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An isometry classification of periodic point sets ?
Olga Anosova1and Vitaliy Kurlin1[0000−0001−5328−5351]
University of Liverpool, Liverpool L69 3BX, UK vitaliy.kurlin@gmail.com
http://kurlin.org
Abstract. We develop discrete geometry methods to resolve the data
ambiguity challenge for periodic point sets to accelerate materials dis-
covery. In any high-dimensional Euclidean space, a periodic point set is
obtained from a finite set (motif) of points in a parallelepiped (unit cell)
by periodic translations of the motif along basis vectors of the cell.
An important equivalence of periodic sets is a rigid motion or an isometry
that preserves interpoint distances. This equivalence is motivated by solid
crystals whose periodic structures are determined in a rigid form.
Crystals are still compared by descriptors that are either not isometry
invariants or depend on manually chosen tolerances or cut-off parameters.
All discrete invariants including symmetry groups can easily break down
under atomic vibrations, which are always present in real crystals.
We introduce a complete isometry invariant for all periodic sets of points,
which can additionally carry labels such as chemical elements. The main
classification theorem says that any two periodic sets are isometric if and
only if their proposed complete invariants (called isosets) are equal.
A potential equality between isosets can be checked by an algorithm,
whose computational complexity is polynomial in the number of motif
points. The key advantage of isosets is continuity under perturbations,
which allows us to quantify similarities between any periodic point sets.
Keywords: lattice
·
periodic set
·
isometry invariant
·
classification
1 Introduction: motivations and problem statement
One well-known challenge in applications is the curse of dimensionality meaning
that any dataset seems sparse in a high-dimensional space. This paper studies the
more basic ambiguity challenge in data representations meaning that equivalent
real-life object can often be represented in infinitely many different ways.
Data ambiguity makes any comparison unreliable. For example, humans
should be not be compared or identified by the average color of their clothes,
though such colors are easily accessible in photos. Justified comparisons should
use only invariant features that are independent of an object representation.
?Supported by the EPSRC grant Application-driven Topological Data Analysis
2 O. Anosova and V. Kurlin
Our objects are periodic point sets, which model all solid crystalline ma-
terials (crystals). Solid crystal structures are determined in a rigid form with
well-defined atomic positions. Atoms form strong bonds only in molecules, while
all inter-molecular bonds are much weaker and have universally agreed defini-
tions. Points at atomic centers can be labeled by chemical elements or any other
properties, e.g. radii. Later we explain how to easily incorporate labels into our
invariants. We start with the most fundamental model of a periodic point set.
The simplest example is a lattice Λ, a discrete set of points that are integer
linear combinations of any linear (not necessarily orthogonal) basis in Rn, see
Fig. 1. More generally, a periodic point set is obtained from a finite collection
(motif ) of points by periodic translations along all vectors of a lattice Λ.
Fig. 1. Left: periodic sets represented by different cells are organized in isometry
classes, which form a continuous space. Right: the new isoset resolves the ambiguity.
The same periodic point set Scan be obtained from infinitely many different
Minkowski sums Λ+M. For example, one can change a linear basis of Λand
get a new motif of points with different coordinates in the new basis.
The above ambiguity with respect to a basis is compounded by infinitely
many rigid motions or isometries that preserve inter-point distances, hence pro-
duce equivalent crystal structures. Shifting all points by a fixed vector changes
all point coordinates in a fixed basis, but not the isometry class of the set.
The curse of ambiguity for periodic sets can be resolved only by a complete
isometry invariant as follows: two periodic sets given by any decompositions
Λ+Minto a lattice and a motif should be isometric if and only if their complete
invariants coincide. Such a complete invariant should have easily comparable
values from which we could explicitly reconstruct an original crystal structure.
The final requirement for a complete invariant is its continuity under pertur-
bations, which was largely ignored in the past despite all atoms vibrate above the
absolute zero temperature. All discrete invariants including symmetry groups are
discontinuous under perturbation of points. A similarity between crystals should
be quantified in a continuous way to filter out nearly identical crystals obtained
as approximations to local energy minima in Crystal Structure Prediction [22].
An isometry classification of periodic point sets 3
Problem 1 formalizes the above curse of ambiguity for crystal structures.
Problem 1 (complete isometry classification of periodic point sets).Find a
function Ion the space of all periodic point sets in Rnsuch that
(1a) invariance : if any periodic sets S, Q are isometric, then I(S) = I(Q);
(1b) continuity :I(S) continuously changes under perturbations of points;
(1c) computability:I(S) = I(Q) is checked in a polynomial time in a motif size;
(1d) completeness : if I(S) = I(Q), then the periodic sets S, Q are isometric.
The main contribution is the new invariant isoset in Definition 9 whose com-
pleteness is proved in Theorem 10. Conditions (1cd) are proved in the recent
work [3] introducing the new research area of Periodic Geometry and Topology.
