Periodic triangulations of $\mathbb{Z}^n$

We consider in this work triangulations of $\mathbb{Z}^n$ that are periodic along $\mathbb{Z}^n$. They generalize the triangulations obtained from Delaunay tessellations of lattices. Other important property is the regularity and central-symmetry property of triangulations. Full enumeration for dimension at most $4$ is obtained. In dimension $5$ several new phenomena happen: there are centrally-symmetric triangulations that are not Delaunay, there are non-regular triangulations (it could happen in dimension $4$) and a given simplex has a priori an infinity of possible adjacent simplices. We found $950$ periodic triangulations in dimension $5$ but finiteness is unknown.


Introduction
Given a positive definite quadratic form A we obtain a tessellation of Z n by taking the projection of the facets of the convex hull of (x, x T Ax) for x ∈ Z n . This triangulation is Z n -periodic, centrally symmetric and is called the Delaunay tessellation [11]. For dimension at most 5 those tessellations are classified and there are 1, 2, 5, 52 and 110244 for 1 ≤ n ≤ 5 [17,9] up to the action of GL n (Z). If one limit oneself to the Delaunay triangulations formed of simplices then the number of types is 1, 1, 1, 3 and 222 [3,15,12] for 1 ≤ n ≤ 5, respectively. For n = 6 Baburin and Engel [2] reported more than 500'000'000 non-equivalent triangulations. A triangulation is called regular if it is obtained as projection of facets of an infinite convex body of vertices (x, f (x)) for f a function defined on Z n . This generalizes the Delaunay tessellations.
In this paper we consider general triangulations of the point set Z n which are invariant under translations by Z n and are face-to-face. Such triangulations can be viewed as decomposition of a torus into a cell complex with one vertex where all cells are simplices. Other triangulations of the torus into simplices were considered in [13,14]. In Section 2 we consider general results on periodic triangulations of Z n in particular on groups, simplices and refinement of periodic tilings. In Section 3 we detail a number of computational tools for testing Delaunayness and regularity that we use in this work.
In Section 4 we prove that for n ≤ 3 all such triangulations are Delaunay. For n = 4 a non-centrally symmetric triangulation named "red-triangular" was described in [1,Example 5.13.1]. We also prove in Section 5 that this triangulation together with the Delaunay ones form all the set of triangulations up to the action of GL 4 (Z).
In dimension n ≥ 5 full enumeration of periodic triangulation appears to be difficult. First of all the finiteness is not proved and may not hold since we prove in Section 6 that given a simplex of volume 1 there are a priori infinitely many possibilities for an adjacent simplex.
In Section 7 we obtain 950 non-isomorphic periodic triangulations of Z 5 but we do not know if this is the complete list. This list allow us to prove that there are centrally symmetric but not Delaunay triangulations and non-regular triangulations.
The first author has been supported by the Humboldt Foundation.
In 8 we list several open questions on enumeration, extensibility and regularity properties of periodic triangulations of Z n that should be of general interest.

