Examples of quasicrystallic rhombic embeddings

An important class of examples of rhombic embeddings of b-quadgraphs can be constructed using ideas of the grid projection method.

Example 3.3 (Quasicrystallic rhombic embedding obtained from a plane inZ^d).Let E={t+λ1v1+λ2v2:λ1, λ2∈R}

be a two-dimensional plane inR^d. Denote the following segments in direction ofe1, . . . ,ed

by sj ={t+λej : λ∈ [0,1]} for j = 1, . . . , d. We assume that E does not contain any

E Ω^L(E)

(a) Straight lineE.

E4

Ω^L(E4)

(b) Modified lineE4with a “hat”.

Figure 3.2: Example for the usage of the grid-projection method inZ². The Voronoi cells which contain Ω^L are marked in red.

of these segments, that is s₁, . . . , s_d 6⊂ E. If E contains two different segments s_j1 and sj₂, the following construction only leads to the standard square grid pattern Z². If E contains exactly one such segment sj, then the following construction may be adapted for the remaining dimensions (excluding ej), but we do not consider this case. We further assume that the orthogonal projections onto E of the two-dimensional facets Ej1,j2 = {λ1ej1+λ2ej2:λ1, λ2∈[0,1]}for 1≤j1< j2≤dare non-degenerate parallelograms. Then we can choose positive constants c1, . . . , cd such that the orthogonal projections PE(cjej) have length 1.

Consider around each vertex p of the lattice L = c1Z× · · · ×cdZ the hypercuboid V = [−c1/2, c1/2]× · · · ×[−cd/2, cd/2], that is the Voronoi cellp+V. These translations of V then coverR^d. We build an infinite monotone two-dimensional surface Ω^L(E) in the latticeLwhich projects to a rhombic embedding onE by the following construction. The basic idea is illustrated in Figure 3.2 (left) for the toy example of a line in Z².

IfE intersects the interior of the Voronoi cell of a lattice point (i.e. (p+V)^◦∩E 6=∅ forp∈ L), then this point belongs to Ω^L(E). Undirected edges correspond to intersections of E with the interior of a (d−1)-dimensional facet bounding two Voronoi cells. Thus we get a connected graph inL. An intersection ofE with the interior of a translated (d− 2)-dimensional facet ofV corresponds to a rectangular two-dimensional face of the lattice. By construction, the orthogonal projection of this graph ontoE results in a planar connected graph whose faces are all of even degree (= number of incident edges or of incident vertices).

A face of degree bigger than 4 corresponds to an intersection ofE with the translation of a (d−k)-dimensional facet of V for some k ≥ 3. Consider the vertices and edges of such a face and the corresponding points and edges in the lattice L. These points lie on a combinatorialk-dimensional hypercuboid contained inL. Enumerate the edgesw1, . . . , w2k

such that consecutive edges have a vertex in common. Then the edges are such that wj is parallel towj+kforj= 1, . . . , k. Also the corresponding edges ˆw1, . . . ,wˆk inLare linearly independent. Therefore there are two points of the k-dimensional hypercuboid which are each incident to kof the given vertices. We choose a point with least distance fromE and add it to the surface. Adding edges to neighboring vertices splits the face of degree 2kinto k faces of degree 4.

Thus we obtain an infinite monotone two-dimensional combinatorial surface Ω^L(E) which projects to an infinite rhombic embedding covering the whole planeE.

Example 3.4 (Modification of the construction in Example 3.3).The method used in the preceding example can be modified to result in similar but different rhombic embeddings.

The basic idea of the modified construction is illustrated in Figure 3.2 (right).

Let E be a two-dimensional plane inR^d as in Example 3.3. We also make the same assumptions, that is s1, . . . , sd 6⊂ E and the orthogonal projections onto E of the

two-dimensional facets Ej1,j2 = {λ1ej1 +λ2ej2 : λ1, λ2 ∈ [0,1]} for 1 ≤ j1 < j2 ≤ d are non-degenerate parallelograms. Furthermore, we choose positive constantsc₁, . . . , c_d such that kPE(cjej)k= 1.

Let N 6= 0 be any vector orthogonal to E. Let 4 be an equilateral triangle in E with verticest1,t2,t3and letsbe the intersection point of the bisecting lines of the angles.

Consider the two-dimensional facets of the three-dimensional tetrahedronT(4,N) spanned by the four verticest1,t2,t3,s+N. Exactly one of these facets is completely contained in E (this is the triangle4). We remove the triangle from E and add instead the remaining facets ofT(4,N). LetE4be the resulting two-dimensional surface. Note thatE4is closed and orientable likeE and thatE4 can be seen as the graph of a real-valued function over E. Defineγ∈(0, π/2) byγ= arctan(2√

3kNk/kt1−t2k), wherek · kdenotes the Euclidean norm of vectors inR^d. Thenγis the acute angle betweenEand one of the two-dimensional facets of the tetrahedronT(4,N) not contained in E.

By assumption, the intersection ofE with each of the (d−1)-dimensional hyperspaces Hj= span{ek :k∈ {1, . . . , d} \ {j}}

for j = 1, . . . , d is a line `j ⊂ E. Let `j = {pj +v`jλ : λ ∈ R} for suitable vectors pj,v`j ∈R^d. Letv^⊥<sub>`j</sub> be orthogonal tov`j such thatpj+v^⊥_`j ∈E. Then theintersection angleθj∈(0, π/2) ofE andHj can be defined by

cosθj= max{hv^⊥<sub>`j</sub>,wi:w∈Hj,w⊥v`j,kwk= 1}.