2 A review of the relevant work on periodic crystals
Despite any lattice can be defined by infinitely many primitive cells, there is
a unique Niggli’s reduced cell, which can be theoretically used for comparing
periodic sets [13, section 9.2]. Niggli’s and other reduced cells are discontinuous
under perturbations in the sense that a reduced cell of a perturbed lattice can
have a basis that substantially differs from that of a non-perturbed lattice [2].
Continuity condition (1b) fails not only for Niggli’s reduced cell, but also for
all discretely-valued invariants including symmetry groups. The 230 crystallo-
graphic groups in R3cut the continuous space of isometry classes into disjoint
pieces. This stratification shows many nearly identical crystals as distant.
The first step towards a complete isometry classification of crystals has re-
cently been done in [19] by introducing two proper distances between arbitrary
lattices that satisfy the metric axioms and are also continuous under perturba-
tions. Also [19, section 3] reviews many past tools to compare crystals.
The world’s largest Cambridge Structural Database (CSD) has more than
1M crystals. Each crystal is represented by one of infinitely many choices of
a unit cell and a motif Min the form of a Crystallographic Information File
(CIF). The CSD is a super-long list of CIFs with limited search tools, mainly by
chemical compositions, and without any organization by geometric similarity.
Quantifying crystal similarities is even more important for Crystal Structure
Prediction (CSP). A typical CSP software starts from a given chemical compo-
sition and outputs thousands of predicted crystals as approximations to local
minima of a complicated energy function. Any iterative optimization produces
many approximations to the same local minimum. These nearly identical crystals
are currently impossible to automatically identify in a reliable way [22].
Crystals are often compared by the Radial Distribution Function (RDF) that
measures the probability of finding one atom at a distance of rfrom a reference
atom, which is computed up to a manually chosen cut-off radius.
4 O. Anosova and V. Kurlin
The new concept of a stable radius in Definition 8 gives exact conditions
for a required radius depending on a complexity of a periodic set. The crystals
indistinguishable by their RDF or diffraction patterns are known as homometric
[20]. The most recent survey [21, Fig. S4] has highlighted pairs of finite atomic
arrangements that cannot be distinguished by any known crystal descriptors.
On a positive side, the mathematical approach in [12] has solved the al-
ready non-trivial 1-dimensional case for sets whose points have only integer (or
rational) coordinates. Briefly, any given points c0, . . . , cm−1on the unit circle
S1⊂Care converted into the Fourier coefficients d(k) =
m−1
P
j=0
cjexp 2πijk
m,
k= 0, . . . , m −1. Then all point sets in the unit circle can be distinguished up
to circular rotations by the n-th order invariants up to n= 6, which are all
products of the form d(k1)··· d(kn) with k1+· · · +kn≡0 (mod m).
The more recent advances in Problem 1 are Density Functions [11] and Aver-
age Minimum Distances [25]. The k-density function ψk[S] of a periodic point set
S⊂Rnmeasures the fractional area of the region within a unit cell Ucovered
by exactly kclosed balls with centers a∈Sand a radius t≥0. The density
functions satisfy conditions (1abc) and completeness (1d) in general position.
However, the density functions do not distinguish the following 1-dimensional
sets S15 ={0,1,3,4,5,7,9,10,12}+15Zand Q15 ={0,1,3,4,6,8,9,12,14}+15Z
with period 15, see [3, Example 11]. The sets S15 , Q15 were introduced at the
beginning of section 5 in [11] as U±V+ 15Zfor U={0,4,9}and V={0,1,3}.
The above sets S15, Q15 are distinguished by the faster Average Minimum
Distances (AMD), see [3, Example 6]. For any integer k≥1, AMDk(S) is the
distance from a point p∈Sto its k-th nearest neighbor, averaged over all points
pin a motif of S. For k→+∞, AMDk(S) behaves as n
√k, see [25, Theorem 14].
3 Necessary concepts from computational geometry
In the Euclidean space Rn, any point p∈Rnis represented by the vector ~p
from the origin of Rnto p. The Euclidean distance between points p, q ∈Rn
is denoted by |pq|=|~p −~q|. For a standard orthonormal basis ~e1, . . . , ~en, the
integer lattice Zn⊂Rnconsists of all points with integer coordinates.
Definition 2 (a lattice Λ, a unit cell U, a motif M, a periodic set S=Λ+M).
For any linear basis ~v1, . . . , ~vnin Rn, a lattice is Λ={
n
P
i=1
λi~vi:λi∈Z}. The
unit cell U(~v1, . . . , ~vn) = n
P
i=1
λi~vi:λi∈[0,1)is the parallelepiped spanned
by the basis. A motif Mis any finite set of points p1, . . . , pm∈U. A periodic
point set is the Minkowski sum S=Λ+M={~u +~v :~u ∈Λ, ~v ∈M}. A unit
cell Uof a periodic set S=Λ+Mis primitive if any vector ~v that translates S
to itself is an integer linear combination of the basis of the cell U, i.e. ~v ∈Λ.
