /2\varLambda \). All Voronoi vectors \(v_S\) are strict if and only if all \(p_{ij}>0\). \(\square \)

Theorem 2.9

(Reduction to an obtuse superbase) Any lattice \(\varLambda \) in dimensions 2 and 3 has an obtuse superbase \(\{v_0,v_1,\dots ,v_n\}\) so that \(v_0=-\sum \nolimits _{i=1}^n v_i\) and all conorms \(p_{ij}=-v_i\cdot v_j\ge 0\) for all distinct indices \(i,j\in \{0,1,\dots ,n\}\). \(\square \)

Proposition 2.10

Any periodic point sets \(\varLambda +M=\{v+p \mid v\in \varLambda ,\, p\in M\}\), where \(\varLambda \subset {\mathbb {R}}^n\) is a lattice, M is a finite set (motif) of points in a unit cell of \(\varLambda \), can be deformed into each other so that coordinates of all points change continuously. \(\square \)

Proof

Starting from any basis \(v_1,\dots ,v_n\), continuously rotate \(v_2,\dots ,v_n\) to make all basis vectors pairwise orthogonal. If given periodic sets have different numbers \(m_1\ne m_2\) 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 \(m=m_1m_2\) points in their cells. If the coordinates of points \(p\in M\) remain constant in the moving basis \(v_1,\dots ,v_n\), they change continuously in a fixed basis of \({\mathbb {R}}^n\). Then we can continuously elongate \(v_1,\dots ,v_n\) 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. \(\square \)

Definition 3.1

(Voform \(\textrm{VF}\), coform \(\textrm{CF}\), ordered root invariant \(\textrm{RI}\)) For any ordered obtuse superbase B in \({\mathbb {R}}^2\), the voform \(\textrm{VF}(B)\) is the cycle on three nodes marked by the vonorms \(v_0^2,v_1^2,v_2^2\), see Fig. 6. The coform \(\textrm{CF}(B)\) is the cycle on three nodes marked by the conorms \(p_{12},p_{02},p_{01}\). Since all conorms \(p_{ij}\ge 0\), we can define the root products \(r_{ij}=\sqrt{p_{ij}}\). The root invariant \(\textrm{RI}(B)\) is obtained by writing the three root products \(r_{12},r_{01},r_{02}\) in the increasing order. \(\square \)

Example 3.2

(a) A lattice \(\varLambda \) with a rectangular cell of sides \(a\le b\) has an obtuse superbase B with \(v_1=(a,0)\), \(v_2=(0,b)\), \(v_0=(-a,-b)\), and \(\textrm{RI}(B)=(0,a,b)\).

(b) For any lattice \(\varLambda \subset {\mathbb {R}}^2\) whose Voronoi domain \(V(\varLambda )\) is a mirror-symmetric hexagon, assume that the x-axis is its line of symmetry. Since \(V(\varLambda )\) is centrally symmetric with respect to the origin 0, the y-axis is also its line of symmetry, see Fig. 7. Then \(\varLambda \) has the centred rectangular (non-primitive) cell with sides \(2a\le 2b\). The obtuse superbase B with \(v_1=(2a,0)\), \(v_2=(-a,b)\), \(v_0=(-a,-b)\) has \(\textrm{RI}(B)=(a\sqrt{2},a\sqrt{2},\sqrt{b^2-a^2})\) for \(b\ge a\sqrt{3}\). For \(a\le b<a\sqrt{3}\), we should swap \(r_{02}=\sqrt{b^2-a^2}\) with \(r_{12}=a\sqrt{2}\) to get an ordered root invariant \(\textrm{RI}(B)\). \(\square \)

Lemma 3.3

Proof

The part if \(\Leftarrow \). Let \(\textrm{RI}(B)\) include a zero, which should be the first root product, say \(0=r_{12}=\sqrt{-v_1\cdot v_2}\). The vectors \(v_1,v_2\) of the superbase B are orthogonal and generate a rectangular lattice, which is mirror-symmetric. If \(\textrm{RI}(B)\) has two equal root products, say \(r_{01}=r_{02}\), the conorms are also equal: \(p_{01}=p_{02}\). Formulae (2.7a) imply that \(v_1^2=p_{01}+p_{12}=p_{02}+p_{12}=v_2^2\). The vectors \(v_1,v_2\) have equal lengths and can be swapped (\(v_1\leftrightarrow v_2\)) by the reflection in the bisector L between \(v_1,v_2\), which preserves \(v_0=-v_1-v_2\), so \(\varLambda (B)\) is mirror-symmetric.

The part only if \(\Rightarrow \). If \(\varLambda (B)\) is mirror-symmetric, then so is its Voronoi domain \(V(\varLambda )\). If \(V(\varLambda )\) is a rectangle or a mirror-symmetric hexagon as in Fig. 7, \(\textrm{RI}(B)\) computed in Example 3.2 contains either a zero or two equal root products. \(\square \)

Definition 3.4

(\(\textrm{sign}(B)\), the oriented root invariant \(\textrm{RI}^o(B)\)) If an obtuse superbase B generates a mirror-symmetric lattice, set \(\textrm{sign}(B)=0\). Else all vectors of B have different lengths and angles not equal to \(90^{\circ }\) by Lemma 3.3. Let \(v_1,v_2\) be the shortest vectors of B so that \(|v_1|<|v_2|\). Then \(\textrm{sign}(B)=\pm 1\) is the sign of the determinant \(\det (v_1,v_2)\) of the matrix with the columns \(v_1,v_2\). The oriented root invariant \(\textrm{RI}^o(B)\) is obtained by adding \(\textrm{sign}(B)\) as a superscript to \(\textrm{RI}(B)\). \(\square \)

Example 3.5

(Signs of lattices) The lattice \(\varLambda ^+\) in the first picture of Fig. 8 has the obtuse superbase B with \(v_1=(3,0)\), \(v_2=(-1,3)\), \(v_0=(-2,-3)\) of lengths \(3,\sqrt{10},\sqrt{13}\), respectively, so \(\varLambda ^+\) is not mirror-symmetric. Since \(v_1,v_2\) are the two shortest vectors of \(B^+\) and \(\det (v_1,v_2)=\det \left( \begin{array}{ccc} 3 &{} -1 \\ 0 &{} 3 \end{array}\right) >0\), we get \(\textrm{sign}(B^+)=+1\). The anticlockwise angle is \(\angle (v_1,v_2)=180^\circ -\arcsin \frac{3}{\sqrt{10}}\approx 108^\circ \).

The lattice \(\varLambda ^-\) in the last picture of Fig. 8 is obtained from \(\varLambda ^+\) by a mirror reflection and has the obtuse superbase \(B^-\) with \(u_1=v_1\), \(u_2=(-2,3)\), \(u_0=(-1,-3)\) of lengths \(3,\sqrt{13},\sqrt{10}\), respectively, so \(\varLambda ^-\) is not mirror-symmetric. Since \(u_1,u_0\) are the shortest vectors, \(\det (u_1,u_0)=\det \left( \begin{array}{ccc} 3 &{} -1 \\ 0 &{} -3 \end{array}\right) <0\), we get \(\textrm{sign}(B^-)=-1\). The anticlockwise angle is \(\angle (u_1,u_0)=\arcsin \frac{3}{\sqrt{10}}-180^\circ \approx -108^\circ \). \(\square \)

Lemma 3.6

(\(\textrm{RI}\) invariance) For an unordered obtuse superbase B in \({\mathbb {R}}^2\), any isometry preserves \(\textrm{RI}(B)\). Any rigid motion preserves \(\textrm{sign}(B)\) and \(\textrm{RI}^o(B)\). \(\square \)

Proof

Any isometry of an ordered obtuse superbase B preserves the lengths and scalar products of the ordered vectors, so \(\textrm{RI}(B)\) is unchanged. Any re-ordering of vectors of B permutes conorms. \(\textrm{RI}(B)\) is unique due to ordered root products.

If a lattice is mirror-symmetric, then so is its image under any rigid motion in \({\mathbb {R}}^2\), hence \(\textrm{sign}(B)=0\) 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 \(\det (v_1,v_2)\), \(\textrm{sign}(B)\). \(\square \)

Theorem 3.7

(Isometric obtuse superbases) Any lattices \(\varLambda ,\varLambda '\subset {\mathbb {R}}^2\) are isometric if and only if any obtuse superbases of \(\varLambda ,\varLambda '\) are isometric. If \(\varLambda ,\varLambda '\) 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. \(\square \)

Proof

Part if (\(\Leftarrow \)): any isometry between obtuse superbases of \(\varLambda ,\varLambda '\) linearly extends to an isometry \(\varLambda \rightarrow \varLambda '\). Part only if (\(\Rightarrow \)) means that any obtuse superbase of \(\varLambda \) is unique up to isometry. By Lemma 2.8 for \(n=2\), if a lattice \(\varLambda \) has a strict obtuse superbase \(B=\{v_0,v_1,v_2\}\), the Voronoi vectors of \(\varLambda \) are the pairs of opposite partial sums \(\pm v_0,\pm v_1,\pm v_2\), see Fig. 3 (left). Hence B is uniquely determined by the strict Voronoi vectors up to a sign. So B is one of only two obtuse superbases \(\pm \{v_0,v_1,v_2\}\) related by central symmetry or rotation through \(180^{\circ }\) around 0. Hence \(\varLambda \) has a unique obtuse superbase up to rigid motion.

If a superbase of \(\varLambda \) is non-strict, one conorm vanishes, say \(p_{12}=0\). Then \(v_1,v_2\) span a rectangular unit cell and \(\varLambda \) has four non-strict Voronoi vectors \(\pm v_1\pm v_2\) with all possible combinations of signs. Hence \(\varLambda \) has four obtuse superbases \(\{v_1,v_2,-v_1-v_2\}\), \(\{-v_1,-v_2,v_1+v_2\}\), \(\{-v_1,v_2,v_1-v_2\}\), \(\{v_1,-v_2,v_2-v_1\}\), see Fig. 3. The first two (and the last two) superbases are obtained from each other by rotation through \(180^\circ \) around the origin. Unless the lattice is square, the resulting two classes of superbases are related by reflection, not by rigid motion. \(\square \)

Lemma 3.8

Proof

(a) An obtuse superbase B of any lattice \(\varLambda \) is unique up to isometry by Theorem 3.7. Lemma 3.6 implies that the root invariant \(\textrm{RI}\) is an isometry invariant of \(\varLambda \), independent of any obtuse superbase B, hence can be denoted by \(\textrm{RI}(\varLambda )\).

Since an obtuse superbase B of any non-mirror-symmetric lattice \(\varLambda \) is unique up to rigid motion by part (a), Lemma 3.6 implies that \(\textrm{sign}(B)\) and \(\textrm{RI}^o(B)\) are invariant up to rigid motion and hence can be denoted by \(\textrm{sign}(\varLambda )\) and \(\textrm{RI}^o(\varLambda )\), respectively. If \(\varLambda \) is mirror-symmetric, then any rigid motion preserves \(\textrm{sign}(\varLambda )=0\) as well as \(\textrm{RI}(\varLambda )\). So \(\textrm{RI}^o(\varLambda )\) is invariant up to rigid motion for all \(\varLambda \subset {\mathbb {R}}^2\).

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 \(\varLambda \) and multiplies the determinant \(\det (v_1,v_2)\) from Definition 3.4 by \(s^2>0\), hence preserving \(\textrm{sign}(\varLambda )\).

(b) By Definition 3.4 any mirror-symmetric lattice has \(\textrm{sign}(\varLambda )=0\). Any basis \(v_1,v_2\) of a non-mirror symmetric lattice \(\varLambda \) has \(\det (v_1,v_2)\ne 0\), so \(\textrm{sign}(\varLambda )\ne 0\). \(\square \)

Lemma 3.9

(Geometry of signs) (a) Up to orientation-preserving similarity, any lattice \(\varLambda \subset {\mathbb {R}}^2\) has an obtuse superbase with \(v_1=(1,0)\) and \(v_2=(x,y)\), where (x, y) belongs to the region \(\textrm{Obt}=\{-1\le x\le 0<y,\; x^2+x+y^2\ge 0\}\). Then \(\textrm{sign}(\varLambda )\) from Definition 3.4 is determined by (x, y) in Fig. 9, see also Table 1.

