Flat (planar) rearrangements

Although the notion of atom admits a large variety of possible constructions, we restrict ourselves to only a few basic constructions, all of which appear in this section. These choices yield a double edged sword: we avoid self-intersections and thus preserve the topology of the original essential partition after rearrangement, as a penalty we create neglected faces (discussed in§4.4).

In the next two sections we discuss local rearrangements, based on centered building blocks in a single n-cube. This section concernsflat rearrangements, in that atoms are extended across a single face of a cube. The following section considers the case that atoms are extended across several faces of a cube.

With this objective in mind, we say that an atomA, which is a pairwise essentially disjoint union of building blocks, consists of building blocks. Note that planar atoms admit unique partitions into building blocks, but essential partitions of non-planar atoms into building blocks are not unique. Indeed, in each corner where two planar parts of a non-planar atom meet, there are two possible partitions if one of the building blocks consists of two cubes. This ambiguity is, however, not significant in our considerations, since in these cases we may take any possible partition. Keeping this ambiguity in mind, we give the following definition.

Definition 4.13. Given an atomAconsisting of building blocks, we denote byeΓ(A) the adjacency graph Γ(B), where B is an essential partition of A into building blocks.

We also set`bb(A)=`(Γ(B)).

Thus, when the essential partition ofAinto building blocks is clear from the context, we denote this adjacency graph byΓ(A). Note thate eΓ(A) is always a tree.

We consider different cases, starting from simple and heading to more complicated constructions.

LetQbe ann-cube of side-length 9 and F be a face of Q. We subdivideQinto 3ⁿ congruentn-cubes of side-length 3, i.e., we considerQ^∗. ThenQ^∗induces a subdivision of Finto 3ⁿ⁻¹congruent (n−1)-cubes of side-length 3. The collection of these (n−1)-cubes isF^∗. LetQ(Q;F) be the subset of cubes inQ^∗ with a face inF^∗.

Definition 4.14. A quadruple (Q, F,Q⁰₀, q0) formsinitial data if

(a) q0 is an n-cube of side-length 3 so that q0∩Q is a face of q0 and q0∩F is an (n−2)-cube; and

(b) Q⁰₀⊂Q(Q;F) is a collection with (i) Γ(Q⁰₀) connected, and

(ii) q0∩|Q⁰₀|=q0∩Q.

Figure 16. An example of an initial data (Q, F,Q⁰₀, q0). The faceF (side-length 9) and cube q0 (side-length 3) are viewed from above, cubes inQ(Q;F)\Q⁰₀ are marked with ‘x’;n=3.

Definition 4.15. Let (Q, F,Q⁰₀, q0) be initial data. A maximal tree Γ⊂Γ(Q⁰₀∪{q0}) is aspanning tree associated with this initial data if Γ has valence less than 2(n−1).

The valence bound 2(n−1) in Definition4.15was already anticipated in our valence bound for building blocks, recall Definition4.7.

The following simple lemma shows the existence of spanning trees in the configura-tions we consider here. LetqF be the unique cube of side-length 3 in Q(Q;F) having valence 2(n−1) in Γ(Q(Q;F)); thus the barycenter ofq_F∩F is the barycenter ofF.

Lemma 4.16. Suppose (Q, F,Q⁰₀, q0) forms initial data and Γ(Q⁰₀\{qF}) is con-nected. Then there exists a spanning tree Γ⊂Γ(Q⁰₀).

Proof. Let Γ⁰ be a maximal tree in Γ(Q⁰₀\{qF}). Since Γ(Q⁰₀\{qF})⊂Γ(Q(Q;F)) andqF is the unique vertex in Γ(Q(Q;F)) having valence 2(n−1), Γ⁰ is a spanning tree of Γ(Q⁰₀\{qF}). IfqF∈Q/ ⁰₀, we may take Γ=Γ⁰.

If qF∈Q⁰₀, let q⁰∈Γ(Q⁰₀) be a vertex adjacent to qF. We extend Γ⁰ to a tree Γ containing qF by adding the edge {q⁰, qF}. Since the valence of q⁰ in Γ⁰ is less than 2(n−1)−1, the claim follows.

Spanning trees repartition Qusing atoms.

Lemma 4.17. Given initial data (Q, F,Q⁰₀, q0)and a spanning tree Γ,there exists a 1-fine atom AΓ in Qwith the following properties:

(1) AΓ∩q⁰ is an F-based building block for every q⁰∈Q⁰₀; (2) the adjacency graph Γ(Ae Γ)of building blocks is Γ\{q0};

(3) A_Γ∪q0 is an n-cell;and (4) A_Γ∩∂Q⊂F∪q0.