An isometry classification of periodic point sets 5
Fig. 2. Left: three primitive cells U, U 0, U00 of the square lattice S. Other pictures
show different periodic sets Λ+M, which are all isometric to the square lattice S.
A primitive unit cell Uof any lattice has a motif of one point (the origin). If
Uis defined as the closed parallelepiped in Rn, hence includes 2nvertices, one
could count every vertex with weight 2−nso that the sum is 1. All closed unit
cells in Fig. 2 are primitive, because four corners are counted as one point in U.
The first picture in Fig. 3 shows a small perturbation of a square lattice. The
new periodic set has a twice larger primitive unit cell with two points in a motif
instead of one. All invariants based on a fixed primitive unit cell such as Niggli’s
reduced cell [13, section 9.2] fail continuity condition (1b) in Problem 1.
Fig. 3. A continuous invariant should take close values on these nearly identical peri-
odic sets, though their symmetry groups and primitive cells substantially differ.
The auxiliary concepts in Definitions 3, 4, 5 follow Dolbilin’s papers [9], [5].
Definition 3 (bridge distance β(S)).For a periodic point set S⊂Rn, the
bridge distance is a minimum β(S)>0 such that any two points a, b ∈Scan
be connected by a finite sequence a0=a, a1, . . . , am=bsuch that any two
successive points ai, ai+1 are close, i.e. the Euclidean distance |~ai−1−~ai| ≤ β(S)
for i= 1, . . . , m. Fig. 4 shows periodic sets with different bridge distances.
Definition 4 (m-regularity of a periodic set).For any point ain a periodic set
S⊂Rn, the global cluster C(S, a) is the infinite set of vectors ~
b−~a for all points
b∈S. Points a, b ∈Sare called isometrically equivalent if there is an isometry
f:C(S, a)→C(S, b) such that f(a) = b. A periodic set S⊂Rnis called regular
if all points a, b ∈Sare isometrically equivalent. A periodic set Sis m-regular
if all global clusters of Sform exactly m≥1 isometry classes.
6 O. Anosova and V. Kurlin
For any point a∈S, its global cluster is a view of Sfrom the position of a,
e.g. how we view all astronomical stars in the universe Sfrom our planet Earth.
Any lattice is 1-regular, because all its global clusters are related by translations.
Though the global clusters C(S, a) and C(S, b) at any different points a, b ∈S
seem to contain the same set S, they can be different even modulo translations.
Fig. 4. Left: the periodic point set Q1has the four points (±2,±2) in the square unit
cell [0,10]2, so Q1isn’t a lattice, but is 1-regular by Definition 4, also β(Q1) = 6. All
local α-clusters are isometric, shown by red arrows for radii α= 5,6,8, see Definition 5.
Right: the periodic point set Q2has the extra point (5,5) in the center of [0,10]2and
is 2-regular with β(Q2) = 3√2. Local clusters have two isometry types.
The first picture in Fig. 4 shows the 1-regular set Q1⊂R2, where all points
have isometric global clusters related by translations and rotations through
π
2, π, 3π
2, so Q1is not a lattice. The global clusters are infinite, hence distin-
guishing them up to isometry is not easier than distinguishing the original sets.
However, m-regularity can be checked in terms of local clusters defined below.
Definition 5 (local α-clusters C(S, a;α) and symmetry groups Sym(S, a;α)).
For a point ain a crystal S⊂Rnand any radius α≥0, the local cluster
C(S, a;α) is the set of vectors ~
b−~a of lengths |~
b−~a| ≤ αfor b∈S. An isometry
f∈Iso(Rn) between clusters should match their centers. The symmetry group
Sym(S, a;α) consists of self-isometries of C(S, a;α) that fix the center a.
If α > 0 is smaller than the minimum distance between any points, then
every cluster C(S, a;α) is the single-point set {a}and its symmetry group O(Rn)
consists of all isometries fixing the center a. When the radius αis increasing,
the α-clusters C(S, a;α) become larger and can have fewer self-isometries, so the
symmetry group Sym(S, a;α) becomes smaller and eventually stabilizes.
An isometry classification of periodic point sets 7
The 1-regular set Q1in Fig. 4 for any point a∈Q1has the symmetry group
Sym(Q1, a;α) = O(R2) for α∈[0,4). The group Sym(Q1, a;α) stabilizes as Z2
for α≥4 as soon as the local α-cluster C(Q1, a;α) includes one more point.
4 The isotree of isometry classes and a stable radius
This section introduces the isotree and a stable radius in Definitions 6 and 8 by
comparing local clusters at radii α−βand β, where βis the bridge distance.
