A Four-Color Theorem for Periodic Tilings

A Four-Color Theorem for Periodic Tilings D.H. Huson September 10, 1996

Abstract There exist exactly 4044 topological types of 4-colorable tile-4-transitive tilings of the plane. These can be obtained by systematic application of two geometric algorithms, edge-contraction and vertex-truncation, to all tile-3transitive tilings of the plane.

1 Introduction It is well established that there exist precisely 93 equivariant types of tiletransitive tilings of the euclidean plane, grouped in 11 topological families (the so-called Laves nets). The 508 topological types of (normal) tile-2transitive tilings are listed in [GLST85]. The 1270 (or 2268, if digons are permitted) equivariant types of tile-2-transitive tilings of the plane are systematically classi ed in [DHZ92]. The theory of Delaney symbols (or -more appropriately- Delaney-Dress symbols) introduced by A. Dress has proven to be a very useful approach to classi cation problems concerning tilings, see [Dre84] or [Dre87]. In [Hus93a] the author uses this method to formulate two geometric algorithms GLUE and SPLIT based on ideas due to M. Stogrin and E. Zamorzeva, see [DDS78], [Zam84] and [Zam88]. Using GLUE and SPLIT one obtains precisely 16774 (or 23712, if digons are permitted) topological types, or 48231 (or 65319, if digons are permitted) equivariant types, of tile-3-transitive tilings of the plane. 1

2 K -COLORABLE TILINGS

2

The number of types of tile-4-transitive tilings of the plane is certainly too large to allow a complete classi cation of such tilings, e.g. using GLUE and SPLIT, in a reasonable time. In this paper we restrict ourselves to the more limited problem of enumerating all 4-colorable tile-4-transitive tilings of the plane, i.e. such tile-4-transitive tilings with the property that no two equivalent tiles share an edge. This can be achieved using the two geometric algorithms vertex-truncation and edge-contraction, formulated for DelaneyDress symbols. (For another application of edge-contraction and vertextruncation, see [Hus93b].) In other words, we classify all tilings of the plane whose tiles can be colored using exactly 4 colors in such a way that two tiles have the same color if and only if they are equivalent, and that no two tiles of the same color share an edge. Furthermore, each tiling is assumed to be minimal , in the sense that it is not topologically equivalent to any tiling with less than 4 classes of tiles. Note that the latter condition is important and excludes most of the colored-tilings considered in the literature, which are nearly always based on tile-transitive \underlying" tilings, see [GS87], pg. 402pp. The main result of this paper is that there exist precisely 4044 topological types of 4-colorable tile-4-transitive tilings of the plane. These can be systematically obtained from the list of all tile-3-transitive tilings using vertex-truncation and edge-contraction. All enumerated tilings are available on a disk encoded as Delaney-Dress symbols from the author, together with the computer program RepTiles, developed in cooperation with Olaf Delgado Friedrichs, which translates Delaney-Dress symbols into tilings and visualizes them, see [DH92] and Fig. 3. I am grateful to Andreas Dress, who suggested to undertake this investigation.

2

k-Colorable

Tilings

A set T of topological disks contained in the euclidean plane E 2 is called a tiling of E 2 , if every point of x 2 E 2 belongs to at least one disk and if no two disks have an inner point in common. The disks of a tiling are called tiles . In this paper all tilings are presumed to be locally nite , that is, any

3 VERTEX-TRUNCATION AND EDGE-CONTRACTION

3

compact disk in E 2 meets only a nite number of tiles. Let t 2 T be a tile. A vertex of t is any point that lies in t and at least two other tiles. Consider the boundary of t with its vertices removed. The closures of the connected components of this set are the edges of t. The degree of a vertex v is the number of edges incident to v and the degree of a tile t is the number of vertices incident to t. Note that it follows from the de nitions that tiles have degree 2 or more. A tiling for which all tiles have degree 3 or more is called proper . In this paper we only consider proper tilings. Let T be a tiling and let ? be a discrete group of isometries of the euclidean plane. If T is ? -invariant, i.e. if T = T := f t j t 2 T g for all 2 ? , then we call the system (T ; ? ) an equivariant tiling . Two tiles t; t0 2 T are called equivalent , if there exists a symmetry 2 ? with t = t0. The equivalence of vertices and edges is de ned similarly. We call (T ; ? ) a tile-transitive (equivariant) tiling, if any two tiles in T are equivalent. If there exist precisely k 2 N equivalence classes of tiles, then (T ; ? ) is called tile-k-transitive . Let x be a vertex, an edge or a tile of T . We call ?x := f 2 ? j x = xg the stabilizer of x. In this paper we will assume that (T ; ? ) is periodic , i.e. that ? is a crystallographic group, in other words, that ? contains translations in two linearly independent directions. Two equivariant tilings (T ; ? ) and (T 0; ? 0) are called topologically equivalent , if there exists a homeomorphism : E 2 ! E 2 with T = T 0. They are called equivariantly equivalent , if there exists a homeomorphism : E 2 ! E 2 with T = T 0 and ? 0 = ?? . Let (T ; ? ) be a tile-k-transitive tiling with k 2 N. Then (T ; ? ) is called k-colorable if no two equivalent tiles in T share an edge and if (T ; ? ) is not topologically equivalent to any tiling with less than k classes of tiles. 1