General properties of periodic triangulations
Definition 2.1. A k-dimensional lattice is a discrete subgroup of R n of rank k, i.e. a set of the form L = Zv 1 + · · · + Zv k with linearly independent vectors v 1 , . . . , v k .
Throughout the paper we will work with the lattice L = Z n . The group of affine transformations preserving a n-dimensional lattice is isomorphic to AGL n (Z).
Definition 2.2. A partial triangulation PT of Z n is a packing of R n by n-dimensional simplices with the vertex set Z n , i.e. representation of R n as a union of countably many simplices with integer vertices such that the intersection of any pair of simplices is a face of both.
A triangulation T is a partial triangulation which is also a tiling.
Definition 2.3. The symmetry group Sym(T ) of a periodic triangulation T of Z n is the group of affine transformations of R n preserving T . The group Sym(T ) contains Z n as a normal subgroup of finite index. The quotient Sym(T )/Z n is called the point group P t(T ).
The symmetry group is split if Sym(T ) is a semi-direct product Z n ⋊ P t(T ). This is equivalent to having P t(T ) being realized as a subgroup of Sym(T ).
Proposition 2.4. The symmetry group of a periodic triangulation T is split.
Proof. Let v 0 = 0, v 1 = e 1 , . . . , v n = e n with (e i ) the standard basis of Z n . Let f be a symmetry of the lattice. Define v ′ i = f (v i ) for 0 ≤ i ≤ n and write the transformation f in matrix form as Ax + b. Then This equation implies that A and b are integral. Thus f is the composition of a transformation preserving the origin and an integral translation. This implies that the symmetry group is split.
Definition 2.5. Let Λ be a d-dimensional lattice with fundamental volume V , and let S be a d-dimensional simplex with vertices from Λ. In this case volume of S is k · V d! for some integer k, and we will say that relative volume of S is k.
In the following we will refer to relative volume of S as just volume of S, or vol(S), unless we need to emphasize the dimension. Proof. If the periodic triangulation is formed by the simplices S 1 , . . . , S p and their Z n translations then we have the equality from which (i) obviously follows. The proof of (ii) follows exactly the same arguments as [6,Proposition 14.2.4].
Definition 2.7. Let T be a periodic tiling of Z n by polytopes having only integer points as vertices. Let A be a positive definite quadratic form on Z n . Then A induces another tiling Ref A (T ) of Z n defined on each polytope P ∈ T as projection of the lower facets of Lemma 2.8. For a n-dimensional periodic tiling T and a positive definite quadratic form A the following properties hold: is a periodic tiling of Z n which is a refinement of T .
(ii) If g ∈ GL n (Z) preserves A and belongs to the point group of T then it belongs to the point group of Proof. (i) Let us consider for each polytope P of T the scaling map used to describe the Delaunay polytopes.
The lower facets of Scal(P ) define a tiling of P into polytopes. It also defines a tiling of the faces of P . If a face F of T is contained into polytopes P 1 , . . . , P m then the induced tilings are compatible. The tiling is periodic since A[x] differ on two different translate of a tile by an affine term.
(iii) If A is generic then the tiling induced by Scal(P ) is a triangulation which proves the claim. Definition 2.9. A triangulation T of Z n is called regular if there exists a function f : Z n → R such that: • The points (x, f (x)) are vertices of the convex polyhedron H(f ) = conv{(x, f (x))|x ∈ Z n } in R n+1 . • The simplices of T are orthogonal projections of the facets of H(f ) onto R n .
Obviously Delaunay triangulations are regular.

Computational tools
3.1. Testing Delaunay property. Given a periodic triangulation T of Z n we can test if it is Delaunay in the following way: We determine all facets F of simplices S of T up to translations. Any such facet F is contained in exactly two Delaunay simplices S 1 = conv(F ∪ {v 1 }) and S 2 = conv(F ∪ {v 2 }). We then form the inequality N S1,v2 (A) ≥ 0 with N S,v being a linear form called the Voronoi regulator (see [18,10] for details). The polyhedral cone defined by all such inequalities is called P. If P is full dimensional then every quadratic form in the interior of P induces T as Delaunay triangulation. Otherwise, it is not a Delaunay triangulation.