(b) Up to similarity, any lattice \(\varLambda \subset {\mathbb {R}}^2\) with an obtuse superbase \(v_0,v_1,v_2\) can be represented by up to six points (x, y) in the subregions of \(\textrm{Obt}\). Swapping \(v_0\leftrightarrow v_2\) is realised by the reflection in the line \(x=-\frac{1}{2}\), so \(v_2=(x,y)\mapsto (-1-x,y)\). Swapping and re-scaling the vectors \(v_1\leftrightarrow v_2\) is realised by the inversion with respect to the circle \(x^2+y^2=1\) so that \(v_2\mapsto v_1\mapsto \dfrac{v_2}{x^2+y^2}\), see Fig. 10 (right). \(\square \)

Proof

The inequality \(p_{12}<p_{01}\) is equivalent to \(-x<1+x\), \(x>-\frac{1}{2}\), see Table 2. The inequality \(p_{01}<p_{02}\) is equivalent to \(1+x<x^2+x+y^2\), \(x^2+y^2>1\), so the point (x, y) is above the circle C(0; 1) with the centre (0, 0) and radius 1 in Fig. 9. The inequality \(p_{12}<p_{02}\) is equivalent to \(-x<x^2+x+y^2\), \((x+1)^2+y^2>1\), so the point (x, y) is above the circle \(C(-1;1)\) with the centre \((-1,0)\) and radius 1.

The inequalities on \(p_{ij}\) from Table 2 justify that the region \(\textrm{Obt}\) splits into six subregions split by the vertical line \(x=-\frac{1}{2}\) 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 \(p_{ij}=\frac{1}{2}(v_i^2+v_j^2-v_k^2)\) from (2.7b) implies that \(v_i,v_j\) are the shortest of three vectors \(v_0,v_1,v_2\).

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 \(|v_2|=\sqrt{x^2+y^2}>1=|v_1|\); hence, by Definition 3.4\(\textrm{sign}(\varLambda )\) equals the sign of \(\det (v_1,v_2)=y>0\). In the last row, \(v_2=(x,y)\) has the length \(|v_2|=\sqrt{x^2+y^2}<1=|v_1|\); hence, by Definition 3.4\(\textrm{sign}(\varLambda )\) 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 \(\textrm{Obt}\) (representing all rectangular lattices in \(|R^2\)) or in one of the lines \(x=-\frac{1}{2}\) 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 \((x,y)=(-\frac{1}{2},\frac{\sqrt{3}}{2})\) represents all hexagonal lattices. All these lattices are mirror-symmetric and have \(\textrm{sign}(\varLambda )=0\).

(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 transpositions \(v_0\leftrightarrow v_2\) and \(v_1\leftrightarrow v_2\).

When we swap \(v_2=(x,y)\) and \(v_0=(-1-x,-y)\), while keeping \(v_1=(x,y)\) fixed, we reflect the lattice \(\varLambda (B)\) generated by B in the x-axis so that \(v_1=(x,y)\) has an obtuse angle to the image \((-1-x,y)\) of \(v_0\). This new vector \((-1-x,y)\) plays the role of \(v_2\) in the reflected lattice and is symmetric to \(v_2=(x,y)\) in the vertical line \(x=-\frac{1}{2}\) within the yellow region \(\textrm{Obt}\) in Fig. 10 (right).

When we swap \(v_1=(1,0)\) and \(v_2=(x,y)\), the second vector is divided by its length \(|v_2|=\sqrt{x^2+y^2}\). Hence the first vector \(v_1\) maps to the vector that is parallel to \(v_2=(x,y)\) and has the length \(1/\sqrt{x^2+y^2}\). 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 \(x=-\frac{1}{2}\) and this inversion generate up to six bright2021proof of (x, y) in the six subregions of \(\textrm{Obt}\), though the point \((x,y)=(-\frac{1}{2},\frac{\sqrt{3}}{2})\) representing all hexagonal lattices is fixed. \(\square \)

Proposition 3.10

Proof

(a) Up to similarity by Lemma 3.9(b), any lattice \(\varLambda \) has an obtuse superbase \(B=\{v_0,v_1,v_2\}\) with \(v_1=(1,0)\) and \(v_2=(x,y)\), where the point (x, y) belongs to the yellow region \(\textrm{Obt}\) in Fig. 9. By Lemma 3.9(b) the six permutations of \(v_0,v_1,v_2\) are realised by internal symmetries of \(\textrm{Obt}\), so we may assume that \(|v_1|\le |v_2|\le |v_0|\). The equivalent inequalities in conorms \(p_{12}\le p_{01}\le p_{02}\) define the 1st subregion of \(\textrm{Obt}\) in Fig. 9, which coincides with the closure of the region \(\overline{\textrm{Red}_+}=\{(x,y)\in {\mathbb {R}}^2\mid x^2+y^2\ge 1, -\frac{1}{2}\le x\le 0<y\}\) in Fig. 11.

Due to uniqueness of B up to isometry by Theorem 3.7, the position of \(v_2=(x,y)\in \overline{\textrm{Red}_+}\) is unique for \(\varLambda \). Up to uniform scaling and reflection \(y\leftrightarrow -y\), the closure \(\overline{\textrm{Red}_+}\) is defined by the same conditions \(v_1^2\le v_2^2=x^2+y^2\) and \(-\dfrac{1}{2}\le \dfrac{v_1\cdot v_2}{v_1^2}=x\le 0\) 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 non-rectangular lattice \(\varLambda \). 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 \(\textrm{Obt}\) in Fig. 9.

For the first open subregion with \(\textrm{sign}(B)=+1\), the conditions \(v_1^2<v_2^2=x^2+y^2\), \(-\frac{1}{2}<x=\dfrac{v_1\cdot v_2}{v_1^2}<0<y=\det (v_1,v_2)\) of Definition 2.3 specify the interior \(\overline{\textrm{Red}_+}\), so the basis \(\{v_1,v_2\}\) is reduced. For the second open subregion with \(\textrm{sign}(B)=-1\), another basis \(\{v_1,v_1+v_2\}\) is reduced by Definition 2.3 because the shifted point \(v_1+v_2=(x+1,y)\) belongs to the interior of the right-half region in Fig. 11 (left): \(\overline{\textrm{Red}_-}=\{(x,y)\in {\mathbb {R}}^2\mid x^2+y^2\ge 1,\, 0<x<\frac{1}{2},\, 0<y\}\).

It remains to consider singular cases. We include the common boundary line \(x=\dfrac{v_1\cdot v_2}{v_1^2}=-\dfrac{1}{2}\) 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 \(x\mapsto x+1\) so that the unique reduced basis \(v_1,v_1+v_2\) satisfies the conditions \(\dfrac{v_1\cdot v_2}{v_1^2}=\dfrac{1}{2}\) and \(v_1\cdot v_2\ge 0\) for \(|v_1|=|v_2|\) in Definition 2.3.

The above boundaries in Fig. 11 (left) include the blue and red points representing the basis vectors \(v_2=(x,y)\) of the lattices \(\varLambda ,\varLambda ''\), respectively.

The final boundary lines \(x=-1\) and \(x=0\) represent rectangular lattices with a unit cell \(a\times b\) for \(0<a<b\) and two obtuse superbases related by reflection, not by rigid motion, for example \(v_1=(a,0)\), \(v_2=(0,\pm b)\). In this case there is no 1-1 correspondence between obtuse superbases and reduced bases. Definition 2.3 selects a unique reduced basis \(v_1=(a,0)\), \(v_2=(0,b)\) due to \(\det (v_1,v_2)>0\). \(\square \)

Lemma 4.1

Proof

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. \(\square \)

Theorem 4.2

(Isometric 2D lattices \(\leftrightarrow \) root invariants) Any lattices \(\varLambda ,\varLambda '\subset {\mathbb {R}}^2\) are isometric if and only if their root invariants coincide: \(\textrm{RI}(\varLambda )=\textrm{RI}(\varLambda ')\). Any lattices \(\varLambda ,\varLambda '\) are related by rigid motion if and only if \(\textrm{RI}^o(\varLambda )=\textrm{RI}^o(\varLambda ')\). \(\square \)

Proof

The part only if (\(\Rightarrow \)) follows from Lemma 3.8(a) saying that \(\textrm{RI}(\varLambda ),\textrm{RI}^o(\varLambda )\) are invariant under isometry and rigid motion, respectively. The part if (\(\Leftarrow \)) follows from Lemma 4.1 reconstructing a superbase from \(\textrm{RI}(\varLambda )\) or \(\textrm{RI}^o(\varLambda )\). \(\square \)

Lemma 4.3

( [11, Theorem 7]) For any obtuse superbase \((v_0,v_1,v_2)\) of a lattice \(\varLambda \subset {\mathbb {R}}^2\), the vonorms \(v_0^2,v_1^2,v_2^2\) are squared lengths of shortest Voronoi vectors. \(\square \)

Definition 4.4

(Triangular cone \(\textrm{TC}\)) All root invariants \(\textrm{RI}(\varLambda )=(r_{12},r_{01},r_{02})\) of lattices \(\varLambda \subset {\mathbb {R}}^2\) live in the triangular cone \(\textrm{TC}=\{0\le r_{12}\le r_{01}\le r_{02}\}\) within the octant \(\textrm{Oct}=[0,+\infty )^3\) excluding the axes in the coordinates \(r_{12},r_{01},r_{02}\), see Fig. 12 (left). The boundary \(\partial (\textrm{TC})\) of the cone \(\textrm{TC}\) consists of root invariants of all mirror-symmetric lattices from Lemma 3.3: the bisector planes \(\{r_{01}=r_{02}\}\) and \(\{r_{12}=r_{01}\}\) within \(\textrm{TC}\). The orange line \(\{0<r_{12}=r_{01}=r_{02}\}\subset \partial (\textrm{TC})\) in Fig. 12 (left) consists of root invariants of hexagonal lattices with a minimum inter-point distance \(r_{12}\sqrt{2}\). The blue line \(\{r_{12}=0<r_{01}=r_{02}\}\subset \partial (\textrm{TC})\) consists of root invariants of square lattices with a minimum inter-point distance \(r_{01}\). \(\square \)

Definition 4.5

(Projected invariants \(\textrm{PI}(\varLambda )\) and \(\textrm{PI}^o(\varLambda )\)) The triangular projection \(\textrm{TP}:\textrm{TC}\rightarrow \{r_{12}+r_{01}+r_{02}=1\}\) divides each coordinate by the size \(\sigma (\varLambda )=r_{12}+r_{01}+r_{02}\) and gives \({{\overline{\textrm{RI}}}}(\varLambda )=({{\bar{r}}}_{12},{{\bar{r}}}_{01},\bar{r}_{02})=\dfrac{(r_{12},r_{01},r_{02})}{r_{12}+r_{01}+r_{02}}\) in \(\textrm{TC}\cap \{r_{12}+r_{01}+r_{02}=1\}\). Then we map \(({{\bar{r}}}_{12},\bar{r}_{01},{{\bar{r}}}_{02})\) to the projected invariant \(\textrm{PI}(\varLambda )=(x,y)\) with \(x={{\bar{r}}}_{02}-{{\bar{r}}}_{01}\in [0,1)\) and \(y=3{{\bar{r}}}_{12}\in [0,1]\) in the quotient triangle \(\textrm{QT}=\{(x,y)\in {\mathbb {R}}^2\mid 0\le x<1,\, 0\le y\le 1,\, x+y\le 1\}\), see Fig. 13.

