PERIODIC TILINGS AND TILINGS BY REGULAR POLYGONS DARRAH PERRY CRAVEY. A thesis submitted in partia1 fu1fi11ment of the. requirements for the degree of

Transcription

1 PERODC TLNGS AND TLNGS BY REGULAR POLYGONS by DARRAH PERRY CRAVEY A thesis submitted in partia1 fu1fi11ment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the UNVERSTY OF WSCONSN - MADSON 1984

2 Copyright by Darrah Perry Chavey 1984 All Rights Reserved

3 PERODC TLNGS AND TLNGS BY REGULAR POLYGONS Darrah Perry Chavey Under the supervision of Professor Donald W. Crowe Abstract: We assume a tiling has, under its symmetry group, v orbits of vertices; e orbits of edges; and t orbits of tiles. nequalities are established relating these parameters, both for arbitrary tilings and for tilings by regular polygons, and we show that some of these inequalities are sharp. n the case of tilings by regular polygons, we classify those tilings with v ~ 3, e ~ 3, or t~ 2, and show that the number of tilings with some fixed number of orbits of vertices [or edges; or tiles] is finite. The edge figures which can occur in a tiling by regular polygons are Classified, as are tilings which contain at most three different types of these edge figures. Progress is made towards classifying those tilings by regular polygons which contain at most two different types of vertex figures. with respect to tilings by regular polygons which contain only two types of tiles (two congruence classes of polygons), the number of possible orbits of each polygon is determined. Tilings by regular polygons in which any two congruent tiles are equivalent under the symmetries of the tiling are classified, as are tilings which satisfy a similar condition on the edges.

4 ii "We're all in it - we're all tiled, here." Olga. The Grand Duke, by Gilbert and Sullivan. "He's got 'em on the list - he's got 'em on the list; And they'll none of them be missed - they'll none of them be missed." Chorus of Men, The Mikado, by Gilbert and Sullivan. Dedicated to the two women love Peggy and Eunice Chavey.

5 iii Acknowledgements Now that it's almost over, it seems amazing to me that my friends and my thesis committee (which are not exclusive) have managed to put up with me for the last month or so. They are among the many people wish to thank for helping to make this thesis possibl.e. None of this work woul.d have been possible without the excellent survey of the subject by GrUnbaum and Shephard, and wish to thank them for making their advance copy avail.abl.eto us. Professors Donald Crowe and Michael Bl.eicher deserve thanks for their efforts in creating and sustaining a seminar covering this work, and it was from this seminar that most of these results developed. Much of the work in this thesis owes a great deal, in ways that are difficult to pin down, to conversations with Don Crowe and Mike Bleicher; but some of the work can be more directly attributed to my colleagues. Mary Leland discovered one class of tilirlgs used in the proof of theorem 2.3 (as mentioned there), and this class helps to extend the known range of realizable parameters in tilings. The nice proof of fact 1 in section 1.3 is a drastic improvement of my original, and this proof was pointed out by John Rosenberg. Elsa Gunter vol.unteered to draw most of the til.ings in figures on a Carnegie-Mellon laser printer, and these figures (one of prettier aspects of the thesis) would have been impossible without her help.

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48 32 graph induced by E, the dual graph induced by E does not in general include all edges of the dual graph which join vertices of this induced subgraph. Lemma 2.1: Assume T is a (v, e, t)-tiling. Then: (1) ~t'helreis a representative set of v vertices whose induced grclph is connected; (2) ThE~re is a representative set of e edges whose induced grclph is connected; (3) Thl~re is a representative set of e edges whose induced dual graph is connected; and (4) Th~!re is a representative set of t tiles whose induced dual graph is connected. Pf: We first prove (1). Let G be a maximal set of inequiva~ent vertices 'whose induced graph is connected. Let V be the set of vertices 'equivalent to some vertex in G and ~et V' contain all other vertices. f Viis non-empty then by connectivity there is an edge [v, Vi] with v in V; Vi in V'. Let S be a symmetry of the tiling taking v to a vertex in G. Then S(v') is not equiva~ent to any vertex of G, and G U { S(v') } induces a connected graph, contradicting the maxlmallty of G. The proofs of (2) and (4) are quite similar, but (3) is less intuitlve and so we a~so prove it. Let G be a maximal set of