3 Vertex-Truncation and Edge-Contraction In this Section we discuss how to obtain all topological types of (k + 1)colorable tile-(k + 1)-transitive tilings of the plane, given the classi cation of all tile-k-transitive tilings, for k 2 N. This can be done using vertextruncation and edge-contraction. Given a tile-k-transitive tiling (T ; ? ) with k 2 N. First let us de ne vertex-truncation. The idea is simple: Choose a vertex v and then \cut it


4

o", i.e. replace it by a small tile. Do this simultaneously to all vertices equivalent to v.

Algorithm 3.1 (Vertex-Truncation) Let (T ; ? ) be a tile-k-transitive tiling with k 2 N, and let v be a vertex of T . Let U E 2 be a small neighborhood around v that intersects no other vertex and only such edges and tiles that are incident to v. The intersection with any of the latter is assumed to be simply connected. Furthermore, assume that U = U or U \U = ; for every

2 ? . Then the system T 0 := ft n S 2? U j t 2 T g is a ? -invariant tiling with K0 = K [f U j 2 ? g and we obtain a new tile-(k +1)-transitive tiling (T 0; ? ). We say that (T 0; ? ) can be derived from (T ; ? ) by vertex-truncation. See Fig. 1.

Now let us de ne edge-contraction. The idea is to contract an edge e to a point, pulling together its two end-vertices. This is done simultaneously to all edges equivalent to e. Note that this is possible if and only if one of the two vertices that is incident to e has the property that it is not incident to any other edge which is equivalent to e.

Algorithm 3.2 (Edge-Contraction) Let (T ; ? ) be a tile-k-transitive tiling with k 2 N, and let e be an edge of T . Let v and w be the two vertices incident to e. Note that v = 6 w. Assume that the number of edges incident to v, say, that are equivalent to e, is one, then ?v ?e . If ?e ?w , then we can \move" v along e to w (bending other edges incident to v), thus contracting e to the point w. If ?w ?e , then e has a unique ?e- xed-point x, and we can move both v and w along e to x (again bending incident edges), thus contracting e to x. The conditions ful lled by v insure that this can be done simultaneously to every edge e0 2 ?e. We thus obtain a new tile-k-transitive tiling (T 0 ; ? ) and say that it can be derived from (T ; ? ) by edge-contraction. See Fig. 2.

Lemma 3.3 Let (T 0; ? ) be a tile-k-transitive tiling with k > 1. If (T 0; ? ) is k-colorable, then (T 0 ; ? ) can be derived from some tile-(k ? 1)-transitive tiling (T ; ? ) by a single application of vertex-truncation, followed by at most p applications of edge-contraction, where p is the degree of the vertex which was truncated.