All oriented root invariants \(\textrm{RI}^o(\varLambda )\) live in the doubled cone \(\textrm{DC}\) that is the union of two triangular cones \(\textrm{TC}^{\pm }\), where we identify any two boundary points representing the same root invariant \(\textrm{RI}(\varLambda )\) with \(\textrm{sign}(\varLambda )=0\). The oriented projected invariant \(\textrm{PI}^o(\varLambda )=(x,y)^{\pm }\) is \(\textrm{PI}(\varLambda )\) with the superscript from \(\textrm{sign}(\varLambda )\). \(\square \)

Corollary 4.6

(Similar lattices \(\leftrightarrow \) projected invariants \(\textrm{PI}\)) Lattices \(\varLambda ,\varLambda '\subset {\mathbb {R}}^2\) are similar (related by an isometry composed with a uniform scaling) if and only if their projected invariants are equal: \(\textrm{PI}(\varLambda )=\textrm{PI}(\varLambda ')\). The lattices \(\varLambda ,\varLambda '\) are related by an orientation-preserving similarity if and only if \(\textrm{PI}^o(\varLambda )=\textrm{PI}^o(\varLambda ')\). \(\square \)

Proof

follows from Theorem 4.2 because a uniform scaling of all basis vectors \(v_i\mapsto sv_i\) by a factor \(s>0\) multiplies all root products \(r_{ij}=\sqrt{-v_i\cdot v_j}\) by s, which is neutralised by the triangular projection \(\textrm{TP}\) from Definition 4.4. \(\square \)

Lemma 4.7

(Criteria of mirror-symmetric lattices in \({\mathbb {R}}^2\)) A lattice \(\varLambda \) in \({\mathbb {R}}^2\) is mirror-symmetric if and only if one of the following equivalent conditions holds: \(\textrm{sign}(\varLambda )=0\) or \(\textrm{RI}(\varLambda )\in \partial \textrm{TC}\) or \(\textrm{PI}(\varLambda )\in \partial \textrm{QT}\). So the boundaries of the triangular cone \(\textrm{TC}\) and the quotient triangle \(\textrm{QT}\) consist of root invariants and projected invariants, respectively, of all mirror-symmetric lattices \(\varLambda \subset {\mathbb {R}}^2\). \(\square \)

Proof

By Lemma 3.8 a lattice \(\varLambda \) is mirror-symmetric if and only if \(\textrm{sign}(\varLambda )=0\). By Lemma 3.3 any mirror-symmetric lattice \(\varLambda \) has \(\textrm{RI}(\varLambda )\) with 0 (rectangular lattice) or with two equal root products (centred rectangular lattices). The last conditions on \(\textrm{RI}\) define the boundary \(\partial \textrm{TC}\) of the triangular cone in Fig. 12 or, equivalently, the projected invariant \(\textrm{PI}(\varLambda )\) belongs to the boundary of \(\textrm{QT}\) in Fig. 13 (left). \(\square \)

Remark 4.8

\(Q_\varLambda \) has a reduced (non-acute) form if \(0\le -2q_{12}\le q_{11}\le q_{22}\) and \(q_{11}>0\), see [17, formula (1.130) on p. 75]. The above conditions define the fundamental domain \(T=\{0\le \xi <1,\; 0\le \eta \le \frac{1}{2},\; \xi +2\eta \le 1\}\), see [42, Fig. 8.1]. This non-isosceles triangle is one of the infinitely many triangular domains within the disk \(\xi ^2+\eta ^2<1\) in [17, Fig. 1.2 on p. 82] or [42, Fig. 6.2]. Choosing one triangular domain is equivalent to choosing a reduced basis up to isometry, not up to rigid motion. For instance, the mirror-symmetric bases \(v_1=(1,0)\), \(v_2^{\pm }=(-\frac{1}{2},\pm 1)\) have the same reduced non-acute form \(x^2-xy+\frac{5}{4}y^2\) represented only by \((\xi ,\eta )=(\frac{1}{9},\frac{4}{9})\). The above ambiguity up to rigid motion is resolved by \(\textrm{sign}(\varLambda )\) in the twice larger space \(\textrm{LSS}^o({\mathbb {R}}^2)\) visualised as the quotient square \(\textrm{QS}\), see Example 7.6.

More importantly, the inverse map from \(\textrm{RI}^o(\varLambda )\) to a reduced basis is discontinuous at any rectangular lattice \(\varLambda \) with a unit cell \(a\times b\). Indeed, slight perturbations of \(\varLambda \) have unique reduced bases that are not close to each other, being close to the distant bases \((a,0),(0,\pm b)\), which are not equivalent up to rigid motion for \(a<b\). This discontinuity of lattice bases will emerge in \({\mathbb {R}}^3\) even up to isometry [21]. In \({\mathbb {R}}^2\), Corollary 7.9 will completely settle the basis discontinuity up to rigid motion.

Another complete invariant is the ordered voform \(v_1^2\le v_2^2\le v_0^2\) or the lengths \(|v_1|\le |v_2|\le |v_0|\) of the three shortest Voronoi vectors from Lemma 4.3. However, this invariant does not extend even to dimension \(n=3\) due to a 6-parameter family of pairs of non-isometric lattices \(\varLambda _1\not \cong \varLambda _2\) that have the same lengths of seven shortest Voronoi vectors in \({\mathbb {R}}^3\), see [21]. The above reasons justify the choice of homogeneous coordinates \(r_{ij}\), which easily extend to higher dimensions. \(\square \)

Proposition 4.9

(Inverse design of 2D lattices) For \(\sigma >0\) and any point (x, y) in the quotient triangle \(\textrm{QT}\), there is a unique (up to isometry) lattice \(\varLambda \) with the projected invariant \(\textrm{PI}(\varLambda )=(x,y)\) and size \(\sigma =r_{12}+r_{01}+r_{02}\). Then

If (x, y) is in the interior of \(\textrm{QT}\), the invariant \(\textrm{RI}\) defines a pair of lattices \(\varLambda ^{\pm }\) that have opposite signs and unique (up to isometry) reduced basis vectors \(v_1,v_2\) with the lengths \(|v_1|=\sqrt{r_{12}^2+r_{01}^2}\), \(|v_2|=\sqrt{r_{12}^2+r_{02}^2}\) and the anticlockwise angle

\(\square \)

Proof

In Definition 4.5 the projected invariant \(\textrm{PI}(\varLambda )=(x,y)\) is obtained from the coordinates \(({{\bar{r}}}_{12},{{\bar{r}}}_{01},\bar{r}_{02})\) of \({{\overline{\textrm{RI}}}}(\varLambda )\) satisfying the equations \(\left\{ \begin{array}{l} x={{\bar{r}}}_{02}-{{\bar{r}}}_{01}, \\ y=3{{\bar{r}}}_{12}, \\ {{\bar{r}}}_{12}+{{\bar{r}}}_{01}+{{\bar{r}}}_{02}=1. \end{array}\right. \) The solution is \({{\overline{\textrm{RI}}}}(\varLambda )=({{\bar{r}}}_{12},{{\bar{r}}}_{01},\bar{r}_{02})=(\frac{y}{3},\frac{1}{2}-\frac{x}{2}-\frac{y}{6},\frac{1}{2}+\frac{x}{2}-\frac{y}{6})\). Multiplying all coordinates by the size \(\sigma =r_{12}+r_{01}+r_{02}\) gives the root invariant in (4.9a).

Example 4.10

(inverse design of 2D lattices) We will inversely design the lattices \(\varLambda _4,\varLambda _6,\varLambda _0,\varLambda _2^{\pm },\varLambda _{\infty }^\pm \), see their visualised invariants in Fig. 14 (right).

(\(\pmb {\varLambda _4}\)) We design the square lattice \(\varLambda _4\) starting from its projected invariant at the origin \(\textrm{PI}(\varLambda _4)=(0,0)\in \textrm{QT}\), which is identified with the top right vertex \((1,1)\in \textrm{QS}\) in Fig. 14 (left). Formula (4.9a) for the size \(\sigma (\varLambda _4)=2\) (only to get simplest integers) gives \(\textrm{RI}(\varLambda _4)=(0,1,1)\). An obtuse superbase \(\{v_0,v_1,v_2\}\) can be reconstructed by Lemma 4.1. The vonorms are \(v_1^2=v_2^2=0^2+1^2=1\), \(v_0^2=1^2+1^2=2\). We can choose the standard obtuse superbase \(v_1=(1,0)\), \(v_2=(0,1)\), \(v_0=(-1,-1)\).

(\(\pmb {\varLambda _6}\)) We design the hexagonal lattice \(\varLambda _6\) starting from the projected invariant at the top left vertex \(\textrm{PI}(\varLambda _6)=(0,1)\in \textrm{QT}\), which is identified with the bottom right vertex \((1,0)\in \textrm{QS}\) in Fig. 14 (left). Formula (4.9a) for the size \(\sigma (\varLambda _6)=3\) (only to get simplest integers) gives \(\textrm{RI}(\varLambda _6)=(1,1,1)\). To reconstruct an obtuse superbase \(\{v_0,v_1,v_2\}\) by Lemma 4.1, find the vonorms \(v_1^2=v_2^2=v_0^2=1^2+1^2=2\). Formula (4.9b) gives the angle \(\angle (v_1,v_2)=\arccos \frac{-4}{\sqrt{(5-6+9)^2}}= \arccos \left( -\frac{1}{2}\right) =120^\circ \). We can choose the superbase \(v_1=(\sqrt{2},0)\), \(v_2=(-\frac{1}{\sqrt{2}},\frac{\sqrt{3}}{\sqrt{2}})\), \(v_0=(-\frac{1}{\sqrt{2}},-\frac{\sqrt{3}}{\sqrt{2}})\).

(\(\pmb {\varLambda _0}\)) We inversely design the lattice \(\varLambda _0\) in Fig. 14 starting from \(\textrm{PI}(\varLambda _0)=(x,y)\) at the centre \((\frac{1}{2},\frac{1}{2})\in \textrm{QS}\). Formula (4.9a) for the size \(\sigma (\varLambda _0)=6\) (only to get simplest integers) gives \(\textrm{RI}(\varLambda _0)=(1,1,4)\). To reconstruct an obtuse superbase \(\{v_0,v_1,v_2\}\) by Lemma 4.1, find the vonorms \(v_1^2=1^2+1^2=2\), \(v_0^2=v_2^2=1^2+4^2=17\). Formula (4.9b) gives the angle \(\angle (v_1,v_2)=\arccos \dfrac{-4}{\sqrt{(\frac{9}{4}+\frac{5}{4}-3+9)^2-9(\frac{5}{4})^2}} =\arccos (-\frac{1}{\sqrt{34}})\approx 99.9^\circ \). We can choose the following superbase, see Fig. 15: \(v_1=(\sqrt{2},0)\), \(v_2=|v_2|(\cos \angle (v_1,v_2),\sin \angle (v_1,v_2)) =(-\frac{1}{\sqrt{2}},\frac{\sqrt{33}}{\sqrt{2}})\), \(v_0=(-\frac{1}{\sqrt{2}},-\frac{\sqrt{33}}{\sqrt{2}})\).

(\(\pmb {\varLambda _2}\)) We inversely design the lattice \(\varLambda _2\) in Fig. 14 starting from their projected invariants \(\textrm{PI}(\varLambda _2)=(\frac{1}{2+\sqrt{2}},\frac{1}{2+\sqrt{2}})\), which will maximise the chiral distance \(\textrm{PC}[D_2]\) in Theorem 6.6(a). Formula (4.9a) for the size \(\sigma (\varLambda _2)=6\) (only to simplify the root invariant) gives \(\textrm{RI}(\varLambda _2)=(2-\sqrt{2},2\sqrt{2}-1,5-\sqrt{2})\). Since all root products are nonzero and distinct, by Lemma 3.3 there is a pair of lattices \(\varLambda _2^\pm \) with \(\textrm{sign}(\varLambda _2^\pm )=\pm 1\). The lattices \(\varLambda _2^\pm \) are related by reflection, not by rigid motion.

