In order to specify for a given rhombic embedding of a b-quad-graph the family of em-beddings for which Lemma 3.21 can be applied, we investigate (local) changes of rhombic embeddings. This also generalizes the construction of Example 3.4.
Definition 3.22. LetDbe a rhombic embedding of a finite simply connected b-quad-graph and let ΩD be the corresponding combinatorial surface inZd.
Let ˆz∈Vint(ΩD) be an interior vertex with exactly
Figure 3.4: A flip of a three-dimensional cube. The red edges are not part of the surface inZd. three incident two-dimensional facets of ΩD. Consider
the three-dimensional cube with these boundary facets.
Replace the three given facets with the three other two-dimensional facets of this cube. This procedure is called a flip; see Figure 3.4 for an illustration.
A vertexz∈Zd can be reached with flips fromΩD if zis contained in a combinatorial surface obtained from ΩD by a suitable sequence of flips.
The set of all vertices which can be reached with flips (includingV(ΩD)) will be denoted byF(ΩD).
Using the correspondence of rhombic embeddings and two-dimensional subcomplexes of Zd, a flip may also be defined in terms of the given rhombic embedding D. Note that performing a flip always leads to a (locally different) rhombic embedding. In particular, the new combinatorial surface inZd is still monotone.
LetD be a rhombic embedding of a finite simply connected b-quad-graph and let ΩD
in Zd be the corresponding combinatorial surface. Assume that the boundary of ΩD is simple in the sense that any two-dimensional face ofZd belongs to the surface ΩD if three of its incident edges do. Then the points of F(ΩD) may be characterized in terms of the boundary curve of ΩD which remains fixed.
Lete1, . . . ,ed be the standard orthonormal basis of Rd spanning the lattice Zd. For 1≤i < j≤ddenote by
Eij :={λ1ei+λ2ej :λ1, λ2∈R}
the two-dimensional planes which are parallel to the two-dimensional facets of Zd. Con-sequently, these planes are parallel or orthogonal to the two-dimensional facets of any combinatorial surface ΩD. Consider the closed curve Γ(ΩD) of the boundary edges. Then the orthogonal projection Γij(ΩD) of Γ(ΩD) onto one of the planesEij is a Jordan curve and thusEij\Γij(ΩD) contains exactly one unbounded componentUij(Γij(ΩD))⊂Eij. Lemma 3.23. A vertex z can be reached with flips if and only if for all 0 ≤ i < j ≤ d the orthogonal projection zij of z to Eij is contained within the projected boundary curve Γij(ΩD), that is zij ∈Eij\Uij(Γij(ΩD)).
In particular,F(ΩD)⊂Π(ΩD)and the inclusion is in general strict.
Proof. The “only if”-part is clear from the Monotonicity Lemma 3.2.
The “if”-part is proven by induction overn, where 2nis the length of the boundary curve of Γ(ΩD). Forn= 2,3 the claim is obvious. So assume that the claim holds for somen≥3 and let ΩD be a monotone combinatorial surface as above with 2(n+ 1) boundary edges.
Letz∈Zd be such that all orthogonal projectionszij ofztoEij are contained within the corresponding projected boundary curves Γij(ΩD). Ifz∈ΩD, the claim is obvious. So we assume that z∈Zd\ΩD. Denote by (n1, . . . , nd) the coordinates ofz. We want to show that zcan be reached with flips from a part of ΩD (or from another monotone surface) with less boundary edges. We distinguish three possible cases.
First, assume that there is an indexj∈ {1, . . . , d} such that min{mj∈Z:
Xd
k=1
mkek∈V(ΩD)}< nj<max{mj ∈Z: Xd
k=1
mkek ∈V(ΩD)}.
Then neither of the half-spacesHSj≤:={Pd
k=1λkek:λj ≤nj}andHS≥j :={Pd
k=1λkek: λj≥nj}contains ΩD. Consider the curve of edges inE(ΩD)∩HSj≤which are incident to a
two-dimensional facet ofF(ΩD) not contained inHSj≤. Due to the Monotonicity Lemma 3.2 and to the fact that ΩD is a simply connected two-dimensional combinatorial surface inZd, this curve is simple and contains exactly two boundary points. Using the choice of the index j, this curve thus divides ΩD in two non-empty, monotone, two-dimensional combinatorial surfaces Ω>D and Ω≤D ⊂HS≤j . Note that Ω≤D is also simply connected. Again due to the Monotonicity Lemma 3.2 and to the choice of the index j, the boundary curve of Ω≤D is strictly shorter than the boundary curve of ΩD. Furthermore, the orthogonal projections ofzto the two-dimensional planesEikfor 0≤i < k≤dare contained within the projected boundary curve of Ω≤D. Now the induction hypothesis implies thatzcan be reached by flips from Ω≤D. As Ω≤D ⊂ΩD, this proves the inductions step and the claim for the first case.