5

Let (T 0; ? ) be a k-colorable tile-k-transitive tiling with k > 1. Choose any tile t 2 T . Let v be a vertex incident to t, whose degree is greater than 3. Let e and e0 be the two edges of t that are incident to v. Picture these two edges as the two lines in the letter V, with the vertex v at the bottom of the letter. Now, \zip-up" the two edges a short distance, starting at v, to obtain something which now looks like the letter Y, with the vertex v still at the bottom and a new vertex w in the middle of the letter Y. Note that v is no longer incident to t, whereas w is. Furthermore, the degree of v has decreased and the degree of w is 3. Note also that this operation can always be performed in a way locally compatible with the symmetries of the tiling, this is because t lies between two tiles not equivalent to t, and hence, e and e0 do not lie on reflectional axes. Furthermore, this step can be applied simultaneously to all pairs ( t; v) in a consistent way. Performing the whole operation at most p times, where p is the degree of t, we obtain a tiling (T ; ? ), containing a tile t with the property that all its vertices have degree 3. Note that, although (T ; ? ) is not necessarily k-colorable, the described construction assures that t does not share an edge with any other tile equivalent to t. This operation is exactly the reversion of edgecontraction. Let (T ; ? ) be a tile-k-transitive tiling and t 2 T a tile incident only to vertices of degree 3, obtained as above. Subdivide the tile t in the following way: Choose a point x in the interior of t with the property that ?x = ?t. This is called the center of t. Join the center of t to each vertex of t by a arc, thus dividing t into q triangles, where q is the degree of t. Now, for each edge e in the boundary of t, form the union of the second tile t0 incident to e and the triangle in t which is incident to e. In other words , the tile t is split into q pieces which are then glued to the q neighboring tiles. This can and should be done in a way locally compatible with the symmetries of the tiling. The operation is applied simultaneously to all tiles equivalent to t, which is always possible because no two tiles equivalent to t share an edge. We thus obtain a new tiling (T ; ? ), with one equivalence class of tiles less. This construction is the reversion of vertex-truncation. As we will see in the next Section, both vertex-truncation and edgecontraction can be formulated in terms of Delaney-Dress symbols and then implemented as computer programs. Using the programs TRUNCATEVERTEX and CONTRACT-EDGE applied to all tile-1-, 2- and 3-transitive Proof.


Group

cm cmm p1 p2 p3 p31m p3m1 p4 p4g p4m p6 p6m pg pgg pm pmg pmm Total

2-colorable

3-colorable

4-colorable

0

14

364

0

18

391

0

1

68

0

3

185

0

0

37

0

9

221

0

3

41

0

3

133

0

8

203

0

9

127

0

8

298

1

21

279

0

8

312

0

20

814

0

2

53

0

15

479

0

1

39

1

143

4044

6

Table 1: For each of the 17 crystallographic plane groups we list the number of 2-colorable, 3-colorable and 4-colorable tile-2-, 3- and 4-transitive tilings, respectively. Note that every tiling counted here has the property that no topologically equivalent tiling exists with a smaller number of equivalence classes of tiles.

4 COMBINATORIAL DESCRIPTION

7

tilings of the plane we obtain the main result of this paper:

Theorem 3.4 There exist precisely 1, 143 and 4044 topological types of 2-,

3-, and 4-colorable tile-2-, 3- and 4-transitive tilings of the plane, respectively. See Table 1 and Fig. 3.

This type of result is of course hard to check. The 2- and 3-colorable tilings have been cross-checked with the tilings enumerated in [DHZ92] and [Hus93a]. Hence the author has very high con dence in the rst two numbers and is quite optimistic about the 4-colorable case. As a futher con rmation of the result, the author has classi ed all proper tile-4-transitive tilings with symmetry group p1, using SPLIT. Of the 381 topological types enumerated, precisely 68 are indeed 4-colorable. Finally, let us remark that the same method can be used to enumerate colorable tilings of the sphere and of the hyperbolic plane.

4 Combinatorial Description In this Section we continue the discussion of geometric algorithms in [Hus93a]. Our aim is to give a combinatorial description of the algorithms vertextruncation and edge-contraction in terms of Delaney-Dress symbols. The two operations have been implemented on a computer and are available as a part of the program RepTiles. The Delaney-Dress symbol of a periodic 2-dimensional tiling (T ; ? ) is a nite, edge-3-colored graph, together with two functions de ned on the vertices of the graph, that encode the tiling up to equivariant equivalence. The relationship between such tilings and their symbols has been studied in great detail, see the references. Therefore, we do not discuss the correspondence between tilings and their Delaney-Dress symbols here. First let us recall the de nition of a Delaney-Dress symbol. Consider an edge-colored n graph (D; E ) consisting of a vertex setoD and a set of colored edges E (fD; D0 g; i) j D; D0 2 D and i 2 f0; 1; 2g , where the component i of an edge (fD; D0 g; i) is called its color . The graph (D; E ) is a DelaneyDress graph , if every vertex is incident to precisely one edge of each color. For a Delaney-Dress graph (D; E ) we de ne for i 2 f0; 1; 2g a permutation i : D ! D, that maps D 2 D onto D0 =: Di , if and only if (fD; D0 g; i) 2 E .