To reconstruct an obtuse superbase \(\{v_0,v_1,v_2\}\) of \(\varLambda _2^\pm \) by Lemma 4.1, find

$$\begin{aligned} v_0^2= & {} (2\sqrt{2}-1)^2+(5-\sqrt{2})^2=(9-4\sqrt{2})+(27-10\sqrt{2})=36-14\sqrt{2}\approx 16.2, \\ v_1^2= & {} (2-\sqrt{2})^2+(2\sqrt{2}-1)^2=(6-4\sqrt{2})+(9-4\sqrt{2})=15-8\sqrt{2}\approx 3.7, \\ v_2^2= & {} (2-\sqrt{2})^2+(5-\sqrt{2})^2=(6-4\sqrt{2})+(27-10\sqrt{2})=33-14\sqrt{2}\approx 13.2, \\ \end{aligned}$$

and the anticlockwise angle \(\angle (v_1,v_2)=\arccos \dfrac{-r_{12}^2}{|v_1|\cdot |v_2|}\approx 92.8^\circ \). Then \(\varLambda _2^\pm \) have the following obtuse superbases in Fig. 15: \(v_1=(\sqrt{15-8\sqrt{2}},0)\approx (1.9,0)\), \(v_2=|v_2|(\cos \angle (v_1,v_2),\sin \angle (v_1,v_2)) \approx (-0.18,3.63)\), \(v_0 \approx (-1.72,-3.63)\).

(\(\pmb {\varLambda _{\infty }}\)) We inversely design the lattice \(\varLambda _{\infty }\) in Fig. 15 starting from \(\textrm{PI}(\varLambda _{\infty })=(x,y)\) at the mid-point \((\frac{1}{4},\frac{1}{4})\) of the segment between \(\textrm{PI}(\varLambda _4),\textrm{PI}(\varLambda _0)\in \textrm{QT}\). Formula (4.9a) for the size \(\sigma (\varLambda _{\infty })=12\) (only to simplify the root invariant) gives \(\textrm{RI}(\varLambda _{\infty })=(1,4,7)\). Since all root products are nonzero and distinct, by Lemma 3.3 there is a pair of lattices \(\varLambda _{\infty }^\pm \) of opposite signs \(\textrm{sign}(\varLambda _{\infty }^\pm )=\pm 1\).

Definition 5.1

(Root metrics \(\textrm{RM}\), projected metrics \(\textrm{PM}\)) Any metric d on \({\mathbb {R}}^3\) defines the root metric \(\textrm{RM}(\varLambda _1,\varLambda _2)=d(\textrm{RI}(\varLambda _1),\textrm{RI}(\varLambda _2))\) on lattices \(\varLambda _1,\varLambda _2\subset {\mathbb {R}}^2\) up to isometry. The Root Invariant Space \(\textrm{RIS}=(\textrm{TC},d)\) is the triangular cone with a fixed metric d. If we use the Minkowski norm \(M_q(v)=||v||_q=(\sum \limits _{i=1}^n |x_i|^q)^{1/q}\) of a vector \(v=(x_1,\dots ,x_n)\in {\mathbb {R}}^n\) for any real \(q\in [1,+\infty ]\), the root metric is denoted by \(\textrm{RM}_q(\varLambda _1,\varLambda _2)=||\textrm{RI}(\varLambda _1)-\textrm{RI}(\varLambda _2)||_q\). The limit case \(q=+\infty \) uses \(||v||_{\infty }=\max \limits _{i=1,\dots ,n}|x_i|\). The projected metric \(\textrm{PM}(\varLambda _1,\varLambda _2)=d(\textrm{PI}(\varLambda _1),\textrm{PI}(\varLambda _2))\) is on lattices up to similarity for any metric d on \({\mathbb {R}}^2\). The space of projected invariants \(\textrm{PIN}=(\textrm{QT},d)\) is the quotient triangle with a metric d. The notation \(\textrm{PM}_q(\varLambda _1,\varLambda _2)=||\textrm{PI}(\varLambda _1)-\textrm{PI}(\varLambda _2)||_q\) includes a parameter \(q\in [1,+\infty ]\) of \(M_q\). \(\square \)

Example 5.2

(metrics \(\textrm{RM}_q,\textrm{PM}_q\)) Table 4 summarises metric computations for the lattices \(\varLambda _4,\varLambda _6,\varLambda _0,\varLambda _{\infty }^{\pm }\), which were inversely designed in Example 4.10. \(\square \)

Lemma 5.3

(Metric axioms for \(\textrm{RM},\textrm{PM}\)) (a) Any metrics \(\textrm{RM}\) and \(\textrm{PM}\) from Definition 5.1 satisfy all metric axioms in (1.1c) on the Lattice Isometry Space \(\textrm{LIS}({\mathbb {R}}^2)\) and the Lattice Similarity Space \(\textrm{LSS}({\mathbb {R}}^2)\), respectively. \(\square \)

Proof

The metric axioms for \(\textrm{RM},\textrm{PM}\) 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 \(\textrm{RM}(\varLambda _1,\varLambda _2)=d(\textrm{RI}(\varLambda _1),\textrm{RI}(\varLambda _2))=0\) if and only if \(\textrm{RI}(\varLambda _1)=\textrm{RI}(\varLambda _2)\). Now Theorem 4.2 says that \(\textrm{RI}(\varLambda _1)=\textrm{RI}(\varLambda _2)\) is equivalent to \(\varLambda _1,\varLambda _2\) being isometric. Corollary 4.6 classifying lattices up to similarity by projected invariants similarly justifies the first axiom for \(\textrm{PM}(\varLambda _1,\varLambda _2)\). \(\square \)

Definition 5.4

(Orientation-aware metrics \(\textrm{RM}^o,\textrm{PM}^o\)) For lattices \(\varLambda _1,\varLambda _2\subset {\mathbb {R}}^2\) with \(\textrm{sign}(\varLambda _1)\textrm{sign}(\varLambda _2)\ge 0\), the orientation-aware root metric is \(\textrm{RM}^o(\varLambda _1,\varLambda _2)=\textrm{RM}(\varLambda _1,\varLambda _2)\) as in Definition 5.1. If any lattices \(\varLambda _1,\varLambda _2\) have opposite signs, set \(\textrm{RM}^o(\varLambda _1,\varLambda _2)=\inf \limits _{\textrm{sign}(\varLambda _3)=0}(\textrm{RM}(\varLambda _1,\varLambda _3)+\textrm{RM}(\varLambda _2,\varLambda _3))\). The orientation-based metric \(\textrm{PM}^o(\varLambda _1,\varLambda _2)\) is defined by the same formula, where we replace \(\textrm{RM}\) by \(\textrm{PM}\).

\(\square \)

Lemma 5.5

(Metric axioms for \(\textrm{RM}^o,\textrm{PM}^o\)) Any root metric \(\textrm{RM}^o\) and projected metric \(\textrm{PM}^o\) from Definition 5.4 satisfy all metric axioms in (1.1c) on the Lattice Isometry Space \(\textrm{LIS}^o\) and the Lattice Similarity Space \(\textrm{LSS}^o\), respectively. \(\square \)

Proof

Lemma 5.3 implies metric axioms in all cases when involved lattices have the same sign or sign 0. For example, the metrics \(\textrm{RM}^o,\textrm{PM}^o\) vanish only if the lattices \(\varLambda _1,\varLambda _2\) in question have equal invariants \(\textrm{RI}^o,\textrm{PI}^o\), respectively, so \(\varLambda _1,\varLambda _2\) are isometric or similar by Theorem 4.2 or Corollary 4.6, respectively. The symmetry axiom for any lattices \(\varLambda ,\varLambda '\) with opposite signs follows from the symmetry of the sum in Definition 5.4: \(\textrm{RM}^o(\varLambda _1,\varLambda _2)=\min \limits _{\textrm{sign}(\varLambda _3)=0}(\textrm{RM}(\varLambda _1,\varLambda _3)+\textrm{RM}(\varLambda _2,\varLambda _3))\).

Without loss of generality it suffices to prove the required triangle inequality \(\textrm{RM}^o(\varLambda _1,\varLambda _2)+\textrm{RM}^o(\varLambda _2,\varLambda _3)\ge \textrm{RM}^o(\varLambda _1,\varLambda _3)\) in the following two cases below.

Lemma 5.6

(Reversed signs) If lattices \(\varLambda _1^{\pm },\varLambda _2^{\pm }\subset {\mathbb {R}}^2\) have specified signs, then \(\textrm{RM}^o(\varLambda _1^+,\varLambda _2^-)=\textrm{RM}^o(\varLambda _1^-,\varLambda _2^+)\) and \(\textrm{PM}^o(\varLambda _1^+,\varLambda _2^-)=\textrm{PM}^o(\varLambda _1^-,\varLambda _2^+)\). \(\square \)

Proof

Lemma 5.7

(Maximum sum of moduli) For any real numbers \(a,b,c,d\ge 0\), the maximum sum \(S(x)=\max \{|a-x|+|x-b|,|c-x|+|x-d|\}\) over \(x\ge 0\) has the minimum value \(\textrm{MS}(a,b,c,d)=\max \{|a-b|,|c-d|,\frac{1}{2}|a+b-c-d|\}\). \(\square \)

Proof

The maximum sum S(x) and the formula for \(\textrm{MS}(a,b,c,d)\) are invariant under permutations \(a\leftrightarrow b\), \(c\leftrightarrow d\) and \((a,b)\leftrightarrow (c,d)\). Without loss of generality one can assume that \(a\le b\) and \(a\le c\le d\). Then Fig. 17 shows the graphs of \(y=|a-x|+|x-b|\), \(y=|c-x|+|x-d|\), \(y=S(x)\) in green, blue, red, respectively.

If \(b\ge c\), then \(a\le c\le b\le d\) or \(a\le c\le d\le b\), see the first two pictures of Fig. 17. In the first case S(x) has the minimum value \(\max \{b-a,d-c\}\) for any \(x\in [c,b]\). In the second case S(x) has the minimum value \(b-a\ge d-c\) for any \(x\in [a,b]\). In both cases the minimum value coincides with \(\max \{|a-b|,|c-d|\}\).

If \(b\le c\), then \(a\le b\le c\le d\), see the last picture of Fig. 17. Then \(S(x)=\max \{2x-a-b,c+d-2x\}\) has a minimum at x such that \(2x-a-b=c+d-2x\), so \(x=\frac{1}{4}(a+b+c+d)\). The minimum value is \(\textrm{MS}(a,b,c,d)=\frac{1}{2}|a+b-c-d|\ge \max \{|a-b|,|c-d|\}\) in this case. The last inequality is reversed in the previous two cases. So \(\textrm{MS}(a,b,c,d)=\max \{|a-b|,|c-d|,\frac{1}{2}|a+b-c-d|\}\) in all cases. \(\square \)

Proposition 5.8

(Root metrics for \(q=2,+\infty \)) Let \(\varLambda _1,\varLambda _2\subset {\mathbb {R}}^2\) be lattices of opposite signs with \(\textrm{RI}(\varLambda _1)=(r_{12},r_{01},r_{02})\) and \(\textrm{RI}(\varLambda _2)=(s_{12},s_{01},s_{02})\). Then

(a) \(\textrm{RM}_2^o(\varLambda _1,\varLambda _2)\) is the minimum of the Euclidean distances from the point \(\textrm{RI}(\varLambda _1)\) to the three points \((-s_{12},s_{01},s_{02})\), \((s_{01},s_{12},s_{02})\), and \((s_{12},s_{02},s_{01})\) in \({\mathbb {R}}^3\).

Proof

(a) By Definition 5.4\(\textrm{RM}^o(\varLambda _1,\varLambda _2)\) is the minimum value of \(\textrm{RM}(\varLambda _1,\varLambda _3)+\textrm{RM}(\varLambda _2,\varLambda _3)\) over mirror-symmetric lattices \(\varLambda _3\). By Lemma 4.7 the root invariant \(\textrm{RI}(\varLambda _3)\) belongs to one of the boundary sectors of the triangular cone \(\textrm{TC}\). Let \(\textrm{RI}'\) be the mirror image of \(\textrm{RI}(\varLambda _2)\) in the boundary sector containing \(\textrm{RI}(\varLambda _3)\).