Any isometry A→Bbetween local clusters should map the center of A
to the center of B. The isotree in Definition 6 is inspired by a dendrogram
of hierarchical clustering, though points are partitioned according to isometry
classes of local α-clusters at different radii α, not by a distance threshold.
Definition 6 (isotree IT(S) of α-partitions).Fix a periodic set S⊂Rnand
α≥0. Points a, b ∈Sare called α-equivalent if their α-clusters C(S, a;α) and
C(S, b;α) are isometric. The α-equivalence class [C(S, a;α)] consists of all α-
clusters isometric to C(S, a;α). The α-partition P(S;α) is the splitting of Sinto
α-equivalence classes of points. The number of α-equivalence classes of α-clusters
is the cluster count |P(S;α)|. When the radius αis increasing, the α-partition
can be refined by subdividing α-equivalence classes of points of Sinto subclasses.
If we represent each α-equivalence class by an abstract point, the resulting points
form the isotree IT(S) of all α-partitions, see Fig. 5, 6.
The α-equivalence and isoset in Definition 9 can be refined by labels of points
such as chemical elements. Theorem 10 will remain valid for labelled points.
Recall that isometries include reflections, however an orientation sign can be
easily added to α-clusters, hence we focus on the basic case of all isometries.
When a radius αis increasing, α-clusters C(S, a;α) include more points,
hence are less likely to be isometric, so |P(S;α)|is a non-increasing function of
α. Fig. 5, 6 show α-clusters and isotrees of non-isometric 1D periodic sets S, Q
[20, p. 197, Fig. 2], which have identical 1D analogs of diffraction patterns.
Any α-equivalence class from P(S;α) may split into two or more classes,
which will not merge at any larger radius α0. Lemma 7 justifies that the isotree
IT(S) can be visualized as a merge tree of α-equivalence classes of clusters.
Lemma 7 (isotree properties).The isotree IT(S) has the following properties:
(7a) for α= 0, the α-partition P(S; 0) consists of one class;
(7b) if α < α0, then Sym(S, a;α0)⊆Sym(S, a;α) for a∈S;
(7c) if α < α0, the α0-partition P(S;α0)refines P(S;α), i.e. any set from the
α0-partition P(S;α0) is included into a set from the α-partition P(S;α).
Proof. (7a) If α≥0 is smaller than the minimum distance rbetween point of S,
every cluster C(S, a;α) is the single-point set {a}. All these single-point clusters
are isometric to each other. So |P(S;α)|= 1 for all small radii α < r.
8 O. Anosova and V. Kurlin
Fig. 5. Left:S={0,1,3,4}+ 8Zhas t= 4 and is 2-regular by Definition 4. Right:
Local clusters with radii α= 0,1,2,3 represent vertices of the isotree IT(S) in Defini-
tion 6. All α-clusters are isometric for α < 2, form two isometry classes for α≥2.
Fig. 6. Left:Q={0,3,4,5}+ 8Zhas t= 3 and is 3-regular by Definition 4. Right:
Local clusters with radii α= 0,1,2,3 represent vertices of the isotree IT(Q) in Defini-
tion 6. All α-clusters are isometric for α < 1, form three isometry classes for α≥1.
(7b) For any point a∈S, the inclusion of clusters C(S, a;α)⊆C(S, a;α0) implies
that any self-isometry of the larger cluster C(S, a;α0) can be restricted to a self-
isometry of the smaller cluster C(S, a;α). So Sym(S, a;α0)⊆Sym(S, a;α).
(7c) If points a, b ∈Sare α0-equivalent at the larger radius α0, i.e. the clusters
C(S, a;α0) and C(S, b;α0) are isometric, then a, b are α-equivalent at the smaller
radius α. Hence any α0-equivalence class is a subset of an α-equivalence class.
Property (7c) can be illustrated by the examples in Fig. 5 and 6. For α= 1,
all points of the periodic set S={0,1,3,4}+ 8Zare in the same α-equivalence
class with 1-cluster {0,1}. For α0= 2, Ssplits in two α0-equivalence classes: one
containing the points from 0+Zand 4+ Zwith the 2-clusters {0,1}and another
one containing 1 + Zand 3 + Zwith 2-clusters {−1,0,2}.
If a point set Sis periodic, the α-partitions of Sstabilize in the sense below.
An isometry classification of periodic point sets 9
Definition 8 (a stable radius).Let a periodic point set S⊂Rnand βbe an
upper bound of its bridge distance β(S) from Definition 3. A radius α≥βis
called stable if both conditions below hold:
(8a) the α-partition P(S;α) coincides with the (α−β)-partition P(S;α−β);
(8b) the symmetry groups stabilize: Sym(S, a;α) = Sym(S, a;α−β) for all points
a∈S, which is enough to check for points only from a finite motif of S.