Second, suppose that the assumption of the first case is not true, but there are an index j ∈ {1, . . . , d}andlj∈Zsuch that
min{mj ∈Z: Xd
k=1
nkek ∈V(ΩD)}< lj<max{mj ∈Z: Xd
k=1
nkek∈V(ΩD)}.
Then nj is equal to the min- or to the max-term of this inequality. Define the half-spaces HSj≤ :={Pd
k=1λkek :λj ≤lj} andHSj≥:={Pd
k=1λkek:λj≥lj} analogously as above.
Again, neither of this half-spaces contains ΩD. Without loss of generality we assume that z ∈ HSj≤. The reasoning for z ∈ HSj≥ is analogous. Construct the two non-empty, monotone, two-dimensional combinatorial surfaces Ω>D and Ω≤D ⊂HSj≤ as in the first case.
If all projection of zto the two-dimensional planes Eik for 0 ≤i < k ≤ dare contained within the corresponding projected boundary curves Γik(Ω≤D) we can conclude as in the first case. If not, let Ek1k2, . . . , Ek2s−1k2s be the different planes where the projection of z is not contained within the projected boundary curve of Ω≤D, for somes ≥1. Consider the point w ∈ Zd with coordinates wj =lj, wki =nki for i = 1, . . . ,2s, and such that all projection of w to the two-dimensional planes Eik are contained within the projected boundary curve of the original surface ΩD. This is possible due to our assumptions on ΩD. Now, the first case applies to the pointw. Using the induction hypothesis, there is a simply connected, monotone, two-dimensional combinatorial surfaces Ωw containing w which is obtained from ΩD by a suitable sequence of flips. Construct the two non-empty, simply connected, monotone, two-dimensional combinatorial surfaces Ω>w and Ω≤w ⊂HSj≤ as above. Then all projections ofzto the two-dimensional planesEikfor 0≤i < k≤dare contained within the corresponding projected boundary curves Γik(Ω≤w) and we can again conclude as in the first case.
If neither the first nor the second case applies, then the two-dimensional combinatorial surface ΩD is contained in ad-dimensional unit hypercube. Without loss of generality we may assume that this cube has all coordinates in the interval [0,1]. First consider the case that ΩDcontains a parallel two-dimensional facet for alld(d−1)/2 different two-dimensional planes Eik for 0≤i < k≤d. Then it is easy to see that all points of the hypercube can be reached by flips. For the remaining case, we proceed by induction on the dimension d.
The cases d= 2 andd= 3 are obvious. LetEik be a two-dimensional plane such that ΩD
does not contain a parallel two-dimensional facet, but contains parallel edges to the vectors ei and ek. Without loss of generality, we assume that ΩD actually contains the vectorsei
andek. The other cases are very similar. Consider the projection T :Rd →Rd−1,
Xd
l=1
λlel7→
Xd
l6=i,kl=1
λlel+ (λi−λj)ei.
Then T(ΩD) is a simply connected, monotone, two-dimensional combinatorial surface in Zd−1by our assumption onEik. Furthermore, all projections ofT(z) to the two-dimensional
. . . . . .
. . . R¾+
6
¾
À
ˆ z
~ej1
~ej2
~ej3
f1
f2
. . . - . . . . . .
. . .
ˆz
. . .
Figure 3.5: An example of an infinite flip.
planes ofZd−1are contained within the corresponding projected boundary curve ofT(ΩD).
Moreover, a flip can be performed for T(ΩD) if and only if this can be done for ΩD. Therefore the proof can be continued as in the previous cases. This completes the induction step ond, and thus completes also the induction step on nand the proof.
For further use, we also consider the effect of flips for an (embedded) quasicrystallic circle pattern. LetD be a quasicrystallic rhombic embedding of a b-quad-graph with associated graph Gbuilt from white vertices. LetC be a circle pattern forGand the corresponding labelling taken fromD. Letw be the comparison function defined in (3.11). As explained in Section 3.4 this function wmay be extended toF(ΩD). Now perform a flip to obtain a locally different combinatorial surface Ω0D from ΩD. By Lemma 3.21, a circle pattern C0 can be obtained using the values of the extension ofw, such thatC0 agrees withC except for the three kites which correspond to the faces which have been flipped. Furthermore, if C is embedded this is also true forC0. Therefore, we may consider a flip not only in terms of the combinatorial surface inZd, but also directly for circle patterns using the kites which correspond to the two-dimensional facets of ΩD.