(b) For \(\textrm{RI}(\varLambda _1)=(r_{12},r_{01},r_{02})\) and \(\textrm{RI}(\varLambda _2)=(s_{12},s_{01},s_{02})\), the required formula \(\textrm{RM}_{\infty }^o(\varLambda _1,\varLambda _2)=\min \{d_0,d_1,d_2\}\) will be proved by minimising the total length \(D=\textrm{RM}_{\infty }(\varLambda _1,\varLambda _3)+\textrm{RM}_{\infty }(\varLambda _2,\varLambda _3)\) of a path from \(\varLambda _1\) to \(\varLambda _2\) via \(\varLambda _3\) whose root invariant \(\textrm{RI}(\varLambda _3)\) can be in one of the three boundary sectors of \(\textrm{TC}\).

Proposition 5.9

(Projected metrics for \(q=2,+\infty \)) Let \(\varLambda _1,\varLambda _2\) be lattices with opposite signs and projected invariants \(\textrm{PI}(\varLambda _1)=(x_1,y_1)\), \(\textrm{PI}(\varLambda _2)=(x_2,y_2)\).

(a) \(\textrm{PM}_2^o(\varLambda _1,\varLambda _2)\) is the minimum of the Euclidean distances from \(\textrm{PI}(\varLambda _1)=(x_1,y_1)\) to the three points \((-x_2,y_2),(x_2,-y_2),(1-y_2,1-x_2)\) in \({\mathbb {R}}^2\).

(b) For \(x_1\le x_2\), \(\textrm{PM}_{\infty }^o(\varLambda _1,\varLambda _2)=\min \{ d_x, d_y, d_{xy} \}\) for \(d_x=\max \{x_2-x_1,y_2+y_1\}\), \(d_y=\max \{x_2+x_1,|y_2-y_1|\}\), \(d_{xy}=\max \{x_2-x_1,1-x_2-y_2+|1-y_1-x_2|\}\). \(\square \)

Proof

(a) By Definition 5.4\(\textrm{PM}^o(\varLambda _1,\varLambda _2)\) is the minimum value of \(\textrm{PM}(\varLambda _1,\varLambda _3)+\textrm{PM}(\varLambda _2,\varLambda _3)\) achieved for a mirror-symmetric lattices \(\varLambda _3\). By Lemma 4.7 the invariant \(\textrm{PI}(\varLambda _3)\) belongs to one of the sides of the quotient triangle \(\textrm{QT}\). Let \(\textrm{PI}'\) be the mirror image of \(\textrm{PI}(\varLambda _2)\) with respect to the side of \(\textrm{QT}\) containing \(\textrm{PI}(\varLambda _3)\).

(b) \(\textrm{PM}^o(\varLambda _1,\varLambda _2)\) is the minimum value of \(\textrm{PM}(\varLambda _1,\varLambda _3)+\textrm{PM}(\varLambda _2,\varLambda _3)\) over mirror-symmetric lattices \(\varLambda _3\). The formula \(\textrm{PM}_{\infty }^o(\varLambda _1,\varLambda _2)=\min \{d_x,d_y,d_{xy}\}\) will be proved by minimising the Minkowski length \(M_{\infty }\) of a path from \(\varLambda _1\) to \(\varLambda _2\) via \(\varLambda _3\) whose projected invariant \(\textrm{PI}(\varLambda _3)\) can be in one of the three sides of the quotient triangle \(\textrm{QT}\). For given \(\textrm{PI}(\varLambda _1)=(x_1,y_1)\) and \(\textrm{RI}(\varLambda _2)=(x_2,y_2)\) with \(x_1\le x_2\), we minimise \(D=\textrm{PM}_{\infty }(\varLambda _1,\varLambda _3)+\textrm{PM}_{\infty }(\varLambda _2,\varLambda _3)\) for each side of \(\textrm{QT}\) below.

Horizontal side: \(\textrm{PI}(\varLambda _3)=(x_3,0)\) for a variable parameter \(x_3\ge 0\). For any \(x_3\in [x_1,x_2]\), the distance \(D=\max \{(x_3-x_1)+(x_2-x_3),y_2+y_1\}\) equals the simpler function \(d_x=\max \{x_2-x_1,y_2+y_1\}\) and can be only larger for any \(x_3\not \in [x_1,x_2]\).

Vertical side: \(\textrm{PI}(\varLambda _3)=(0,y_3)\) for a variable parameter \(y_3\). For any \(y_3\) between \(y_1,y_2\), the distance \(D=\max \{x_1+x_2,|y_1-y_3|+|y_3-y_2|\}\) has the minimum \(d_y=\max \{x_1+x_2,|y_1-y_2|\}\) and is larger for any \(y_3\) that is not between \(y_1,y_2\).

Hypotenuse: \(\textrm{PI}(\varLambda _3)=(x_3,y_3)\) for variables \(x_3,y_3\) such that \(x_3+y_3=1\). For \(x_1\le x_2\), we aim to minimise \(D=\max \{|x_1-x_3|+|x_3-x_2|,|y_1-y_3|+|y_3-y_2|\}\). Figure 18 shows all three different cases how we can find an optimal chiral lattice \(\varLambda _3\) with a projected invariant \(\textrm{PI}(\varLambda _3)=(x_3,y_3)\) for given \(\textrm{PI}(\varLambda _1)=(x_1,y_1)\) and \(\textrm{PI}(\varLambda _2)=(x_2,y_2)\). The sum \(|x_1-x_3|+|x_3-x_2|\) has the minimum value \(x_2-x_1\) for any \(x_3\in [x_1,x_2]\). Similarly, \(|y_1-y_3|+|y_3-y_2|\) has the minimum value \(|y_1-y_2|\) for any \(y_3\) between \(y_1,y_2\). If the point \((x_3,y_3)\) moves along the hypotenuse so that both \(x_3,y_3\) are outside their minimum ranges above, then \(|x_1-x_3|+|x_3-x_2|=|2x_3-x_1-x_2|\) increases at the same rate as \(|y_1-y_3|+|y_3-y_2|=|2y_3-y_1-y_2|\) decreases during this movement because \(x_3+y_3=1\). Hence \(x_3\) can be chosen as \(x_1\) or \(x_2\) to minimise D. In the first two pictures of Fig. 18 we choose \(x_3=x_2\).

Definition 6.1

Lemma 6.2

(Properties of chiral distances) (a) A lattice \(\varLambda \) is mirror-symmetric if and only if \(\textrm{RC}[D_2](\varLambda )=0\) or, equivalently, \(\textrm{PC}[D_2](\varLambda )=0\).

(b) For any crystallographic point group G in \({\mathbb {R}}^2\), mirror reflections \(\varLambda ^{\pm }\subset {\mathbb {R}}^2\) have equal G-chiral distances: \(\textrm{RC}[G](\varLambda ^+)=\textrm{RC}[G](\varLambda ^-)\), \(\textrm{PC}[G](\varLambda ^+)=\textrm{PC}[G](\varLambda ^-)\).

\(\square \)

Proof

(a) By Definition 6.1\(\textrm{RC}[D_2](\varLambda )=\min \limits _{\textrm{sign}(\varLambda ')=0}\{\textrm{RM}(\varLambda ,\varLambda ')\}=0\) means that \(\textrm{RI}(\varLambda )=\textrm{RI}(\varLambda ')\) for some mirror-symmetric lattice \(\varLambda '\) due to Lemma 4.7. Then \(\varLambda \) is isometric to \(\varLambda '\) by Theorem 4.2 and is mirror-symmetric.

(b) The G-chiral distances from Definition 6.1 depend only on a root invariant \(\textrm{RI}(\varLambda )\) and projected invariant \(\textrm{PI}(\varLambda )\), which are the same for mirror bright2021proof \(\varLambda ^{\pm }\). \(\square \)

Lemma 6.3

Proof

(b) If \(\varLambda _1=\varLambda _2\), the sum \(\textrm{RM}(\varLambda _1,\varLambda ')+\textrm{RM}(\varLambda _2,\varLambda ')\) in part (a) consists of equal terms. Hence the inequality becomes equality giving the double distance. \(\square \)

Lemma 6.4

(Maximum modulus) For any fixed points \(a,b\in {\mathbb {R}}\), the function \(D_{\infty }(x)=\max \{|a-x|,|x-b|\}\) has the minimum value \(\dfrac{1}{2}|a-b|\) for \(x=\dfrac{a+b}{2}\). \(\square \)

Proof

Without loss of generality assume that \(a\le b\). For \(x\in [a,b]\), the function \(D_{\infty }(x)=\max \{x-a,b-x\}\) has the minimum value \(\dfrac{b-a}{2}\) at the mid-point x of the interval [a, b] and takes values larger than \(b-a\) for any \(x\not \in [a,b]\). \(\square \)

Proposition 6.5

Proof

(a) The chiral distance \(\textrm{RC}_2[D_2](\varLambda )=\min \nolimits _{\textrm{sign}(\varLambda ')=0}||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_2\) by Definition 6.1 is the minimum Euclidean distance from \(\textrm{RI}(\varLambda )\) to the boundary of \(\textrm{TC}\). This boundary of three triangular sectors consists of root invariants of all mirror-symmetric lattices by Lemma 4.7. Any point \(\textrm{RI}(\varLambda )=(r_{12},r_{01},r_{02})\) with \(0\le r_{12}\le r_{01}\le r_{02}\) has Euclidean distances \(r_{12}\), \(\dfrac{r_{01}-r_{12}}{\sqrt{2}}\), \(\dfrac{r_{02}-r_{01}}{\sqrt{2}}\) to the horizontal boundary \(r_{12}=0\), inclined boundary \(r_{12}=r_{01}\), and the vertical boundary \(r_{01}=r_{02}\), respectively, see Fig. 14(right). Then \(\textrm{RC}_2[D_2](\varLambda )\) is the minimum of the above Euclidean distances to the three boundary sectors of \(\textrm{TC}\).

\(\textrm{RC}_2[D_4](\varLambda )\) is the Euclidean distance from \(\textrm{RI}(\varLambda )=(r_{12},r_{01},r_{02})\) to a closest root invariant \(\textrm{RI}(\varLambda ')=(0,s,s)\) of a square lattice \(\varLambda '\). The square of \(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_2\) is \(d_4(s)=r_{12}^2+(s-r_{01})^2+(s-r_{02})^2\). The quadratic function \(d_4(s)\) is minimised when \(0=d'_4(s)=2(s-r_{01})+2(s-r_{02})\), so \(s=\frac{1}{2}(r_{01}+r_{02})\). Then \(d_4(\frac{1}{2}(r_{01}+r_{02}))=r_{12}^2+\frac{1}{4}(r_{02}-r_{01})^2\) and \(\textrm{RC}_2[D_4](\varLambda )=\sqrt{r_{12}^2+\frac{1}{4}(r_{02}-r_{01})^2}\).

(b) \(\textrm{RC}_{\infty }[D_2](\varLambda )=\min \limits _{\textrm{sign}(\varLambda ')=0}||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }\) is minimised over mirror-symmetric lattices \(\varLambda '\) whose root invariants by Lemma 4.7 belong to one of the three boundary sectors of the triangular cone \(\textrm{TC}\). We consider them below one by one.

Horizontal boundary : \(\textrm{RI}(\varLambda ')=(0,s_{01},s_{02})\) for \(0<s_{01}\le s_{02}\). Then \(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }=||(r_{12},r_{01},r_{02})-(0,s_{01},s_{02})||_{\infty }=\max \{r_{12},|r_{01}-s_{01}|,|r_{02}-s_{02}|\}\) has the minimum \(r_{12}\) for \(s_{01}=r_{01}\), \(s_{02}=r_{02}\) or any \(s_{01},s_{02}\) close to \(r_{01},r_{02}\).