8

A system (D; m) := (D; E ); m ; m ; m ) consisting of a nite, connected Delaney-Dress graph D := (D; E ) and functions m ; m ; m : D ! N, is called a (2-dimensional) Delaney-Dress symbol if and only if for all D 2 D and 0 i < j 2 the following hold: (DS1) mij (D) = mij (Di ) = mij (Dj ) (DS2) D(i j )m D = D(j i)m D = D (DS3) m (D) = 2 (DS4) m (D) 3 and m (D) 3. Two Delaney-Dress symbols (D; m) and (D0 ; m0) are called isomorphic if and only if there exists a bijection : D ! D0 with (D)k = (Dk ) and m0ij (D) = mij (D) for all D 2 D, 0 k 2 and 0 i < j 2. It is not dicult to prove the following result, which, however, only holds for simply-connected spaces ([Dre86]): Lemma 4.1 Two equivariant tilings (T ; ? ) and (T 0; ? 0) are (equivariantly) equivalent if and only if the corresponding Delaney-Dress symbols (D; m) and (D0; m0) are isomorphic. Let D := (D; E ) be a Delaney-Dress graph. If we delete all edges of color k in the graph, then we obtain a 2-colored graph Dij , with fi; j; kg = f0; 1; 2g. The connected components of Dij are called the fi; j g-orbits of D. An fi; j gorbit O is called a cycle , if Di 6= D and Dj 6= D for all D 2 O, and a chain , otherwise. Now we discuss how to formulate the two operations edge-contraction and vertex-truncation in terms of Delaney-Dress symbols. For both we need to consider a number of dierent cases, depending on the stabilizer group of the vertices or edges concerned. To avoid a tedious written description of the operations, we use a number of Figures. Algorithm 4.2 (Vertex-Truncation for Symbols) Let (T ; ? ) be a periodic tiling with Delaney-Dress symbol (D; m). Let v be a vertex of T . Let (T 0; ? ) be the tiling that one obtains by applying vertex-truncation to v and let (D0; m0) be the corresponding Delaney-Dress symbol. To obtain (D0 ; m0) directly from (D; m), transform the Delaney-graph (D; E ) as indicated in Fig. 4{ 7. Determine the functions m0ij as described below. 01

12

02

01

ij (

)

ij (

02 01

12

)

12

02


9

Algorithm 4.3 (Edge-Contraction for Symbols) Let (T ; ? ) be a periodic tiling with Delaney-Dress symbol (D; m). Let e be an edge of T that

is incident to a vertex v, which is not incident to any other edge equivalent to e. Let (T 0; ? ) be the tiling that one obtains by applying edge-contraction to e and let (D0 ; m0) be the corresponding Delaney-Dress symbol. To obtain (D0; m0) directly from (D; m), transform the Delaney-graph (D; E ) as indicated in Fig. 8{12. Determine the functions m0ij as described below. Finally, let us discuss how to determine the functions m0ij from mij . In the case of vertex-truncation, one needs to know how many (equivalent) vertices will be truncated from any given tile t 2 T . If this number is n, then m001(D) = m01(D) + n for any D 2 D0 associated with t. In the case of edge contraction one rst must decide whether the operation is possible for a given choice of edge (or f0; 2g-orbit), by determining how many edges equivalent to e are incident to the one, and to the other vertex incident to e. Furthermore, one must use Lemma 4.4 to determine how many edges equivalent to e lie in any one tile t 2 T . If this number is n, then m001(D) = m01(D) ? n for any D 2 D0 associated with t. For a Delaney-Dress symbol (D; m) de ne rij (D) := minfr 2 N j D (i j)r = D g for all D 2 D and 0 i < j 2. It is easily seen that rij (D) divides mij (D). Hence we can de ne the branching number function vij : D ! N as vij (D) := mr ((DD)) for all D 2 D and 0 i < j 2. Note that both rij and vij are constant on fi; j g-orbits, and we write vij (O) := vij (D), where O is an fi; j g-orbit and D 2 O. Lemma 4.4 Let (T ; ? ) be an equivariant tiling. Let (D; m) be the associated Delaney-Dress symbol. Let x be a vertex, an edge or a tile of T and let Ox denote the corresponding fi; j g-orbit. Let y also be a vertex, an edge or a tile of T and let Oy denote the corresponding fi; kg-orbit, such that fi; j; kg = f0; 1; 2g. The number N of members of ? y that are incident to x is given by N = 21 #(Ox \ Oy )vij (Ox)pij (Ox); where pij (Ox) equals 2, if the fi; j g-orbit Ox is a chain, and equals 1, if it is a cycle. Proof. From left to right, the formula rst takes the cardinality of the intersection of the two orbits into account, then any rotational symmetry ij

ij

REFERENCES

10

associated with x and nally any reflectional symmetry associated with x.