A minimum radius αsatisfying the above conditions for the bridge distance
β(S) from Definition 3 can be called the minimum stable radius and denoted by
α(S). Upper bounds of α(S) and β(S) will be enough for all results below.
Due to Lemma (7bc), conditions (8ab) imply that the α0-partitions P(S;α0)
and the symmetry groups Sym(S, a;α0) remain the same for all α0∈[α−β, α].
Condition (8b) doesn’t follow from condition (8a) due to the following ex-
ample. Let Λbe the 2D lattice with the basis (1,0) and (0, β) for β > 1. Then
βis the bridge distance of Λ. Condition (8a) is satisfied for any α≥0, because
all points of any lattice are equivalent up to translations. However, condition
(8b) fails for any α < β + 1. Indeed, the α-cluster of the origin (0,0) contains
five points (0,0),(±1,0),(0,±β), whose symmetries are generated by the two
reflections in the axes x, y, but the (α−β)-cluster of the origin consists of only
(0,0) and has the symmetry group O(2).
Condition (8b) might imply condition (8a), but in practice it makes sense to
verify (8b) only after checking much simpler condition (8a). Both conditions are
essentially used in the proofs of Isometry Classification Theorem 10.
For the set S={0,1,3,4}+ 8Zin Fig. 5 with the bridge distance β(S) = 4,
any α≥6 is a stable radius, because the partition P(S;α−4) splits Sinto the
same two classes for any α≥6. For the periodic set Q={0,3,4,5}+ 8Zin
Fig. 6 with the bridge distance β(Q) = 3, any α≥4 is a stable radius.
Any periodic set S⊂Rnwith mmotif points has at most m α-equivalence
classes, because any point of Scan be translated to a motif point. Hence it suffices
to check condition (8a) about α-partitions only for the mmotif points. Condi-
tion (8b) can be practically checked by testing if the inclusion Sym(S, a;α0)⊂
Sym(S, a;α) from (7b) is surjective, which is needed only for one representative
cluster from at most misometry classes (exactly mis Sis m-regular).
A stable radius in [5] was defined by using the notations ρand ρ+t. This pair
changed to α−βand α, because subsequent Theorem 10 is more conveniently
stated for the larger radius α. Any 1-regular set in R3with a bridge distance β
has a stable radius α= 7βor ρ= 6tin the past notations of [9].
5 Isosets completely classify periodic sets up to isometry
A criterion of m-regular sets [10, Theorem 1.3] has inspired us to introduce
the new invariant isoset in Definition 9, whose completeness (injectivity) in the
isometry classification of periodic sets will be proved in main Theorem 10.
10 O. Anosova and V. Kurlin
Definition 9 (isoset I(S;α) of a periodic point set Sat a radius α).Let a
periodic point set S⊂Rnhave a motif Mof mpoints. Split all points a∈M
into α-equivalence classes. Then each α-equivalence class consisting of (say) k
points in Mcan be associated with the isometry class of σ= [C(S, a;α)] of an
α-cluster centered at one of these kpoints a∈M. The weight of the class σis
defined as w=k/m. Then the isoset I(S;α) is defined as the unordered set of
all isometry classes with weights (σ;w) over all points a∈M.
All points aof a lattice Λ⊂Rnare α-equivalent for any α≥0, because all
α-clusters C(Λ, a;α) are isometrically equivalent to each other by translations.
Hence the isoset I(Λ;α) is one isometry class of weight 1 for any α.
All isometry classes σ∈I(S;α) are in a 1-1 correspondence with all α-
equivalence classes in the α-partition P(S;α) from Definition 6. So I(S;α) with-
out weights is a set of points in the isotree IT(S) at the radius α. The size of
the isoset I(S;α) equals the cluster count |P(S;α)|. Formally, I(S;α) depends
on α, because α-clusters grow in α. To distinguish periodic point sets S, Q up
to isometry, we will compare their isosets at a common stable radius α.
An equality σ=ξbetween isometry classes of clusters means that there is
an isometry ffrom a cluster C(S, a;α) representing σto a cluster C(Q, b;α)
representing ξsuch that f(a) = b, i.e. frespects the centers of the clusters.
The set S={0,1,3,4}+ 8Zin Fig. 5 has the isoset I(S; 6) of two isometry
classes of 6-clusters represented by {−4,−3,−1,0,1,4,5}and {−3,−2,0,1,5,6}
centered at 0. The set Q={0,3,4,5}+ 8Zin Fig. 6 has the isoset I(Q; 4) of
three isometry classes of 4-clusters represented by {−4,−3,0,3,4},{−3,0,1,2},
{−4,−1,0,1,4}. To conclude that S, Q are not isometric, Theorem 10 will require
us to compare their isosets at a common stable radius α≥6. In the above case
it suffices to say that the stabilized cluster counts differ: 2 6= 3.