Inclined boundary : a variable root invariant \(\textrm{RI}(\varLambda ')\) has equal root products \(s_{12}=s_{01}\). By Lemma 6.4 the \(M_{\infty }\)-distance \(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }=||(r_{12},r_{01},r_{02})-(s_{01},s_{01},s_{02})||_{\infty }=\max \{|r_{12}-s_{01}|,|r_{01}-s_{01}|,|r_{02}-s_{02}|\}\) has the minimum \(\dfrac{r_{01}-r_{12}}{2}\) for \(s_{02}=r_{02}\) and \(s_{01}\) at the mid-point of the interval \([r_{12},r_{01}]\).

Vertical boundary : \(\textrm{RI}(\varLambda ')\) has \(s_{01}=s_{02}\). By Lemma 6.4\(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }=||(r_{12},r_{01},r_{02})-(s_{12},s_{01},s_{01})||_{\infty }=\max \{|r_{12}-s_{12}|,|r_{01}-s_{01}|,|r_{02}-s_{01}|\}\) has the minimum \(\dfrac{r_{02}-r_{01}}{2}\) for \(s_{12}=r_{12}\) and \(s_{01}\) at the mid-point of the interval \([r_{01},r_{02}]\). Finally, \(\textrm{RC}_{\infty }[D_2](\varLambda )\) is the minimum of the three \(\varLambda _{\infty }\) distances.

\(\textrm{RC}_{\infty }[D_4](\varLambda )\) is the distance \(M_{\infty }\) from \(\textrm{RI}(\varLambda )\) to a closest root invariant \(\textrm{RI}(\varLambda ')=(0,s,s)\) of a square lattice \(\varLambda '\). Then \(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }=\max \{r_{12},|s-r_{01}|,|s-r_{02}|\}\). If we ignore \(r_{12}\), by Lemma 6.4 the minimum of the largest value among the last two is \(\frac{1}{2}(r_{02}-r_{01})\), so \(\textrm{RC}_{\infty }[D_4](\varLambda )=\min \left\{ r_{12}, \frac{r_{02}-r_{01}}{2}\right\} \).

\(\textrm{RC}_{\infty }[D_6](\varLambda )\) is the distance \(M_{\infty }\) from \(\textrm{RI}(\varLambda )=(r_{12},r_{01},r_{02})\) to a closest root invariant \(\textrm{RI}(\varLambda ')=(s,s,s)\) of a hexagonal lattice \(\varLambda '\). Then \(||\textrm{RI}(\varLambda )-\textrm{RI}(\varLambda ')||_{\infty }=\max \{|s-r_{12}|,|s-r_{01}|,|s-r_{02}|\}\). Since \(r_{12}\le r_{01}\le r_{02}\), we could ignore \(|s-r_{01}|\) in the maximum. By Lemma 6.4 the final maximum is \(\frac{r_{02}-r_{01}}{2}=\textrm{RC}_{\infty }[D_6](\varLambda )\). \(\square \)

Proposition 6.6

Proof

(a) By Definition 6.1\(\textrm{PC}_2[D_2](\varLambda )\) is the minimum Euclidean distance from \(\textrm{PI}(\varLambda )\) to the three boundary sides of the quotient triangle \(\textrm{QT}\). These sides consist of projected invariants of all mirror-symmetric lattices by Lemma 4.7. Any point \(\textrm{PI}(\varLambda )=(x,y)\in \textrm{QT}\) has distances x, y, \(\dfrac{1-x-y}{\sqrt{2}}\) to the vertical side \(x=0\), horizontal side \(y=0\), and the hypotenuse \(x+y=1\), respectively, see Fig. 14 (left). Hence \(\textrm{PC}_2(\varLambda )\) is the minimum of the above Euclidean distances.

\(\textrm{PC}_q[D_4](\varLambda )=(x^q+y^q)^{1/q}\) is the Minkowski \(M_q\) distance from \(\textrm{PI}(\varLambda )=(x,y)\in \textrm{QT}\) to the single-point subspace \(\textrm{LSS}[D_4]=(0,0)\) for any \(q\in [1,+\infty )\).

\(\textrm{PC}_q[D_6](\varLambda )=(x^q+(1-y)^q)^{1/q}\) is the Minkowski \(M_q\) distance from \(\textrm{PI}(\varLambda )=(x,y)\in \textrm{QT}\) to the single-point subspace \(\textrm{LSS}[D_6]=(0,1)\) for any \(q\in [1,+\infty )\).

(b) \(\textrm{PC}_{\infty }(\varLambda )=\min \limits _{\textrm{sign}(\varLambda ')=0}||\textrm{PI}(\varLambda )-\textrm{PI}(\varLambda ')||_{\infty }\) is minimised over lattices \(\varLambda '\) whose projected invariant is in one of the three sides of the quotient triangle \(\textrm{QT}\).

Horizontal side : \(\textrm{PI}(\varLambda ')=(s,0)\) for a variable s. The distance \(||\textrm{PI}(\varLambda )-\textrm{PI}(\varLambda ')||_{\infty }=||(x,y)-(s,0)||_{\infty }=\max \{|x-s|,y\}\) has the minimum value y for \(s=x\).

Vertical side : \(\textrm{PI}(\varLambda ')=(0,t)\) for a variable t. The distance \(||\textrm{PI}(\varLambda )-\textrm{PI}(\varLambda ')||_{\infty }=||(x,y)-(0,t)||_{\infty }=\max \{x,|y-t|\}\) has the minimum x for \(t=y\).

Hypotenuse : \(\textrm{PI}(\varLambda ')=(s,t)\) for variables \(s,t\ge 0\) such that \(s+t=1\). To compute the \(M_{\infty }\) distance from (x, y) to (s, t), first assume that \(s\ge x\), \(t\ge y\). Then \(||(x,y)-(s,t)||_{\infty }=\max \{s-x,t-y\}\) is minimised when \(s-x=t-y\). Substituting \(t=1-s\), we get \(s-x=1-s-y\), \(s=\dfrac{1+x-y}{2}\), \(t=\dfrac{1-x+y}{2}\). One can check that \(s+t=1\) and \(s\ge x\), \(t\ge y\) due to \(x+y\le 1\) as \((x,y)\in \textrm{QT}\). Then \(s-x=t-y=\dfrac{1-x-y}{2}\). It remains to show that the minimum \(M=\dfrac{1-x-y}{2}\) of the distance from (x, y) to (s, t) cannot have a smaller value for \(s\le x\) or \(t\le y\).

If \(s\le x\), then \(||(x,y)-(s,t)||_{\infty }=\max \{x-s,t-y\}=\max \{x-1+t,t-y\}\le t-y\), whose minimum value \((1-x)-y\) is not less than \(M=\dfrac{1-x-y}{2}\) as \(x+y\le 1\).

If \(t\le y\), then \(||(x,y)-(s,t)||_{\infty }=\max \{s-x,y-t\}=\max \{s-x,y-1+s\}\le s-x\), whose minimum value \((1-y)-x\) is not less than \(M=\dfrac{1-x-y}{2}\) as \(x+y\le 1\).

So \(\textrm{PC}_{\infty }(\varLambda )\) is the minimum of the above three \(M_{\infty }\) distances.

\(\textrm{PC}_{\infty }[D_4](\varLambda )=x\) is the distance \(M_{\infty }\) from (x, y) to \(\textrm{LSS}[D_4]=(0,0)\).

\(\textrm{PC}_{\infty }[D_6](\varLambda )=1-y\) is the distance \(M_{\infty }\) from (x, y) to \(\textrm{LSS}[D_6]=(0,1)\).

(c) The quotient triangle \(\textrm{QT}\) in Fig. 13 (left) is parameterised by \(0\le x<1\) and \(0\le y\le 1\) such that \(x+y\le 1\), By Theorem 6.6(a) the distance \(\textrm{PC}_{2}[D_2](\varLambda )=\min \{x,y,\dfrac{1-x-y}{\sqrt{2}}\}\) is maximal when \(x=y=\dfrac{1-x-y}{\sqrt{2}}\), so \(x=y=\dfrac{1}{2+\sqrt{2}}\). By Theorem 6.6(b) \(\textrm{PC}_{\infty }[D_2](\varLambda )=\min \{x,y,\dfrac{1-x-y}{2}\}\) is maximal when \(x=y=\dfrac{1-x-y}{2}\), \(x=y=\dfrac{1}{4}\). Then \(x^q\le x\), \(y^q\le y\) and \((x^q+y^q)^{1/q}\le (x+y)^{1/q}\le 1\) for any \(q\in [1,+\infty )\). Hence the upper bound \(\textrm{PC}_{q}[D_4](\varLambda )\le 1\) holds for any \(q\in [1,+\infty ]\) and is achieved for any hexagonal lattice with \(\textrm{PI}=(0,1)\). Similarly, the upper bound \(\textrm{PC}_{q}[D_6](\varLambda )<2^{1/q}\) holds for any \(q\in [1,+\infty )\) and is approached but not achieved as \(x\rightarrow 1\), \(y=0\). The bound \(\textrm{PC}_{\infty }[D_6](\varLambda )=1-y\le 1\) holds for any \(\varLambda \) and is achieved for any square and rectangular lattice with \(y=0\). \(\square \)

Example 6.7

(distances \(\textrm{RC}_q,\textrm{PC}_q\)) Table 5 shows the chiral distances computed by Propositions 6.5, 6.6 for the prominent lattices \(\varLambda _2^{\pm }\), \(\varLambda _{\infty }^{\pm }\) in Example 4.10. \(\square \)

Example 6.8

(metrics \(\textrm{RM}_q^o,\textrm{PM}_q^o\)) Table 6 has \(\textrm{RM}_q^o,\textrm{RM}_q^o\) for \(q=2,+\infty \) and the prominent lattices \(\varLambda _2^{\pm }\), \(\varLambda _{\infty }^{\pm }\), which were inversely designed in Example 4.10.

If lattices have the same sign, then \(\textrm{RM}^o,\textrm{PM}^o\) coincide with their unoriented versions by Definition 5.4. For example, \(\textrm{PM}_q^o(\varLambda _2^+,\varLambda _{\infty }^+)\) is the distance \(M_q\) between the invariants \(\textrm{PI}(\varLambda _{\infty })=(\frac{1}{4},\frac{1}{4})\) and \(\textrm{PI}(\varLambda _2)=(\frac{1}{2+\sqrt{2}},\frac{1}{2+\sqrt{2}})=(1-\frac{1}{\sqrt{2}},1-\frac{1}{\sqrt{2}})\), so \(\textrm{PM}_{\infty }^o(\varLambda _2^+,\varLambda _{\infty }^+)=\frac{3}{4}-\frac{1}{\sqrt{2}}\approx 0.04\) and \(\textrm{PM}_{2}^o(\varLambda _2^+,\varLambda _{\infty }^+)=\frac{3}{4}\sqrt{2}-1\approx 0.06\).

Similarly, \(\textrm{RM}_q^o(\varLambda _2^+,\varLambda _{\infty }^+)\) is the \(M_q\) distance between the root invariants \(\textrm{PI}(\varLambda _{\infty })=(1,4,7)\) and \(\textrm{RI}(\varLambda _2)=(2-\sqrt{2},2\sqrt{2}-1,5-\sqrt{2})\), so \(\textrm{RM}_{\infty }^o(\varLambda _2^+,\varLambda _{\infty }^+)=\max \{\sqrt{2}-1,5-2\sqrt{2},2+\sqrt{2}\}=2+\sqrt{2}\approx 3.41\) and \(\textrm{RM}_{2}^o(\varLambda _2^+,\varLambda _{\infty }^+)=\sqrt{6(7-3\sqrt{2})}\approx 4.1\).