References

[DDS78] B.N. Delone, N.P. Dolbilin, and M.I. Stogrin. Combinatorial and metric theory of planigons (in Russian). Tr. Mat. Inst. Steklov Akad. Nauk SSSR, 148:109{140, 1978. [DH87] A.W.M. Dress and D.H. Huson. On tilings of the plane. Geometriae Dedicata, 24:295{310, 1987. [DH92] O. Delgado Friedrichs and D.H. Huson. RepTiles. University of Bielefeld, 1992. Shareware macintosh-program. [DHZ92] O. Delgado Friedrichs, D.H. Huson, and E. Zamorzaeva. The classi cation of 2-isohedral tilings of the plane. Geometriae Dedicata, 42:43{117, 1992. [Dre84] A.W.M. Dress. Regular polytopes and equivariant tessellations from a combinatorial point of view. In Algebraic Topology, pages 56{72. SLN 1172, Gottingen, 1984. [Dre86] A.W.M. Dress. The 37 combinatorial types of regular \heaven and hell" patterns in the euclidean plane. In H.S.M. Coxeter et al., editor, M.C. Escher: Art and Science, pages 35{43. Elsevier Science Publishers B.V., North-Holland, 1986. [Dre87] A.W.M. Dress. Presentations of discrete groups, acting on simply connected manifolds. Adv. in Math., 63:196{212, 1987. [FH92] R. Franz and D.H. Huson. The classi cation of quasi-regular polyhedra of genus 2. Discrete and Computational Geometry, 7:347{ 357, 1992. [GLST85] B. Grunbaum, H.D. Lockenho, G.C. Shephard, and A. Temesvari. The enumeration of normal 2-homeohedral tilings. Geometriae Dedicata, 19:177{196, 1985.

REFERENCES [GS87] [Hus93a] [Hus93b] [Zam84] [Zam88]

11

B. Grunbaum and G.C. Shephard. Tilings and Patterns. W.H. Freeman and Company, New York, 1987. D.H. Huson. The generation and classi cation of tile-k-transitive tilings of the euclidean plane, the sphere and the hyperbolic plane. to appear in: Geometriae Dedicata, 1993. D.H. Huson. Tile-transitive partial tilings of the plane. to appear in: Beitrage zur Geometrie und Algebra, 1993. E. Zamorzaeva. The classi cation of 2-regular tilings for 2dimensional similarity symmetry groups (in Russian). Akad. Nauk MSSR, Institut matematiki s VC, 1984. Kishinev. E. Zamorzaeva. On delone sorts of multiregular tilings (in Russian). Dep. v VINITI 22.04.88, No. 3132-V88, 1988. Kishinev.

REFERENCES

12

Captions Figure 1: Vertex-truncation of the vertex v in the left-hand tiling yields the right-hand tiling. Figure 2: Edge-contraction of the edge e in the left-hand tiling yields the right-hand tiling. Figure 3: The 8 types of 4-colorable tile-4-transitive tilings that consist only of four-sided tiles. Figure 4{12: In each of these Figures we depict a small section of a periodic tiling and the corresponding section of the Delaney-Dress graph. The vertices of the graph are represented by circles and the edges of the graph are drawn as heavy lines, each labeled 0, 1, or 2, indicating the edge's color. On the left we depict the initial tiling and graph, and on the right we show the result of the operation. Vertices that are added to the graph, in Figures 4{7, or removed from the graph, in Figures 8{12, are shaded grey. Figure 4: In this Figure we depict the situation where the stabilizer group ?v of the vertex v is of type Ck, with k 2 f1; 2; 3; 4; 6g. Here k = 2, but it is obvious how to adapt this type of picture to other values of k. Figure 5: The stabilizer group ?v is of type Dk, with k 2 f1; 2; 3; 4; 6g, and every reflection in ?v xes two edges incident to v. Figure 6: The stabilizer group ?v is of type Dk, with k 2 f1; 2; 3; 4; 6g, and no reflection in ?v xes an edge incident to v. Figure 7: The stabilizer group ?v is of type Dk, with k 2 f1; 2; 3; 4; 6g, and every reflection in ?v xes exactly one edge incident to v. Figure 8: The stabilizer group ?e of the edge e is of type C1. Figure 9: The stabilizer group ?e is of type C2. Figure 10: The stabilizer group ?e is of type D1 and the reflection in ?e leave e point-wise xed. Figure 11: The stabilizer group ?e is of type D1 and the reflection in ?e doesn't leave e point-wise xed. Figure 12: The stabilizer group ?e is of type D2.