An equality σ=ξbetween isometry classes means that there is an isometry
ffrom a cluster in σto a cluster in ξso that frespects the centers of the
clusters. This equality is checked in time O(kn−2log k) for any dimension n≥3
by [1, Theorem 1(a)], where kis the maximum number of points in the clusters.
Theorem 10 (complete isometry classification of periodic point sets).For any
periodic point sets S, Q ⊂Rn, let αbe a common stable radius satisfying Defi-
nition 8 for an upper bound βof β(S), β (Q). Then S, Q are isometric if and only
if there is a bijection between their isosets respecting weights: I(S;α) = I(Q;α)
means that any isometry class (σ;w)∈I(S;α) of a weight wcoincides with a
class (ξ;w)∈I(Q;α) of the same weight wand vice versa.
Theoretically a complete invariant of Sshould include isosets I(S;α) for all
sufficiently large radii α. However, when comparing two sets S, Q up to isometry,
it suffices to build their isosets only at a common stable radius α.
The α-equivalence and isoset in Definition 9 can be refined by labels of points
such as chemical elements, which keeps Theorem 10 valid for labeled points.
An isometry classification of periodic point sets 11
Recall that isometries include reflections, however an orientation sign can be
easily added to α-clusters, hence we focus on the basic case of all isometries.
The proposed complete invariant for classification Problem 1 is the function
S7→ I(S;α) from any periodic point set Sto its isoset at a stable radius α, which
doesn’t need to be minimal. All points aof a lattice Λ⊂Rnare α-equivalent to
each other for α≥0, because all α-clusters C(Λ, a;α) are related by translations,
hence the isoset I(Λ;α) of any lattice is a single isometry class for any α.
Lemmas 11 and 12 help to extend an isometry between local clusters to full
periodic sets to prove the complete isometry classification in Theorem 10.
Lemma 11 (local extension).Let periodic sets S, Q ⊂Rnhave bridge distances
at most βand a common stable radius αsuch that α-clusters C(S, a;α) and
C(Q, b;α) are isometric for some a∈S,b∈Q. Then any isometry f:C(S, a;α−
β)→C(Q, b;α−β) extends to an isometry C(S, a;α)→C(Q, b;α).
Proof. Let g:C(S, a;α)→C(Q, b;α) be any isometry, which may not coincide
with fon the (α−β)-subcluster C(S, a;α−β). The composition f−1◦giso-
metrically maps C(S, a;α−β) to itself. Hence f−1◦g=h∈Sym(S, a;α−β)
is a self-isometry. Since the symmetry groups stabilize by condition (8b), the
isometry hmaps the larger cluster C(S, a;α) to itself. Then the initial isometry
fextends to the isometry g◦h−1:C(S, a;α)→C(Q, b;α) as required.
Lemma 12 (global extension).For any periodic point sets S, Q ⊂Rn, let αbe
a common stable radius satisfying Definition 8 for an upper bound βof both
β(S), β(Q). Assume that I(S;α) = I(Q;α). Fix a point a∈S. Then any local
isometry f:C(S, a;α)→C(Q, f (a); α) extends to a global isometry S→Q.
Proof. We shall prove that the image f(b) of any point a0∈Sbelongs to Q,
hence f(S)⊂Q. Swapping the roles of Sand Qwill prove that f−1(Q)⊂S,
i.e. fis a global isometry S→Q. By Definition 3 the above points a, a0∈S
are connected by a sequence of points a=a0, a1, . . . , am=a0∈Ssuch that
|~ai−1−~ai| ≤ β,i= 1, . . . , m, where βis an upper bound of both β(S), β(Q).
The cluster C(S, a;α) is the intersection S∩B(a;α). The ball B(a;α) contains
the smaller ball B(a1;α−β) around the closely located center a1. Indeed, since
|~a −~a1| ≤ β, the triangle inequality for the Euclidean distance implies that any
c∈B(a1;α) with |~a1−~c| ≤ α−βsatisfies |~a −~c|≤|~a −~a1|+|~a1−~c| ≤ α.
Due to I(S;α) = I(Q;α) the isometry class of C(S, a1;α) coincides with
an isometry class of C(Q, b;α) for some b∈Q, i.e. C(S, a1;α) is isometric to
C(Q, b;α). Then the clusters C(S, a1;α−β) and C(Q, b;α−β) are isometric.
By condition (8a), the splitting of Qinto α-equivalence classes coincides
with the splitting into (α−β)-equivalence classes. Take the (α−β)-equivalence
class [C(Q, b;α−β)] containing b. This class includes the point f(a1)∈Q,
because frestricts to the isometry f:C(S, a1;α−β)→C(Q, f (a1); α−β) and
C(S, a1;α−β) was shown to be isometric to C(Q, b;α−β).