3.2.
Adjacency of simplices. Suppose we have two simplices ∆ 1 and ∆ 2 and we want to check if the Z n translates of ∆ 1 and ∆ 2 are a priori admissible as parts of a periodic triangulation of Z n . That is we want to check that the translates do not intersect in their interior and that the intersection is always a face of both. If F is a facet of ∆ 1 represented by an inequality f (x) ≥ 0 with an affine function f then if we have f (∆ 2 + v) < 0 then there is no intersection. That is for some facet f we have max x∈T2 f (x) + f (v) < 0 then there is no intersection. So, the feasible vectors v are the ones that satisfies max x∈T2 f (x + v) ≥ 0 for all facet inequalities f of ∆ 1 .
This defines a convex body C and the integer points can be obtained by using exhaustive enumeration. Then for each integer point v ∈ C we check if ∆ 1 ∩ (∆ 2 + v) is n-dimensional or not. If it is not then we check that the intersection is a face of both.
With this method we can find the possible simplices adjacent to a given simplex ∆. That is for a facet F of ∆ and a point v ∈ Z n we consider whether the pair ∆ and conv(F ∪ {v}) is admissible for a periodic tessellation. By iterating over the facets and vectors v we have a list of possible candidates. However, we have no method for restricting the set of possible vectors v and in Section 6 we show that the set of such vectors can be infinite.
3.3. Testing regularity. Given a periodic triangulation T we want to check if it is regular. According to definition 2.9 the condition that the simplices correspond to facets of the convex body H(f ) translate into linear inequalities on the values of f (v) for v ∈ Z n . Thus testing regularity is equivalent to checking if an infinite dimensional linear program has a feasible solution.
We do not have a general method for working with infinite dimensional linear programs and thus we cannot check regularity of triangulations easily. What we can do instead is prove in some cases that a periodic triangulation is not regular. Let us take a triangulation T of Z n and select a finite set S of simplices. We can consider the function f on the set of vertices V corresponding to the simplices of S. If we have two adjacent simplices S 1 and S 2 then denote by φ S1 the affine linear function coinciding with f on S 1 . For a vertex v of S 2 which is not in S 1 we must have the inequality Now if the function f does exist, and therefore T is regular, then by rescaling there exists a function f on V satisfying This strengthened inequalities define a polyhedral convex body Q. If Q is proven to be empty by linear programming, then we have proved that T is not regular.

Equivalence and stabilizer.
For enumeration purposes, we need to be able to check that two triangulations are equivalent and to compute the stabilizer of a triangulation. The method is simply to take one simplex in a triangulation and to consider all ways in which it may be mapped in a simplex of another or the same triangulation. While computationally expensive, this method is adequate for the cases that we consider which are low-dimensional.

Enumeration of periodic triangulations in dimension 3
The main goal of this and the next section is to give complete enumeration of periodic triangulations in dimensions 3 and 4.
Let S be a simplex in a triangulation (not necessary full-dimensional), denote by tr(S) the translation class of S, i.e. the set of all translations of S by integer vector.
The following lemma is true for arbitrary dimension. Proof. Assume contrary, then there is one more facet F ′ of S 1 which is parallel to some facet of S 2 . If F and F ′ are in the same copy of S 2 in tr(S 2 ), then S 1 =S 2 . Otherwise, facets F and F ′ have a common ridge, so S 2 should have two parallel ridges which is impossible.
In dimension 3, it appears to be known that there is only one periodic triangulation. See work of Alexeev, [1, Sect. 5.13]. Here we give another straightforward approach.
In order to obtain the full classification, we show how one can find the upper bound on the relative volume of a simplex in a periodic triangulation. We apply it for dimensions 3 here and for dimension 4 in the following section. For dimension 2 it is clear that each simplex should have volume 1 (for example, from Pick's formula).
Let T denote an arbitrary periodic triangulation of Z n . We will use a more careful approach compared to Proposition 2.6 in order to show, that if the relative volume of a simplex exceeds a certain number (actually it is 1 for dimensions 3 and 4), then this simplex can not be included in a periodic triangulation, and in T particularly.