49 33 inequiva1ent edges whose induced dua1 graph is connected; let E be the set of edges equivalent to an edge in G, and let E' be the complement of E. Assume E' is non-empty. Let D be the set of tiles incident with E and D' be the set of tiles incident with E'. D and D' cannot be disjoint (else, look at an edge where D and D' meet, whic,h exists by connectedness of the dual graph) so let r be a tile in both D and 0', i.e. T is incident with both E in E, and E' in E'. Pick a symmetry S so that SeE) is in G, then as above G U { S{E') } induces a connected dual graph, contradicting the maximality of G. Although we will have no need of it, it is worth mentioning that this 1emma can be strengthened to state that the representative sets all induce "simply connected" graphs or dua1 graphs. Here simply connected has the obvious definitions: a dual graph is simply connected if the union of the tiles dual to its vertices forms a simply connected set, and a graph is simply connected if its vertices and edges do not separate any pair of vertices which are in the tiling but not in the graph. The proof of lemma 2.1 uses quite heavily the SYmmetries of the tiling, and one would not expect results like this to hold for, say, a k-gonal tiling as opposed to a k-isogonal tiling. Some results in this direction do hold, as shown by the following lemma.

50 34 Lemma 2.2: Let T be a tiling without singular points. Then: (1) f T is 2-gonal, there is a set of 2 incongruent vertices whose induced graph is connected. Similar conclusions do not necessarily hold if T is 3-gonali (2) f T is 2-hedral, there is a set of 2 incongruent tiles whose induc:ed dual subgraph is connected. Similar conclusions do not necessarily hold if T is 3-hedrali (3) f T is 2-toxal, there are 2 sets of pairs of edges whose Lnduced graph and induced dual graph (respectively) are conne ct.ed, Similar conclusions do not necessarily hold if T l is 5 toxal. Pf: The proofs for the 2-gonal, 2-hedral, and 2-toxal assertions all follow imnediately from connectedness of the graph and dual graph, and from arguments similar to those of lemma 2.1. Counterexamples for the 3-gonal, 3-hedral, and 5-toxal cases are given in figure 2.1 (a, b, and c respectively). The mos1t:surprising fact contained in this lemma is that it remains open whether conclusions like those of lemma 2.1 hold in the case of 3- or 4-toxal tilings. Counter-examples in the case of 3-hedral and 3-gonal tilings can be constructed as edge-to-edge tilings by relgular polygons, but no such nice counter-example can be constructed for the 3-toxal case (see theorem 4.8). n fact, for

51 35 l 1 Figure 2.1 (a) Figure 2.1 (b) l 2 Figure 2.1 (c) Figure 2.1 (d) Figure 2.1: Examples of (respectively) 3-gona1, 3-hedra1, S-toxa1, and 6-toxal tilings which have no representative sets of elements of the relevant class which induce connected graphs or dual graphs (as appropriate). Examples of representative sets are marked in each tiling. The bold edge in (a) is referred to in section 3.1.

52 36 edge-to-edge tilings by regular polygons. the smallest value of k with a k-toxal counter-example that we know of is the 6-toxal example of figure 2.1 (d). This situation is an example of a phenomenon that recurs in chapter 4. namely that imposing local regularity on the edge figures is a stronger condition than imposing local regularity on the tiles or the vertex figures. This is especially true for tilings by regular polygons. As another example. although it is fairly easy to find monohedral. non~isohedral tilings and monogonal. non-isogonal tilings. we do not know of an example of a monotoxal tiling that 1s not isotoxal. We are now ready to establish the main result of this section, a partial class:ification of the realizable triples (v, e. t). Theorem 2.3: Let T be a (v. e. t)-tiling. Then (1) v < e+ 1 with equality only if the graph of the til.ing is bipartite; (2) t < e+1 with equality only if the dual graph of the tiling is bipartite; hence only if every vertex of the tiling has even valence; (3) (v, E~) is realizable if it satisfies condition 1; (4) (v. t) is realizable if t > v. (5) (e. t) is realizable if it satisfies condition 2 and t > e - 3 (and t.?- 1); (6) No upper bound can be set on the value of e or t in terms of v.

53 37 Pf: We note that (6) is implicit in (3) and (4). (1) By lemma 2.1, there is a representative set E of e edges which induces a connected subgraph G. Every vertex is incident with some edge e, and e is equivalent to an edge in E, hence each vertex is equivalent to a vertex of G. A connected graph with e edges has at most e+1 vertices, with equality only if the graph is a tree, so v < e+1. f v = e+1, then G is a tree and no two vertices of G are equivalent, i.e. every vertex of T is equivalent to a unique vertex of G. Since G is bipartite, we may 2-color 1~he vertices of G so that no edge joins two vertices of the same color. We now extend this coloring to T by coloring vertex orbits monochromatically. Since every edge of T is equivalent to an edge in G, this is a 2-coloring of the vertices of T, hence the graph of T is bipartite. (2) The definition of the dual graph implies that two dual vertices have only one edge joining them, even- if the tiles they represent intersect in more than one edge. With this in mind, the proof of the first part of (2) duplicates that of (1), using the dual graph induced by E. t remains to note that if a vertex v of the tiling had odd valence, then the tiles incident to v (which are distinct) form an odd cycle in the dual graph, contradicting the biparti tieproperty of the dual graph. (3) and (4) f each edge in the regular tiling by squares is replaced by e pairs of arcs, with each pair placed symmetrically about the omitted edge, we get a (1, e, e+l)-tiling. Figure 2.2 shows an