By Lemma 6.3(b) the distance between mirror bright2021proof of the same lattice equals the doubled \(D_2\)-chiral distance. For example, \(\textrm{PM}_q^o(\varLambda _{\infty }^+,\varLambda _{\infty }^-)=2\textrm{PC}_q[D_2](\varLambda _{\infty })=\frac{1}{2}\) and \(\textrm{PM}_q^o(\varLambda _2^+,\varLambda _2^-)=2\textrm{PC}_q[D_2](\varLambda _2)=\frac{2}{2+\sqrt{2}}=2-\sqrt{2}\approx 0.59\) for \(q=2,+\infty \).

Lemma 6.3(b) and Table 5 also give \(\textrm{RM}_q^o(\varLambda _{\infty }^+,\varLambda _{\infty }^-)=2\textrm{RC}_q[D_2](\varLambda _{\infty })=2\) and \(\textrm{RM}_q^o(\varLambda _2^+,\varLambda _2^-)=2\textrm{RC}_q[D_2](\varLambda _2)=2(2-\sqrt{2})\approx 1.17\) for \(q=2,+\infty \).

Lemma 5.6 says that \(\textrm{RM}^o(\varLambda _2^+,\varLambda _{\infty }^-)=\textrm{RM}^o(\varLambda _2^-,\varLambda _{\infty }^+)\) and \(\textrm{PM}^o(\varLambda _2^+,\varLambda _{\infty }^-)=\textrm{PM}^o(\varLambda _2^-,\varLambda _{\infty }^+)\). Using the above properties, it remains to find four distances.

Definition 7.1

(Space \(\textrm{OSI}\) of obtuse superbases up to isometry) (a) Let \(B=\{v_i\}_{i=0}^n\) and \(B'=\{u_i\}_{i=0}^n\) be any obtuse superbases in \({\mathbb {R}}^n\). The Superbase Isometry Metric \(\textrm{SIM}_{\infty }(B,B')=\min \nolimits _{f\in \textrm{O}({\mathbb {R}}^n)}\max \nolimits _{i=0,\dots ,n}|f(u_i)-v_i|\) minimises vector differences over orthogonal maps f from the group \(\textrm{O}({\mathbb {R}}^n)\). Let \(\textrm{OSI}({\mathbb {R}}^n)\) denote the space of all obtuse superbases up to isometry with the metric \(\textrm{SIM}_{\infty }\).

Lemma 7.2

(Metric axioms for \(\textrm{SIM}_{\infty }\)) The metrics from Definition 7.1 on \(\textrm{OSI}({\mathbb {R}}^n), \textrm{OSI}^o({\mathbb {R}}^n),\textrm{OSS}({\mathbb {R}}^n),\textrm{OSS}^o({\mathbb {R}}^n)\) satisfy all metric axioms in (1.1c). \(\square \)

Proof

Let \(B_j=\{v_{j0},\dots ,v_{jn}\}\), \(j=1,2,3\), be any obtuse superbases in \({\mathbb {R}}^n\). The first axiom: if \(0=\textrm{SIM}_{\infty }(B_1,B_2)=\min \limits _{f\in \textrm{O}({\mathbb {R}}^n)}\max \limits _{i=0,\dots ,n}|f(v_{1i})-v_{2i}|\), there is an isometry \(f\in \textrm{O}({\mathbb {R}}^n)\) such that \(f(B_1)=B_2\). The superbases \(B_1,B_2\) are isometric.

Since any isometry f preserves Euclidean distance, we get \(|f(v_{1i})-v_{2i}|=|f^{-1}(f(v_{1i})-v_{2i})|=|v_{1i}-f^{-1}(v_{2i})|\) and the second axiom: \(\textrm{SIM}_{\infty }(B_1,B_2)=\min \limits _{f\in \textrm{O}({\mathbb {R}}^n)}\max \limits _{i=0,\dots ,n}|f(v_{1i})-v_{2i}|=\min \limits _{f^{-1}\in \textrm{O}({\mathbb {R}}^n)}\max \limits _{i=0,\dots ,n}|v_{1i}-f^{-1}(v_{2i})|=\textrm{SIM}_{\infty }(B_2,B_1)\).

To prove the triangle inequality in the third axiom for \(\textrm{SIM}_{\infty }\), let \(f,g\in \textrm{O}({\mathbb {R}}^n)\) be rotations that minimise the distances \(\textrm{SIM}_{\infty }(B_1,B_2)=\max \nolimits _{i=0,\ldots ,n}|f(v_{1i})-v_{2i}|\) and \(\textrm{SIM}_{\infty }(B_2,B_3)=\max \nolimits _{i=0,\ldots ,n}|g(v_{2i})-v_{3i}|\), respectively. Then we get \(\textrm{SIM}_{\infty }(B_1,B_3)\le \max \nolimits _{i=0,\ldots ,n}|g(f(v_{1i}))-v_{3i}|\le \max \nolimits _{i=0,\dots ,n}|g(f(v_{1i}))-g(v_{2i})|+\max \nolimits _{i=0,\ldots ,n}|g(v_{2i})-v_{3i}| =\) \(\max \nolimits _{i=0,\dots ,n}|f(v_{1i})-v_{2i}|+\textrm{SIM}_{\infty }(B_2,B_3) =\textrm{SIM}_{\infty }(B_1,B_2)+\textrm{SIM}_{\infty }(B_2,B_3)\). The proof for other spaces is identical after replacing \(\textrm{O}({\mathbb {R}}^n)\) with the relevant groups. \(\square \)

Lemma 7.3

Proof

\(|u_1\cdot u_2-v_1\cdot v_2|=|(u_1-v_1)\cdot u_2+v_1\cdot (u_2-v_2)|\le |(u_1-v_1)\cdot u_2|+|v_1\cdot (u_2-v_2)|\le \) \(\le |u_1-v_1|\cdot |u_2|+|v_1|\cdot |u_2-v_2|\le \delta (|u_2|+|v_1|)\le 2l\delta \) as required.

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 \(\varLambda \subset {\mathbb {R}}^2\) have a maximum length l, the size \(\sigma (\varLambda )=r_{12}+r_{01}+r_{02}\) of the lattice \(\varLambda \) has the lower bound \(l\le \sigma (\varLambda )\). \(\square \)

Proof

Let \(|v_1|=l\). Formula 2.7(a) gives \(p_{12}+p_{01}=v_1^2=l^2\) and \(r_{12}+r_{01}+r_{02}=\sqrt{(r_{12}+r_{01}+r_{02})^2}\ge \sqrt{r_{12}^2+r_{01}^2+r_{02}^2}=\sqrt{p_{12}+p_{01}+p_{02}}\ge l\). \(\square \)

Theorem 7.5

(Continuity of \(\textrm{OSI}\rightarrow \textrm{LIS}=\textrm{RIS}\)) (a) Let \(q\in [1,+\infty ]\) and lattices \(\varLambda ,\varLambda '\subset {\mathbb {R}}^2\) have obtuse superbases B and \(B'\) whose vectors have a maximum length l. If \(\textrm{SIM}_{\infty }(B,B')=\delta \ge 0\), then \(\textrm{RM}_q(\varLambda ,\varLambda ')\le 3^{1/q}\sqrt{2l\delta }\). Hence the bijection \(\textrm{OSI}({\mathbb {R}}^2)\rightarrow \textrm{LIS}({\mathbb {R}}^2)\) is continuous in the metrics \(\textrm{SIM}_{\infty }\) and \(\textrm{RM}_q\).

(b) In the conditions above, the projected metric satisfies \(\textrm{PM}_q(\varLambda ,\varLambda ')\le 2^{1/q}3\sqrt{2\delta /l}\), so the bijection \(\textrm{OSS}({\mathbb {R}}^2)\rightarrow \textrm{LSS}({\mathbb {R}}^2)\) is continuous in the metrics \(\textrm{SIM}_{\infty },\textrm{PM}_q\).

(c) In the oriented case, if \(\delta \rightarrow 0\) then \(\textrm{RM}_{q}^o(\varLambda ,\varLambda ')\rightarrow 0\) and \(\textrm{PM}_{q}^o(\varLambda ,\varLambda ')\rightarrow 0\), so both \(\textrm{OSI}^o({\mathbb {R}}^2)\rightarrow \textrm{LIS}^o({\mathbb {R}}^2)\) and \(\textrm{OSS}^o({\mathbb {R}}^2)\rightarrow \textrm{LSS}^o({\mathbb {R}}^2)\) are continuous. \(\square \)

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 \(|u_i-v_i|\le \delta \) for \(i=0,1,2\) after applying a suitable isometry to \(B'\) by Definition 7.1. Lemma 7.3 implies that the root products \(r_{ij}=\sqrt{-v_i\cdot v_j}\) and \(\sqrt{-u_i\cdot u_j}\) differ by at most \(2l\delta \) for any pair (i, j) of indices. The \(M_q\)-norm of the vector difference in \({\mathbb {R}}^3\) is \(\textrm{RM}_q(\varLambda ,\varLambda ')\le 3^{1/q}\sqrt{2l\delta }\), \(q\in [1,+\infty ]\).

(b) The coordinates \(x={{\bar{r}}}_{02}-{{\bar{r}}}_{01}\) and \(y=3{{\bar{r}}}_{12}\) have an error bound that is at most three times larger than the error for \({{\bar{r}}}_{ij}\). In Definition 4.5 each \({{\bar{r}}}_{ij}\) is obtained by dividing the root product \(r_{ij}\) by the sizes with the lower bound \(l\le \sigma =r_{12}+r_{01}+r_{02}\) by Lemma 7.4. The above error bound \(\sqrt{2l\delta }\) for \(r_{ij}\) gives the error bound \(3\sqrt{2\delta /l}\) for x, y. Then \(\textrm{PM}_q(\varLambda ,\varLambda ')\le 2^{1/q}3\sqrt{2\delta /l}\), \(q\in [1,+\infty ]\).

(c) In the oriented case, if \(\textrm{sign}(\varLambda )>0\), then \(\textrm{RI}(\varLambda )\) is strictly inside the triangular cone \(\textrm{TC}\). The continuity of \(\textrm{RM}\) implies that under any continuous motion of \(\delta \)-close superbases \(B\rightarrow B'\), if \(\delta \) is sufficiently small, then all intermediate lattices have their unoriented root invariants inside \(\textrm{TC}\), so their signs are positive and \(\textrm{sign}(\varLambda ')>0\). Hence \(\textrm{sign}(\varLambda ),\textrm{sign}(\varLambda ')\) coincide or one of them is 0. In all cases by Definition 5.4 the metric \(\textrm{RM}^o(\varLambda ,\varLambda ')\) coincides with \(\textrm{RM}(\varLambda ,\varLambda ')\) whose convergence to 0 as \(\delta \rightarrow 0\) was proved above. The proof of \(\textrm{PM}^o(\varLambda ,\varLambda ')\rightarrow 0\) is similar. \(\square \)

Example 7.6

(Continuity of root invariants) The obuse superbase \(v_1=(1,0)\), \(v_2(t)=(-t,2)\), \(v_0(t)=(t-1,-2)\) continuously deforms for \(t\in [0,1]\) in Fig. 4. The basis of \(v_1,v_2(t)\) is reduced (non-acute) for \(t\in [0,\frac{1}{2}]\) and at the critical moment \(t=\frac{1}{2}\) changes to its mirror image \(v_1,v_0(t)\), which remains reduced for \(t\in [\frac{1}{2},1]\). The obtuse superbase of unordered vectors \(\{v_1,v_2(t),v_0(t)\}\) keeps changing continuously because \(v_2(t),v_0(t)\) only swap their places at \(t=\frac{1}{2}\). The discontinuity of the obtuse superbases in the above deformation emerges at \(t=1\) when the final superbase of \((1,0),(-1,2),(0,-2)\) becomes a mirror image of the initial superbase of \((1,0),(0,2),(-1,-2)\), not related by rigid motion, though both (unordered) superbases at \(t=0,1\) generate the same lattice with the rectangular cell \(1\times 2\).