Definition 3.24. Furthermore, we can define flips for simply oder doubly infinite stripsof the following form. See Figure 3.5 for an illustration. Let ΩD ⊂Zdbe a simply connected monotone combinatorial surface. Let ˆz∈ Vw(ΩD) be a white vertex. Let ej1,ej2,ej3 be three different edges incident to ˆzsuch that there are two-dimensional faces f1, f2 of ΩD
incident toej1 and ej2, and to ej2 andej3, respectively. Let α1 =α(f1) and α2=α(f2) be the intersection angles associated to these faces. Let α3 be the intersection angles associated to the two-dimensional facet ofZd incident toej1 andej3. ThenP3
i=1αi= 2π orP2
i=1(π−αi)+α3= 2π. In the first case, consider the half-axisR+={ˆz+λ~ej2:λ≥0}, where~ej2 is the vector corresponding to the edge ej2 and pointing away from ˆz. In the second case, consider the other half-axis R− ={ˆz+λ~ej2 :λ≤0}. In both cases we may also consider the whole axis R = {ˆz+λ~ej2 : λ ∈ R}. Assume that the translations of f1 and f2 along these (half-)axis, that is the facesfj+n~ej2 + ˆzfor j = 1,2 and n∈ N, n∈Z\N, or n∈Z respectively, are contained in ΩD. We only consider the case of the positive half-axisR+ further. For R− the argumentation is analogous and the case of the whole axis R is a simple consequence. Replace each face f1+n~ej2 + ˆz by its translate f1+n~ej2+ ˆz+~ej3 forn∈N0 and similarlyf2+n~ej2+ ˆzbyf2+n~ej2+ ˆz+~ej1 forn∈N0, where~ej1 and~ej3 are the vectors corresponding to the edgesej1 andej3 respectively and pointing away from ˆz. Adding the face incident to ˆz, ej1, and ej3, we obtain a different, but still monotone simply connected combinatorial surface. The definition of an infinite flip for a black vertex ˆz∈Vb(ΩD) is similar.
Note that all points of Π(ΩD) may be obtained performing usual flips or flips for such infinite strips.
These types of infinite flips may also be defined for corresponding circle patterns as for simple flips using the values of the extended comparison functionw.
Lemma 3.25. Let ΩD ⊂Zd be a simply connected monotone combinatorial surface and let Ω0D be the simply connected monotone combinatorial surface obtained from ΩD after performing a flip for a simply infinite strip. Let C be a circle pattern for D and the
corresponding labelling and let C0 be the corresponding circle pattern after performing the corresponding infinite flip as for Ω0D. Then the resulting circle pattern C0 is embedded if the original one C is.
Proof. The proof is based on similar arguments as the proof of Lemma 3.21.
Using the same notation as in Definition 3.24 for a quasicrystallic rhombic embedding ΩD containing an infinite strip, we assume without loss of generality thatP3
i=1αi = 2π.
The other case is analogous. Let ˆv be a black point on R+ with incident white vertices zˆ1= ˆv+~ej1, ˆz2= ˆv+~ej2, ˆz3= ˆv+~ej3. Then the second intersection point of the circles of C corresponding to ˆz1 and ˆz3 as well as the kite built by this intersection point, the intersection point corresponding to ˆv, and the centers of circles corresponding to ˆz1 and zˆ3 is contained in the region covered by the four kites which contain the center of circle corresponding to ˆz2. This is due to geometric reasons and the simple combinatorics at ˆz2. Finally, consider in Ω0D two white vertices ˆz1 = ˆv+~ej1 ∈V(Ω0D) and ˆz3 = ˆv+~ej3 ∈ V(Ω0D) which are incident to the same black vertex ˆv∈R+. Take any simple closed curve in Ω0Dcontaining the edges at ˆz1and ˆz3which are parallel toej3 and toej1respectively and containing no other points incident to R+. Consider the region of the new circle pattern C0, after performing the infinite flip, which corresponds to the faces of Ω0D lying within this boundary curve. Then the above reasoning and the arguments in the proof of Lemma 3.21 imply that the kites of this region are embedded. As this is true for arbitrary large regions, we conclude that the whole circle pattern is embedded.