Assume that γ < θ_min/2 where θ_min = min{θ_j : j = 1, . . . , d} > 0 is the smallest possible intersection angle betweenE andHj.

Lemma 3.5. The intersection ofE4with one of the translated(d−1)-dimensional hyper-spacesp0+Hj forj = 1, . . . , dandp0∈R^d is exactly one curve consisting of at most four different parts of straight lines.

The intersection ofE4with one of the translated (d−2)-dimensional hyperspacesp0+ Hj1,j2 =p0+span{ek :k∈ {0, . . . , d} \ {j1, j2}}for0≤j1< j2≤dandp0∈R^d is exactly one point.

Proof. Both properties are consequences of the suitable choice of the angleγ.

The inequality γ < θmin/2 < θj implies that the intersection of the boundary of the tetrahedronT(4,N) with the hyperspacesHj+p0is either empty or exactly one point of the triangle4or a closed simple curve which has non-empty intersection with the triangle 4. This implies the first claim.

Note that by our assumptions onE the intersection of a translated (d−2)-dimensional hyperspacesHj1,j2+p0withE is exactly one point. The choice of the angleγimplies that Hj1,j2+p0 cannot intersectE\ 4^◦ andE4\E at the same time, more precisely

(Hj1,j2+p0)∩(E\ 4^◦)6=∅ =⇒ (Hj1,j2+p0)∩(E4\E) =∅.

Thus if (Hj1,j2+p0)∩(E4\E)6=∅ we also have (Hj1,j2+p0)∩ 4^◦ ={q} 6=∅. Now if (Hj1,j2+p0)∩(E4\E)6=∅ contains two different points {p1,p2}, we can construct an affine lineq+λ((p1−q)hp2−q,Ni −(p2−q)hp1−q,Ni) lying in (Hj1,j2+p0)∩E. This is a contradiction as the intersection ofE andHj1,j2+p0contains exactly one point.

Our goal is to construct a monotone two-dimensional surface in the lattice L in an analogous way as in the preceding example. Therefore we are interested in properties of the intersection of a Voronoi cellp0+V withE4.

Corollary 3.6. Assume that(V+p0)^◦∩E46=∅. Then the intersection with the boundary of the Voronoi cell,(∂V +p0)∩E4, is a simple closed curveκwhich intersects each of the (d−2)-dimensional facets of V +p0 in at most one point.

Furthermore, assume that the intersection of κ with a (d−1)-dimensional facet F of V +p₀ is non empty. Then the intersection ofκwith the boundary of F (i.e. the union of all incident(d−2)-dimensional facets ofV +p0) consists of at most two different points.

Proof. The first claim is a direct consequence of Lemma 3.5.

For the second claim, assume the contrary. Then there are three different points p1,p2,p3 ofκon the boundary of the (d−1)-dimensional facetF. By definition we have F ⊂Hj+p1for somej∈ {1, . . . , d}. By Lemma 3.5 the intersectionE4∩Hj+p1defines a curveg. Ifgwas a straight line,gwould intersect the boundary ofF in at most two points.

Thus g has the form . The assumption on the intersection angles (γ < θmin/2) implies that the acute angles between different parts of straight lines are strictly smaller than θmin. We can assume that p1 < p2 < p3 using an orientation of g. Connect pj

and pj+1 by straight edges forj= 1,2 to obtain a curve ˆκwhich is contained in the facet F. Moreover, the acute angle between the vectors (p3−p2) and (p2−p1) is also strictly smaller than θmin. Therefore p2 has to lie in the interior of a (d−2)-dimensional facet.

But now a translation of this facets has two different intersection points with ˆκ and by construction there is also a translation of the corresponding (d−2)-dimensional hyperspace which has two different intersection points withκ. This is impossible by Lemma 3.5.

We now apply the same construction algorithm toE4as for the planeEin the previous example. As above, if E4 intersects a (d−2)-dimensional facet of a Voronoi cell in its interior, we get the corresponding two-dimensional face of the latticeL. By Corollary 3.6 this face has exactly one neighboring face for each of its incident edges. If E4 intersects a (d−k)-dimensional facet of a Voronoi cell for k≥3 then we apply the same reasoning as in the previous example. Thus we obtain a closed two-dimensional surface Ω^L(E4) in the lattice L. As every (d−2)-dimensional hyperspace intersects E4 exactly once by Lemma 3.5, this surface is monotone.

Furthermore, E and E4 can be oriented. We choose the orientations such that they agree on E\ 4. Also the monotone surface Ω^L(E) can be oriented. We choose the ori-entation such that positively oriented orthogonal vector to E are are positively oriented with respect to the orientation of Ω^L(E). This is possible because of the monotonicity of Ω^L(E). As Ω^L(E4) is also monotone and coincides with Ω^L(E) except for finitely many faces, we can take the corresponding orientation. Due to the monotonicity, every positively oriented orthogonal vector toEcan intersect Ω^L(E₄) at most once and the same is true for negatively oriented orthogonal vectors. Thus the orthogonal projection of Ω^L(E4) ontoE is a rhombic embedding which coincides with the rhombic embedding from Ω^L(E) except for a finite part.

Remark 3.7. The construction used for Example 3.4 can be generalized further using non-equilateral triangles and less restrictions on the angles of the tetrahedron built from this triangle. Moreover, the combinatorial surfaces Ω^L(E4) can also be obtained using a suitable sequence of flips which is a more general proceedure explained below in Section 3.5.

Definition 3.8. An infinite rhombic embedding which can be obtained by the constructions and methods presented in Examples 3.3 and 3.4 or indicated in Remark 3.7 is called aplane based quasicrystallic rhombic embedding.

3.3 Properties of discrete Green’s function and some consequences