We call AΓ the (unique)atom associated with the spanning tree Γ (and initial data (Q, F,Q⁰₀, q₀)).

Remark 4.18. Note that the atom AΓ in Lemma 4.17is on the boundary of Qas defined in§3.1. ThusQ−AΓ is a dented cube and, in particular, ann-cell.

Figure 17. A spanning tree (left) and the corresponding atom (right) associated with the initial data in Figure16.

Proof of Lemma 4.17. To obtain the building blocks, we make the following obser-vation.

Supposeq⁰∈Γ is a vertex other thanq0. Let Γq⁰ be the star ofq⁰ in Γ, that is, the subgraph of Γ containing only edges connecting toq⁰ and all vertices on these edges. We letEq⁰=|Γq⁰|. ThenEq⁰ is a building block.

To eachq⁰∈Q⁰₀ corresponds a uniqueF-based centered building blockBq⁰⊂q⁰ which is a translation of ¹₃Eq⁰. These building blocks form an essential partition of the atom AΓ=S

q⁰∈Q⁰₀Bq⁰, whose adjacency graphΓ(Ae Γ)=Γ({Bq⁰:q⁰∈Q⁰₀}) is isomorphic to Γ.

Conditions (1), (2), and (4) are clearly satisfied by the construction. Since Γ is a tree,AΓis an atom. Asq0is a leaf in Γ andAΓ∩q0is an (n−1)-cube,AΓ∪q0is ann-cell and (3) holds.

Atoms associated with initial data and spanning trees immediately yield a local tripod property.

Lemma4.19. Let Qand Q⁰be n-cubes of side-length 9sharing the faceF. Suppose that (Q, F,Q(Q;F), q0)forms initial data with spanning treeΓ. LetAΓbe the atom asso-ciated with Γ and (Q, F,Q(Q;F), q0). Then the essential partition U=(Q−AΓ, AΓ, Q⁰) of Q∪Q⁰ has the tripod property.

Proof. Letqbe a cube inQ(Q;F) and letq−be the unique cube inQ⁰sharing a face with q. Denote by Bq the building blockq∩AΓ. By Proposition 4.10, (q−Bq, Bq, q−) satisfies the tripod property. Let ∆q be an essential partition of (∂_∪U)∩q as in Defini-tion4.4. SinceQ(Q;F) is an essential partition of a cubical set having∂_∪U(essentially) in its interior, ∆=S

q∈Q(Q;F)∆q is a required essential partition of∂_∪U.

More generally, we may consider initial data (Q, F,Q⁰₀, q₀), whereq₀∈Q(Q;F); this means thatq₀⊂Qwithq₀∩F being a face ofq₀. Initial data of this type is calledinternal initial data. This notion of initial data is especially useful for extending a 3-fine building block inside a cube of side-length 9. We formulate now this rearrangement procedure.

Corollary 4.20. Let Qbe a cube of side-length 9 and F be a face of Q. Let also q₁, ..., q_p be pairwise essentially disjoint cubes in Q(Q;F). Suppose, for 16r6p, each

Figure 18. Some examples of atomsAr forr=1, ..., peach associated with an internal initial data; herep=1,2,2,3.

(Q, F,Q⁰_r, qr) forms internal initial data with Q⁰_r⊂Q(Q;F) and Q⁰_t∩Q⁰_s=∅ for t6=s.

SupposeΓ1, ...,Γp,respectively,are spanning trees for these initial data. Then there exist pairwise disjoint1-fine atoms Arassociated with initial data(Q, F,Q⁰_r, qr)for r=1, ..., p.

It is easy to obtain a local tripod property for these repartitions. We leave the details, similar to those of the proof of Lemma4.19, to the interested reader.

Corollary 4.21. Let Q and Q⁰ be n-cubes of side-length 9 sharing the face F, and suppose that, for each16r6p, (Q, F,Q⁰_r, q_r)forms internal initial data as in Corol-lary 4.20so that in addition

B:=|Q(Q;F)|−

p

[

r=1

|Q⁰_r|

is a building block of side-length 3. For each 16r6p, let Γ_r be a spanning tree for (Q, F,Q⁰_r, q_r), associate an atom A_r with Γ_r as in Corollary 4.20 and define A as the (disjoint)union of the atoms A_r. Then the essential partition

U= (Q−(B∪A), B∪A, Q⁰) of Q∪Q⁰ has the tripod property.

Convention. Henceforth we do not differentiate between initial data and internal initial data, and refer to both as initial data.