Proposition 4.2.
If T is three-dimensional, then the relative volume of each three-dimensional simplex is 1.
Proof. Let ABCD be an arbitrary simplex of T with volume at least 2. It is clear, that relative volume of every facet of ABCD in corresponding sublattice is 1. So, we can choose a coordinate system (with matrix transformation from GL 3 (Z)) such that vertices of ABCD will have coordinates represented by columns of the following matrix where a, b, c are non-negative, c ≥ 2, a < c, and b < c.
If a is non-zero (similarly for non-zero b), then the point belongs to translations of two faces of ABCD, namely to the two-dimensional face ABD (the left-hand side of the formula) and to the edge AC (the right-hand side of the formula), and is not an integer point. Thus this is a contradiction with the face-to-face property of the tiling T . If a = b = 0 then we have an integer point (0, 0, 1) in the interior of the edge AD which is impossible. Remark 4.3. From this proof we can see, that each 3-dimensional face of T should have the relative volume 1, otherwise we will find a contradiction in 3-dimensional affine space spanned by this face. Indeed, if a relative volume of a 3-dimensional simplex is more than 1, then according to the proof of the previous proposition, its lattice translates will intersect in a non-face-to-face manner.
Next we establish all possible neighbors of a given simplex in a periodic triangulation T of Z 3 .
Lemma 4.4. If n = 3, then given a simplex S 1 of T and its facet F , we have 3 options for a simplex S 2 of T adjacent to S 1 by F . More precisely, two other facets of S 1 and S 2 must form a parallelogram.
At least one of numbers x, y, z is even, assume z. Then the midpoint of EB has coordinates x+1 2 , y 2 , z 2 . Among numbers x + 1 and y one is even, so this midpoint has two integer coordinates, and one half-integer. Therefore, this midpoint is a translation of one of midpoints: AB or AC. Therefore the edge EB is a translation of AB or AC. Similarly, the edge EC is a translation of AB or AC.
There are two options remaining for point E: E = (0, 0, 0) = A, or E = (1, 1, 0). In the first case simplices S 1 and S 2 coincide, which is impossible. In the second case faces ABC and EBC form a parallelogram.
We proceed with the classification of periodic triangulations of Z 3 . We continue to use all notations of Lemma 4.4 and its proof.
Proof. From the proof of Lemma 4.4 we have a pair of simplices S 1 and S 2 , and a parallelogram ABEC (actually a unit square). With translations of this parallelogram we can tile an arbitrary plane z = k for integer k, so any simplex of the triangulation should be between a pair of consecutive planes parallel to z = 0.
Currently we have six "unpaired" facets of tiling (i.e. facets that belong to only one simplex currently determined): ABC, ABD, ACD, EBC, EBD, ECD. No translational class of full-dimensional simplex can contain more than two of these facets, because otherwise it will have two common facets with one of simplices S 1 or S 2 . The second simplex S 3 incident to the facet ABC has the fourth vertex on the plane z = −1, because S 3 has three vertices on z = 0, and ABCD has fourth vertex on z = 1). Similarly, the simplex S 4 incident to EBC has the fourth vertex on the plane z = −1. So, S 3 can not have a facet which is a translation of any of five remaining "unpaired" facets, since four of these facets (except EBC) have two vertices on lower plane and one on the upper, and the facets EBC is parallel to facet ABC. Therefore, classes tr(S 3 ) and tr(S 4 ) contain only facets ABC and EBC from these six classes.
The remaining unpaired facets are: ABD, ACD, EBD, ECD. These classes should be contained in two translational classes of simplices (we already found four classes generated by S 1 , S 2 , S 3 , and S 4 , and we must have six translational classes in total due to volume argument). No class can cover more than two, so these four facets should be divided in pairs, and each pair should belong to one translational class. Pairs can not be from one simplex S 1 or S 2 . So ABD should be paired with EBD or ECD. The first case is impossible, because the edge BD can belong only to one simplex from this class, so this class should be tr(ABDE), but ABDE intersects with interior of ABCD.
Therefore, the class tr(S 5 ) contains facets ABD and ECD, and the class tr(S 6 ) contains remaining facets ACD and EBD.
From four completely defined classes tr(S 1 ), tr(S 2 ), tr(S 5 ), and tr(S 6 ) we have the following facets that do not belong to the second full-dimensional simplex so far: ABC, EBC, ADF , BDF , ADG, CDG. No class tr(S 3 ) or tr(S 4 ) can cover more than three of these facets, otherwise it will cover two facets from one simplex (facets ABC and EBC cannot be covered simultaneously because they are parallel). So each class covers exactly three.
We apply Lemma 4.4 for simplices ABCD and S 3 with common facet ABC. We have three options for the fourth vertex of Similarly with other "unpaired" facets, except EBC, but S 3 already has a facet parallel to EBC, which is ABC.
So, there is only one possible case for S 3 which is ABCH 1 (the translation class contains ABC, BDF , CDG). Similarly, there is only one case for S 4 = EBCH 1 (the translation class contains EBC, ADF , ADG).
We reconstructed the whole triangulation which is unique up to GL 3 (Z)-transformation.