54 l 38 example with e=2. This shows that (3) and (4) hold if v=1 and, in fact, that these parameters can be realized by tilings that are extremal with respect to inequality (2). f we take the regular tiling by squares and replace each 2k-th left-to-right diagonal of squares (k ~ 2) by circles which pass through the same vertices, we get a (k+1, k, k+1)-tiling. Figure 2.3 (without the dashed edges) shows an example with k=3. (n this figure, representative sets of edges and vertices are marked, and the tiles to the lower right of the circled vertices form a representative set of tiles.) This class of tilings not only shows that inequalities (1) and (2) are best possible, but also shows that for any e both bounds can be achieved simultaneously (this example works for e > 1; for an example of a (2, 1, 2)-tiling see Grunbaum & Shephard [1978b], figure 4, #14). Since this class of tilings is used for further constructions, we list the symmetries of these tilings. n addition to the obvious translations, there are: a) reflections in the line of the circles; b) reflections in parallel lines midway between the circles; c) 2-fold rotation centers at the center of each tile on the lines of type a or b; and d) reflections in lines perpendicular to those of type a passing through the center of each circle. These symmetries are all preserved by our modifications below. Dissecting the circles in these tilings by m pairs of arcs

55 39 Figure 2.2: A (1,2,3)-tiling representative of a class of (1, e, e+l)-tilings. Figure 2.3: A (4,3,4)-tiling without the dashed edges, or a (4,4,5)- tiling with these edges. This represents a class of (v, e, e+l)- tilings; 1 < v < e+l. Figure 2.4: A (4,4,4)-tiling representative of a class of (k+l, k+m, k+l)-tilings with 0 < m < k+l. Figure 2.5~ A (2k, 2k+l, k+l)-tiling (with k=2) used to construct a class of (2k, 2k+m+l, k+m+l)- tilings; m < k+3.

56 40 placed symmetrically around the reflection lines of type a gives a class of (k+1, k+m, k+m+1)-tilings. Figure 2.3 (including the dashed edges) is an example for m=1. When k=1 and m > 0, this construction still yields examples whose parameters are (k+1, k+m, k+m+1). These examples prove (3) and (4) when v > 1; again with examples that are extremal with respect to inequality (2). (5) Starting with the class of (k+1, k, k+1)-tilings of (3), we take any m of the tile orbits and bisect the tiles along the ref~ection lines of type d, to get (k+1, k+m, k+1)-tilings where m < k+1. Figure 2.4 is an example with k=3 and m=1. Bisecting one or both of the orbits of tiles which lie on the reflection lines of type a and b increases the value of e while leaving t unchanged. This creates (k+x, k+m, k+l)-tillngs for any m with o ~ m ~ k+3, where x is 1, 2, or 3. Thus e = k+m < 2k+3 = 2t+1, showing that the pair of parameters (e, t) is realizable if e - 2 < t < e+1. This construction works for k > 1; hence for e > 1. For the case where e=1, the tilings #1 and #10 in l figure 4.1 give examples with t=1 and t=2 respectively. TO complete (5), we build a similar class of tilings from the regular hexagonal tiling. 1 f the edges between every 2k-th 1 This class of examples was pointed out to us by Mary Diane Palmer Leland, A. T. & T. Bell Laboratories.

57 41 vertical column of hexagons and the adjacent hexagons are replaced by circular arcs, the result is a class of (1k, 2k+1, k+1)-tilings. Figure 2.5 is an example with k=2. As above, dissecting any of the tile orbits along the reflection lines leaves t unchanged, but increases e by m, where o < m ~ k+3. This gives tilings where: hence: 2t - 1 = 2k+1 < e < 3k+4 = 3t+1; e - 3 < t < e + 2, establishing (5). Although the statement of theorem 2.3, and its proof, assume that v, e, and t are finite (i.e. that the tiling has no singular points), it is worth remarking that the conclusions hold even if we allow them to be infinite. f either t or e is finite, then by theorem 1.3 there are no singular points, so the tiling is a (v, e, t)-tiling. t is conceivable that v is finite and that there are singular points, in which case t and e must be infinite. The existence of such a tiling would imply that no bounds are possible on the values of t or e as a function of v, but theorem 2.3 shows that this is true even without the existence of singular points! All the inequalities of theorem 2.3 (6) involve only two of the parameters v, e and t. t seems likely that a more complete classification of the realizable triples would include relations between all three parameters. Of special interest would be bounds on the Euler characteristic of a periodic tiling T, Eu(T), which we define by Eu(T) = v - e + t. We do not know any lower bounds for