12 O. Anosova and V. Kurlin
The α-equivalence class [C(Q, b;α)] includes both band f(a1). The isometry
class [C(Q, b;α)] = [C(S, a1;α)] can be represented by the cluster C(Q, f(a1); α),
which is now proved to be isometric to C(S, a1;α).
We apply Lemma 11 for frestricted to C(S, a1;α−β)→C(Q, f (a1), α −β)
and conclude that fextends to an isometry C(S, a1;α)→C(Q, f (a1); α).
Continue applying Lemma 11 to the clusters around the next center a2and
so on until we conclude that the initial isometry fmaps the α-cluster centered
at am=a0∈Sto an isometric cluster within Q, so f(a0)∈Qas required.
Lemma 13 (all stable radii of a periodic set).If αis a stable radius of a periodic
point set S⊂Rn, then so is any larger radius α0> α. Then all stable radii form
the interval [α(S),+∞), where α(S) is the minimum stable radius of S.
Proof. Due to Lemma (7bc), conditions (8ab) imply that the α0-partition P(S;α0)
and the symmetry groups Sym(S, a;α0) remain the same for all α0∈[α−β, α].
We need to show that they remain the same for any larger α0> α.
Below we will apply Lemma 12 for the same set S=Qand β=β(S).
Let points a, b ∈Sbe α-equivalent, i.e. there is an isometry f:C(S, a;α)→
C(S, b;α). By Lemma 12 the local isometry fextends to a global self-isometry
S→Ssuch that f(a) = b. Then all larger α0-clusters of a, b are isometric,
i.e. a, b are α0-equivalent and P(S;α) = P(S, α0). Similarly, any self-isometry of
C(S, a;α) extends to a global self-isometry, i.e. the symmetry group Sym(S, a;α0)
for any α0> α is isomorphic to Sym(S, a;α0).
Proof of Theorem 10. The part only if ⇒follows by restricting any given global
isometry f:S→Qbetween the infinite sets of points to the local α-clusters
C(S, a;α)→C(Q, f (a); α) for any point ain a motif Mof S.
Hence the isometry class [C(S, a;α)] is considered equivalent to the class
[C(Q, f (a); α)], which can be represented by the α-cluster C(Q, b;α) centered at
a point bin a motif of Q. Since fis a bijection and the point a∈Mwas arbitrary,
we get a bijection between isometry classes with weights in I(S;α) = I(Q;α).
The part if ⇐. Fix a point a∈S. The α-cluster C(S, a;α) represents a class
with a weight (σ, w)∈I(S;α). Due to I(S;α) = I(Q;α), there is an isometry
f:C(S, a;α)→C(Q, f (a); α) to a cluster from an equal class (σ, w)∈I(Q;α).
By Lemma 12 the local isometry fextends to a global isometry S→Q.
6 A discussion of further properties of isosets
This paper has resolved the ambiguity challenge for crystal representations,
which is common for many data objects [3]. Crystal descriptors [15] are often
based on ambiguous unit cells or computed up to a manual cut-off radii. Rep-
resentations of 2-periodic textiles [6] should be similarly studied up to periodic
isotopies [3, section 10] without fixing a unit cell. Definition 8 gives conditions
for a stable radius so that larger clusters will not bring any new information.
An isometry classification of periodic point sets 13
The recent survey of atomic structure representations [21] confirmed that
there was no complete invariant that distinguishes all crystals up to isometry.
Theorem 10 provides a complete invariant for the first time. The follow-up pa-
per [3] discusses computations and continuity of the new invariant isoset. Isosets
consisting of different numbers of isometry classes will be compared by the Earth
Mover’s Distance [14]. We thank all reviewers for their time and suggestions.
The recent developments in Periodic Geometry include complete classifica-
tion of periodic sequences [4,17], continuous maps of Lattice Isometry Spaces
in dimension two [18,7] and three [16,8], and applications to materials science
[23,26]. The latest ultra-fast and generically complete Pointwise Distance Dis-
tributions [24] justified the Crystal Isometry Principle (CRISP) saying that all
real periodic crystals live in a common space of isometry classes of periodic point
sets continuously parameterised by their complete invariants such as isosets.