Enumeration of periodic triangulations in dimension 4
As with dimension 3, we first bound the relative volume of a four-dimensional simplices.
Proposition 5.1. If T is a periodic triangulation of Z 4 , then volume of each four-dimensional simplex is 1.
Proof. Let ABCDE be an arbitrary simplex of T with volume at least 2. We can choose a coordinate system (with matrix transformation from GL 4 (Z)) such that vertices of ABCDE will have coordinates represented by columns of the following matrix If c = 0, then the point (0, 0, 0, 1) lies in the interior of AE which is impossible, so c ≥ 1.
but these points lie on different faces of ABCDE and the tiling will be non face-to-face. If b + c > d, then which is again a contradiction.
So, a + b > d ≥ b + c, which contradicts with the inequality a ≤ c.
Note that the proofs of this proposition and of the similar proposition 4.2 for dimension 3 can be combined in the following corollary. This corollary allows us to formulate a local approach to enumeration of all periodic tilings. We used this approach in dimension 3 in the previous section and now we are going to use it in dimension 4. We can analyze local structure of the tiling T and show that given a simplex S and its facet F , there are only finitely many options to attach another simplex T at F without violating the face-to-face property. Unfortunately this method doesn't work if dimension n ≥ 5 as shown in Section 6. Proof. We already know, that all simplices have volume 1. We fix one simplex S 1 with vertices A = (0, 0, 0, 0), B = (1, 0, 0, 0), C = (0, 1, 0, 0), D = (0, 0, 1, 0), E = (0, 0, 0, 1), and find all possibilities for the vertex F of the simplex S 2 = BCDEF adjacent to S 1 by facet BCDE. We know that F has coordinates (x, y, z, t) with x + y + z + t = 2. We will show that there are only 10 options for the vertex F . We will do that by analyzing all possible remainders of coordinates of F modulo powers of 2. First, assume t is even, then at least one more number among x, y, z is even, say z. Then midpoint of BF has coordinates x + 1 2 , y 2 , 0, 0 (mod 1) and it is an integer translation of the midpoint of AB (if x and y are odd) or AC (if x and y are even). The only case that will not contradict that T is face-to-face is when BF is parallel and equal to AC. Similarly we get that CF is parallel and equal to AB, so F = (1, 1, 0, 0). Also we can get five more coordinate permutations of this point in the case F has an even coordinate, in all other cases x, y, z, t are odd. We know that x, y, z, t are odd and their sum is 2, so possible cases for modulo 4 remainders are (3, 3, 3, 1) and (1, 1, 1, 3) (x, y, z, t are equivalent, so we will treat these cases as coordinates for (x, y, z, t) modulo 4). In the first case B + C + D + F 4 = 3A + E 4 (mod 1), and the tiling is non face-to-face, so only the case (1, 1, 1, 3) of remainders modulo 4 is possible.
Proof. We prove the statement by induction on k. The basis of induction is true for k = 2 and a = 1. Suppose the lemma is true for k and we will prove it for k + 1. All the coordinates in our proof could be permuted, and when we consider a coordinate modulo n we usually take a representative from the interval [0, n).
We proceed with the proof of the theorem. We can take k such that 2 k > 2 max(|x|, |y|, |z|, |t|), then the only possibility for coordinates with remainder 1 modulo 2 k is 1, so two coordinates of F are 1's and two other add up to 0, so F = (1, 1, a, −a) for some positive odd number a (or permutation). If a ≥ 3, then and if the tiling is face-to-face then the edge BF is a translation of the edge AB. In that case F = (0, 0, 0, 2), so it doesn't have all odd coordinates. Therefore a = 1 and F = (1, 1, 1, −1). In total we get 10 options for the point F : (1, 1, 0, 0) (all six permutations), and (1, 1, 1, −1) (all four permutations).
Theorem 5.5. (i) There are exactly four periodic triangulations of Z 4 up to GL 4 (Z) equivalence.
(ii) Any partial triangulation of Z 4 is extensible to a full triangulation of Z 4 .
Proof. We use Lemma 5.3 with exact classification of neighbors for an exhaustive computerassisted search. We start from one simplex of volume 1 and add adjacent simplices one by one by considering all possibilities. The number of cases to consider is kept down by keeping only non-isomorphic partial tilings in memory. The software is available at [8] as a GAP package. In the end we get four non-equivalent triangulations three of which are Delaunay triangulations and the "red-triangular" triangulation [1, Example 5.13.1] which proves (i). The intermediate object of the enumeration are exactly the partial triangulations of Z 4 . It turns out that in the enumeration it never happenned that a partial triangulation had no extensions by adding simplex which proves (ii).