58 4,2 EU(T), but we have found no classes of tilings for which Eu(T) < O. Sharp upper bounds are established by Corollary 2.4. Corollary 2.4: Eu(T) ~ v+1, and equality is possible for any value of v; Eu(T) ~ t+1, and equality is possible for any value of t; Eu(T) ~ e+2, and equality is possible for any value of e. Pf: The inequalities follow by replacing v and/or t in the definition of Eu(T) by the upper bounds of e+1 from theorem 2.3. The upper bounds are all achieved by the (k+1, k, k+1)-tilings constructed in the proof of theorem 2.3, parts (3) and (4).

59 43 l Chapter 3: Ti1ings by Regu1ar POlygons. The remainder of this thesis considers only edge-to-edge tilings by regular polygons. Much more can be said about this special class of tilings. Such tilings cannot contain singu1ar points, so the results of section '.3 apply. The current chapter considers general questions involving such tilings, while the next two chapters deal with more specific classification theorems. Section 3.1 classifies the elements of a tiling that can appear in an edge-to-edge tiling by regular polygons. Section 3.2 shows that the number of k-isogona1, k-isotoxa1, or k-isohedra1 ti1ings for a fixed k is finite. Finally, section 3.3 is devoted to the improvements that are possib1e in the bounds of theorem 2.3 for this special class of tilings. Section 3.': Classification of the Elements of a Ti1ing. n much of the notation developed in this section, the number n is used to refer to a regular n-gon. n an edge-to-edge tiling by regular polygons a vertex figure is completely determined by knowing what polygons are incident to the vertex, and in what order. For these tilings the type of a vertex is n,.n 2 if the vertex is incident, in cyclic order, to an n,-gon, an n 2 -gon, etc.. Thus the regular tilings by squares and hexagons have vertices of types

60 and respectively. For brevity, we write these symbols as 44 and 6 3, with similar abbreviations in other cases. n order to obtain a unique symbol for each vertex type, we shall choose that symbol which is lexicographically first among all possible expressions. We do not view a reversal of the cyclic order as a distinct type. Consequently, two vertices have congruent vertex figures if and only if they have the same type. t is readily established (e.g. Grunbaum and Shephard [1977a]) that there are only 15 vertex types which can occur in an edge-toedge tiling by regular polygons (6 other vertex figures can be built from regular polygons, but cannot be extended to tilings of the plane). These 15 vertex types are listed in table 1 and pictured in figure 3.1. One consequence of this classification of vertex types is that the only tiles which can occur in these tilings are n-gons where n is 3, 4, 6, 8 or 12. t is well known that the only such tiling which contains an octagon is the unique isogonal tiling in which every vertex has type (see, for example, Grunbaum and Shephard [1977a] or theorem 4.3). n much of this thesis we will have occasion to refer to tilings which contain only a few vertex types. Such tilings are indicated by a list of the vertex types which occur, enclosed in parenthesis. Thus, the 2-gonal tiling of figure 3.4, which contains only the vertex types and , is a (3 4.6; ) tiling. Occasionally, we will abuse this notation to refer to, say, a 2-gonal tiling which is 3-isogonal. n this case the tiling of figure 3.4

61 J J Vertex Type sogonal Tiling Yes Yes Yes Yes No No No No Yes Y.es Yes Yes Yes Yes Yes Valence Table 1: The vertex types which can occur in edge-to-edge tilings by regular polygons, with their valence and whether an isogonal tiling of that type exists CJjffiz:K>(tJ e=> <X){D EE Figure 3.1: The 15 vertex figures which occur in edge-to-edge tl1ings by regular polygons. ",. n