References
1. Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity, and sym-
metries of geometric objects. Discrete & Comp. Geometry 3, 237–256 (1988)
2. Andrews, L., Bernstein, H., Pelletier, G.: A perturbation stable cell comparison
technique. Acta Crystallographica A 36(2), 248–252 (1980)
3. Anosova, O., Kurlin, V.: Introduction to periodic geometry and topology.
arXiv:2103.02749 (2021)
4. Anosova, O., Kurlin, V.: Density functions of periodic sequences (2022), https:
//arxiv.org/abs/22
5. Bouniaev, M., Dolbilin, N.: Regular and multi-regular t-bonded systems. J. Infor-
mation Processing 25, 735–740 (2017)
6. Bright, M., Kurlin, V.: Encoding and topological computation on textile structures.
Computers & Graphics 90, 51–61 (2020)
7. Bright, M., Cooper, A.I., Kurlin, V.: Geographic-style maps for 2-dimensional
lattices. arxiv:2109.10885 (early draft) (2021), http://kurlin.org/projects/
periodic-geometry-topology/lattices2Dmap.pdf
8. Bright, M., Cooper, A.I., Kurlin, V.: Welcome to a continuous world of 3-
dimensional lattices. arxiv:2109.11538 (early draft) (2021), http://kurlin.org/
projects/periodic-geometry-topology/lattices3Dmap.pdf
9. Dolbilin, N., Bouniaev, M.: Regular t-bonded systems in R3. European Journal of
Combinatorics 80, 89–101 (2019)
10. Dolbilin, N., Lagarias, J., Senechal, M.: Multiregular point systems. Discrete &
Computational Geometry 20(4), 477–498 (1998)
11. Edelsbrunner, H., Heiss, T., Kurlin, V., Smith, P., Wintraecken, M.: The density
fingerprint of a periodic point set. In: Proceedings of SoCG. pp. 32:1–32:16 (2021)
12. Gr¨unbaum, F., Moore, C.: The use of higher-order invariants in the determination
of generalized patterson cyclotomic sets. Acta Cryst. A 51, 310–323 (1995)
13. Hahn, T., Shmueli, U., Arthur, J.: Intern. tables for crystallography, vol. 1 (1983)
14. Hargreaves, C.J., Dyer, M.S., Gaultois, M.W., Kurlin, V.A., Rosseinsky, M.J.:
The earth mover’s distance as a metric for the space of inorganic compositions.
Chemistry of Materials (2020)
14 O. Anosova and V. Kurlin
15. Himanen, Land J¨ager, M., Morooka, E., Canova, F., Ranawat, Y., Gao, D., Rinke,
P., Foster, A.: Dscribe: Library of descriptors for machine learning in materials
science. Computer Physics Communications 247, 106949 (2020)
16. Kurlin, V.: A complete isometry classification of 3-dimensional lattices.
arxiv:2201.10543 (early draft) (2022), http://kurlin.org/projects/
periodic-geometry-topology/lattices3Dmaths.pdf
17. Kurlin, V.: A computable and continuous metric on isometry classes of high-
dimensional periodic sequences (2022), https://arxiv.org/abs/22
18. Kurlin, V.: Mathematics of 2-dimensional lattices.
https://arxiv.org/abs/2201.05150 (early draft) (2022), http://kurlin.org/
projects/periodic-geometry-topology/lattices2Dmaths.pdf
19. Mosca, M., Kurlin, V.: Voronoi-based similarity distances between arbitrary crystal
lattices. Crystal Research and Technology 55(5), 1900197 (2020)
20. Patterson, A.: Ambiguities in the x-ray analysis of crystal structures. Physical
Review 65, 195 (1944)
21. Pozdnyakov, S., Willatt, M., Bart´ok, A., Ortner, C., Cs´anyi, G., Ceriotti, M.:
Incompleteness of atomic structure representations. Phys. Rev. Lett. 125, 166001
(2020), arXiv:2001.11696
22. Pulido, A., Chen, L., Kaczorowski, T., Holden, D., Little, M., Chong, S., Slater,
B., McMahon, D., Bonillo, B., Stackhouse, C., Stephenson, A., Kane, C., Clowes,
R., Hasell, T., Cooper, A., Day, G.: Functional materials discovery using energy–
structure–function maps. Nature 543, 657–664 (2017)
23. Ropers, J., Mosca, M.M., Anosova, O., Kurlin, V., Cooper, A.I.: Fast predictions of
lattice energies by continuous isometry invariants of crystal structures. In: Proceed-
ings of DACOMSIN: Data and Computation for Materials Science and Innovation.
https://arxiv.org/abs/2108.07233
24. Widdowson, D., Kurlin, V.: Pointwise distance distributions of periodic sets (2021),
http://kurlin.org/projects/periodic-geometry-topology/PDD.pdf
25. Widdowson, D., Mosca, M., Pulido, A., Kurlin, V., Cooper, A.: Average
minimum distances of periodic point sets - fundamental invariants for map-
ping all periodic crystals. MATCH Communications in Mathematical and
in Computer Chemistry 87, 529–559 (2022), http://kurlin.org/projects/
periodic-geometry-topology/AMD.pdf
26. Zhu, Q., Johal, J., Widdowson, D., Pang, Z., Li, B., Kane, C., Kurlin, V., Day,
G., Little, M., Cooper, A.: Analogy powered by prediction and structural invari-
ants: Computationally-led discovery of a mesoporous hydrogen-bonded organic
cage crystal. Journal of Amer Chem Soc (to appear)