Local approach in higher dimensions
In this section we show that local approach we used in Lemma 4.4 and Theorem 5.3 can not prove finiteness of non-equivalent triangulations in dimension at least 5.
Theorem 6.1. For n ≥ 5 there exist a simplex S of volume 1 and an infinite sequence S k of simplices of volume 1 such that S ∩ S k is a facet and the translates of S and S k are not intersecting.
Proof. We first consider the case n = 5.
Let X ′ = (1, 1, 1, 1, k + 1) for any k ≥ 0, then T = OABCDX ′ will satisfy this condition. For any n both simplices S and T have volume 1, so S doesn't intersect translations of S, and T doesn't intersect translations of T . It is enough to show that an arbitrary integer translation of S doesn't intersect T other than by vertices or by the facet x 1 = 0.
For n > 5 the idea is simply to take the pair of simplices and simply add another point.

Flipping and five-dimensional partial enumeration
Let us consider a periodic triangulation T of Z n . Given a simplex S and a facet F of S, we can consider the adjacent simplex S(F ) to S. The union of the vertex sets of S and S(F ) is a set of n + 2 points and we call the convex hull of those Cv(S, F ).
Proof. Suppose that the vertices are v 1 , . . . , v n+2 then there is exactly one linear relation of the form a 1 v 1 + · · · + a n+2 v n+2 = 0 up to a non-zero multiple. For 1 ≤ j ≤ n + 2 we define S j the simplex formed by v i for i ∈ {1, . . . , n + 2} − {j}. The first triangulation is formed by the simplices S j for j such that a j ≥ 0 and the second triangulation by the simplices S j for j such that a j ≤ 0.  Proof. Given a periodic triangulation of Z 5 we consider all ways to do a coherent flipping on it. We thus obtain a set of new periodic triangulations. We insert element of this list into the list of known periodic triangulations if they are not isomorphic to a triangulation already known. We start from one arbitrary Delaunay triangulation of Z 5 . We finish when all periodic triangulations in the list have been treated. Since the finiteness of the set of periodic triangulation is not proved in dimension 5 this process was not guaranteed to terminate. But it did and yielded 950 periodic triangulations. The code is available at [8].
The list of 950 periodic triangulations (222 of them Delaunay) is interesting in its own right and is available at [8]. The volumes of the simplices in the list of 950 triangulations are 1 or 2 which corresponds to the possible volume of simplices in Delaunay tessellations. Given a simplex S of volume 1 and vertices v 0 , . . . , v 5 we can consider which simplices S ′ can be adjacent to S. Their vertex set will be of the form The symmetry of the tiling varies widely with one of the periodic tiling having a point group symmetry isomorphic to the symmetric group Sym(6). Proof. For n = 5 it suffices to take one of the 23 triangulations out of 950 known in dimension 5 that are not Delaunay but are centrally symmetric. For n > 5 this tiling T can be extended with tiles of the form ∆ × [0, 1] n−5 for ∆ a 5-dimensional simplex of T . By applying Lemma 2.8 (iii) for an arbitrary generic quadratic form we obtain a Z n -periodic triangulation. This triangulation is centrally symmetric since x → −x is a symmetry of the original tiling but also of the quadratic form.