The root invariants are \(r_{12}=\sqrt{t}\), \(r_{01}=\sqrt{1-t}\), \(r_{02}=\sqrt{4-t+t^2}\). Since \(4-t+t^2\ge \frac{15}{4}\ge \max \{t,1-t\}\) for \(t\in [0,1]\), the root invariant can be written as

$$\begin{aligned} \textrm{RI}(\varLambda (t))=\left\{ \begin{array}{ll} (\sqrt{t},\sqrt{1-t},\sqrt{4-t+t^2}) \text { for } t\in [0,\frac{1}{2}],\\ (\sqrt{1-t},\sqrt{t},\sqrt{4-t+t^2}) \text { for } t\in [\frac{1}{2},1]. \end{array} \right. \end{aligned}$$

Theorem 7.7

(Continuity of \(\textrm{LIS}\rightarrow \textrm{OSI}\)) (a) For \(q\in [1,+\infty ]\), let lattices \(\varLambda ,\varLambda '\) in \({\mathbb {R}}^2\) satisfy \(\textrm{RM}_q(\varLambda ,\varLambda ')\le \delta \). For any obtuse superbase B of \(\varLambda \), there is an obtuse superbase \(B'\) of \(\varLambda '\) such that \(\textrm{SIM}_{\infty }(B,B')\le \textrm{SIM}_{\infty }^o(B,B')\rightarrow 0\) as \(\delta \rightarrow 0\).

(b) The bijection \(\textrm{LIS}({\mathbb {R}}^2)\rightarrow \textrm{OSI}({\mathbb {R}}^2)\) is continuous in the metrics \(\textrm{RM}_q,\textrm{SIM}_{\infty }\). \(\textrm{LIS}^o({\mathbb {R}}^2)\rightarrow \textrm{OSI}^o({\mathbb {R}}^2)\) is continuous in \(\textrm{RM}_q^o,\textrm{SIM}_{\infty }^o\) at non-rectangular lattices.

(c) The above conclusions hold for lattices and superbases up to similarity. \(\square \)

Proof

(a) The obtuse superbase \(B=(v_0,v_1,v_2)\) of \(\varLambda \) is already given. Let \(B'=(u_0,u_1,u_2)\) be any obtuse superbase of \(\varLambda '\) found from \(\textrm{RI}(\varLambda ')\) by Lemma 4.1.

Up to rigid motion in \({\mathbb {R}}^2\), one can assume that \(\varLambda ,\varLambda '\) 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 \(v_i^2=r_{ij}^2+r_{ik}^2\) and \(u_i^2=s_{ij}^2+s_{ik}^2\) for distinct indices \(i,j,k\in \{0,1,2\}\), for example if \(i=0\) then \(j=1\), \(k=2\).

(b) By Theorem 3.7, the obtuse superbases \(B,B'\) of \(\varLambda ,\varLambda '\) are unique up to isometry, also up to rigid motion if \(\varLambda ,\varLambda '\) are not rectangular. Since we can start with any obtuse superbase B, the convergence \(\textrm{SIM}_{\infty }^o(B,B')\rightarrow 0\) for a unique obtuse superbase \(B'\) proves the continuity of \(\textrm{LIS}({\mathbb {R}}^2)\rightarrow \textrm{OSI}({\mathbb {R}}^2)\). The continuity of \(\textrm{LIS}^o({\mathbb {R}}^2)\rightarrow \textrm{OSI}^o({\mathbb {R}}^2)\) similarly follows for all non-rectangular lattices.

(c) To extend part (a) to the similarity equivalence, we use the size \(\sigma =r_{12}+r_{01}+r_{02}\) of the given superbase B to reconstruct an obtuse superbase \(B'\) of \(\varLambda '\) from \(\textrm{PI}(\varLambda ')\) with the same size \(\sigma \) by Proposition 4.9. By formula (4.9) the given condition \(\textrm{PM}_q(\varLambda ,\varLambda ')\rightarrow 0\) implies that \(\textrm{RM}_q(\varLambda ,\varLambda ')\rightarrow 0\); hence, part (a) implies the required conclusions for the smaller metrics \(\textrm{SSM}_{\infty }\le \textrm{SIM}_{\infty }\) and \(\textrm{SSM}_{\infty }^o\le \textrm{SIM}_{\infty }^o\). The same argument extends part (b) to the similarity case. \(\square \)

Lemma 7.8

(Lower bound for \(\textrm{SIM}_{\infty }^o\)) For any ordered obtuse superbases \(B,B'\) of lattices in \({\mathbb {R}}^2\) with coforms \(\textrm{CF}(B)=(p_{12},p_{01},p_{02})\) and \(\textrm{CF}(B')=(p'_{12},p'_{01},p'_{02})\), let \(\textrm{CM}_{\infty }(B,B')=\min \limits _{\zeta \in A_3} \max \limits _{i\ne j}\{|p_{ij}-p'_{\zeta (i)\zeta (j)}|\}\) be minimised over three cyclic permutations \(\zeta \in A_3\) of indices 0, 1, 2. Let l be a maximum length of all vectors from \(B,B'\). Then we have the lower bound \(\textrm{SIM}_{\infty }^o(B,B')\ge \textrm{CM}_{\infty }(B,B')/2l\). \(\square \)

Proof

For the superbase \(B=\{v_0,v_1,v_2\}\), by Definition 7.1 find an optimal rotation around the origin so that the resulting image \(\{v'_0,v'_1,v'_2\}\) of \(B'\) satisfies \(|u_i-v_i|\le \textrm{SIM}_{\infty }^o(B,B')\), \(i=0,1,2\). Lemma 7.3 implies that \(|p_{ij}-p'_{ij}|\le 2l\cdot \textrm{SIM}_{\infty }^o(B,B')\) for all distinct \(i,j\in \{0,1,2\}\). The above rotation might have cyclically shifted the coforms of \(B'\), but \(\textrm{CM}_{\infty }(B,B')\) is minimised over cyclic permutations. Then \(\textrm{CM}_{\infty }(B,B')\le \max \limits _{i\ne j}\{|p_{ij}-p'_{ij}|\}\le 2l\cdot \textrm{SIM}_{\infty }^o(B,B')\) gives the lower bound. \(\square \)

Corollary 7.9

(Partial discontinuity of \(\textrm{RIS}^o\rightarrow \textrm{OSI}^o\)) The bijection \(\textrm{LIS}^o\rightarrow \textrm{OSI}^o\) is discontinuous in the metrics \(\textrm{RM}_{\infty },\textrm{SIM}_{\infty }^o\) at any rectangular lattice. \(\square \)

Proof

Proposition 7.10

(Continuous chiral distances) For a crystallographic point group G and lattices \(\varLambda _1,\varLambda _2\) in \({\mathbb {R}}^2\), we have \(|\textrm{RC}[G](\varLambda _1)-\textrm{RC}[G](\varLambda _2)|\le \textrm{RM}(\varLambda _1,\varLambda _2)\) and \(|\textrm{PC}[G](\varLambda _1)-\textrm{PC}[G](\varLambda _2)|\le \textrm{PM}(\varLambda _1,\varLambda _2)\) for any metrics \(\textrm{RM}\) and \(\textrm{PM}\). \(\square \)

Proof

Definition 8.1

(RSD invariant) For any lattice \(\varLambda \subset {\mathbb {R}}^n\), both \(\textrm{AMD}\) and \(\textrm{PDD}\) invariants reduce to the sequence of distances \((d_1,d_1,d_2,d_2,d_3,d_3,\dots )\) from the origin \(0\in \varLambda \) to its k-th nearest neighbour in \(\varLambda \) for \(k\ge 1\). Since any \(\varLambda \) is symmetric with respect to 0, define the Reduced Sequence of Distances \(\textrm{RSD}(\varLambda )=(d_1,d_2,d_3,\dots )\) containing one distance from each pair of equal distances above. \(\square \)

Proposition 8.2

(\(\textrm{RSD}\) completeness) Any lattices \(\varLambda ,\varLambda '\subset {\mathbb {R}}^2\) are isometric if and only if \(\textrm{RSD}(\varLambda )=\textrm{RSD}(\varLambda ')\). \(\square \)

Proof

The lengths \(|v_1|\le |v_2|\le |v_0|\) of shortest Voronoi vectors from Fig. 3 are not necessarily the first three distances in \(\textrm{RSD}\). For example, if \(\varLambda \) has the basis \(v_1=(1,0)\), \(v_2=(0,3)\), then \(\textrm{RSD}(\varLambda )=(1,2,3,\dots )\), where \(2=2|v_1|\ne |v_2|=3\). We will extract \(|v_1|\le |v_2|\le |v_0|\) from \(\textrm{RSD}(\varLambda )\), which proves completeness.

For any integer \(k>1\), the shortest Voronoi vector \(v_1\in \varLambda \) of length \(d_1=|v_1|\) generates a single integer multiple \(kv_1\in \varLambda \) of length \(kd_1\), which can be removed from \(\textrm{RSD}(\varLambda )\). The second shortest Voronoi vector \(v_2\in \varLambda \) may accidentally have the same length \(|v_2|=d_1\). If not, the next distance \(d_2\) in the resulting sequence equals \(|v_2|\) as in \(\textrm{RSD}(\varLambda )=\{1,\sqrt{3},\dots \}\) for the lattice \(\varLambda \) in Fig. 21 (right). If yes, we recognise the repeated value of \(d_1\) as \(d_2=|v_2|\) and do not confuse \(d_2\) with any multiple \(kd_1\) for \(k>1\). For example, if \(\varLambda \) has the basis \(v_1=(1,0)\), \(v_2=(0,2)\), then \(\textrm{RSD}(\varLambda )=\{1,2,2,\dots \}\), where one distance 2 is \(2d_1\), another distance 2 is \(d_2\), so we remove only one distance \(2=2d_1\). Again for any \(k>1\), we remove one multiple \(kd_2\) and find the next distance \(d_3\) equal to the length of the third shortest Voronoi vector \(v_0=-v_1-v_2\) as in Fig. 3. Since \((v_1^2,v_2^2,v_0^2)=(d_1^2,d_2^2,d_3^2)\) is a complete invariant by Theorem 4.2 and Lemma 4.3, then so is \(\textrm{RSD}(\varLambda )\). \(\square \)

Remark 8.3

(Linear structure, scalar product on lattices) Since the triangular cone \(\textrm{TC}\) in Fig. 12 is convex, we can consider any convex linear combination of root invariants \(t\textrm{RI}(\varLambda _1)+(1-t)\textrm{RI}(\varLambda _2)\in \textrm{TC}\), \(t\in [0,1]\). The resulting root invariant determines (an isometry class of) the new lattice that can be denoted by \(t\varLambda _1+(1-t)\varLambda _2\). The average of the square and hexagonal lattices with \(\textrm{RI}(\varLambda _4)=(0,1,1)\), \(\textrm{RI}(\varLambda _6)=(1,1,1)\) has \(\textrm{RI}=(\frac{1}{2},1,1)\). The new lattice \(\frac{1}{2}(\varLambda _4+\varLambda _6)\) is centred rectangular and has the basis \(v_1=(\sqrt{\frac{3}{2}},0)\) and \(v_2=(-\frac{1}{9}\sqrt{\frac{3}{2}},\frac{4}{9}\sqrt{\frac{15}{2}})\). We can define similar sums in \(\textrm{LSS}({\mathbb {R}}^2)\) due to the convexity of the triangle \(\textrm{QT}\).

The usual scalar product of vectors in \({\mathbb {R}}^3\) defines the positive product of root invariants: \(\textrm{RI}(\varLambda _4)\cdot \textrm{RI}(\varLambda _6)=(0,1,1)\cdot (1,1,1)=2\). The lattice spaces \(\textrm{LIS}^o({\mathbb {R}}^2)\) and \(\textrm{LSS}^o({\mathbb {R}}^2)\) up to rigid motion and orientation-preserving similarity are geodesic metric spaces, even CAT(0) spaces and flat manifolds (locally Euclidean). \(\square \)