Note that existence of a periodic centrally symmetric non-Delaunay triangulation for n = 8 was established in [16]. Proof. For n = 5 we apply the method of subsection 3.3 to one of the 950 triangulations of Theorem 7.2. The list of 3264 simplices of the triangulation number 430 that cannot be part of a regular triangulation is available at [8]. For n > 5 this tiling T can be extended with tiles of the form ∆ × [0, 1] n−5 with ∆ a 5-dimensional simplex of T . By applying Lemma 2.8 (iii) for an arbitrary generic quadratic form we obtain a Z n -periodic triangulation which is necessarily non-regular.

Open problems
In this section we list a number of interesting questions that showed up in the course of this research.

Finiteness and enumeration.
A natural question that we were unable to resolve is whether there are finitely many Z n -periodic triangulations of Z n up to the action of GL n (Z)? Theorem 6.1 shows that a local approach considering only pairs of simplices will not work.
There are many related question. For example in a fixed dimension n, is the set of all periodic triangulations of Z n connected by flipping? The resolution of such questions is certainly very hard since analogue questions about triangulations of the hypercube are still unsolved [5]. The resolution of the above connectedness would imply that the number of triangulations in dimension 5 is exactly 950.
A proof of finiteness in dimension 5 would not a priori give an algorithm for the enumeration since we do not know the possible volume of simplices nor the adjacencies between them.
8.2. Extensibility of partial triangulations. In a lot of contexts of this search we reach a point where we had a partial triangulation of Z n and we wanted to extend it to a full triangulation. Is this always possible? If so what would be a process for obtaining such a triangulation? If this extensibility were true then we would have an infinity of types of periodic triangulation in dimension 5. Note that Theorem 5.5 proves that this extensibility holds in dimension n ≤ 4.
One possible way to consider the problem would be following [4] to consider constrained Delaunay triangulations and see if the relevant notion could be extended to our case. It would require a twofold generalization: a generalization from dimension 2 to any dimension and a generalization to the periodic case. 8.3. Regularity. Is every periodic regular triangulations also Delaunay? The answer is not known. As we saw in Section 3 we can test regularity on finite subsets of Z n by linear programming. But we need actually to define the height function all over Z n . Finding such explicit function is difficult since as soon as we impose some translational invariance on the function f we obtain a function that is actually quadratic.
Is the "red-triangular" [1, Example 5.13.1] Z 4 -periodic triangulation regular? If this triangulation is restricted to a set of 12864 simplices containing 1224 points then we can found a corresponding function f which indicates that this triangulation is likely to be regular.
8.4. Volume of simplices. What is the maximum volume of a simplex in a periodic triangulation? So far in all cases considered, we found that the volumes of the simplices occurring was not higher than the volume of the simplices of the Delaunay triangulations in the same dimension which are 1, 2, 3 and 5, respectively in dimension n ≤ 4, 5, 6 and 7, respectively [7]. We see no reason why this should always be the case.

Acknowledgments
We thank Francisco Santos and Achill Schürmann for interesting discussions on this work.