62 46 might also be called a (3 4.6; 3 4.6; ) tiling to emphasize the fact that the set of vertices of type splits into two orbits of vertices. As with vertex figures, an edge figure is completely determined by knowing what polygons meet the edge, and in what order. Consequently, in many of the arguments used from here on, we abuse the definitions of edge and vertex figures by including all of the edges of these polygons within the figure (as in figure 3.1). To describe an edge figure, we use the idea of an edge type. This is, essentially, a listing in cyclic order of the polygons which meet the edge, with the list starting and ending at one of the polygons incident to the edge. Specifically, the edge type of an edge starts with one of the two polygons incident to ; lists the polygons incident with one of its endpoints in the (unique) cyclic order around that vertex which ends at the other polygon incident to Ei and then lists the polygons around the other endpoint, ending at the original polygon. For convenience we separate the tiles incident with each vertex by a "/" and list the tiles incident with the edge twice, once with each endpoint. As an example, the bold edge marked in figure 2.1a is of type 36 / Again, we use exponent notation for brevity. There are in general 4 distinct symbols we could use for a given edge figure. As with the vertex type, we can choose that symbol which lexicographically precedes the others, and two edge figures will then be congruent if and only if they have the same edge type. We will not always use this convention,.however, since we will some-

63 47 times wish to list one endpoint before the other to emphasize the first vertex. There are times when we need, or have, less information about an edge than its edge type. Sometimes, all that is necessary is to know what polygons are incident with the edge. Thus an edge is said to have simple edge type n 1.n 2 if the edge is incident with both an n,-gon and an n 2 -gon. As an example, the bold edge in figure 2.1a has simple type 3.3, or 3 2 When there is danger of confusion as to whether we are referring to a "simple edge type" or a standard "edge type"1 we will refer to the latter as a full edge type. As a step towards solving the classification problems considered later, it is useful to know what vertex and edge types can occur in a tiling by regular polygons. As mentioned before, the possible vertex types are listed in table 1. The possible edge types have not been '\ previously compiled, and so this is achieved by the following theorem. n the classification of edge figures it is useful to distinguish between two types of edge figures. A monogonal edge figure (or edge type) is one where both endpoints have the same vertex type. f the endpoints have different types, the edge figure or edge type is non-monogonal. Theorem 3.1: The only edge types which occur in edge-to-edge tilings by regular polygons are the 41 monogonal edge types listed in table 2 and the 57 non-monogonal edge types listed in table 3.

64 48 Pf: Under the symmetries of the vertex figure (see figure 3.1) certain edges incident to a vertex are equivalent. Using the inequivalent edges it is straight-forward to write down all combinations of vertex figures which could meet along, say, an edge of simple type 3 2 Such a list includes separate entries for inequivalent edges of the same simple type in a vertex figure. n listing the edge figures for edges of simple types 32, 42, or 62 it is also necessary to include two entries if there are two inequivalent orientations for matching a pair of vertex figures along that edge; i.e. a reflection of one vertex. This list will include some duplicates in the case of monogonal edge types, but when these are deleted, the resulting list contains 101 edge types. Three of these, , , and , cannot occur because such an edge type forces the existence of a vertex incident to a triangle, hexagon, and dodecagon and yet table 1 indicates that no such vertex exists. To show that the other edge types do occur in tilings, we must demonstrate tilings which contain them. The 3rd column in table 2 and the 4th column in table 3 give the minimum values of v for which there is a v-isogonal tiling containing that edge type. The 1- and 2-isogonal tilings are pictured in figures 4.1 and 4.2 respectively,.and an entry in the tables of the form 2-17 indicates that the edge type exists in the 17th tiling shown in the collection of 2-isogonal tilings. The 3-isogonal, vertexhomogeneous tilings are shown in figure 5.1 and an entry in the

65 49 tables of the form 3-19 refers to these tilings. Similarly. an entry of the form 4-19 refers to the 4-isogonal. vertexhomogeneous tilings of figure 5.2. An entry of the form 4'-2 refers to a 4-isogonal tiling which is not vertex-homogeneous. The three non-homogeneous. 4-isogonal tilings referred to by the table are shown in figure 3.2. The only edge types remaining are the three in table 3 where v = 6, all of which occur in the tiling of figure 3.3 (b) (marked as bold edges). The statement of theorem 3.1 does not claim that the "minimum v" columns of tables 2 and 3 are correct. but this is easily verified. Since the 1-, 2-, and 3-isogonal tilings have all been classified (or are classified in Chapter 5), a straight-forward, (tedious) search of these tilings combined with the 4-isogonal tilings of figure 3.2 verifies all of the values in these columns except for those where v = 6. n these three cases it is easily seen that any of the edges forces the configuration of figure 3.3(a). The vertices marked 1 to 4 must be in different orbits (1 and 2 have the same vertex types, but vertex 1 is adjacent to a vertex of type , and vertex 2 cannot be). Vertex 5 could only be equivalent to one of the others if it had type in which case vertex x would be incident to a triangle, hexagon, and dodecagon. which is impossible. Thus vertices 1 to 5 are inequivalent and either the value v = 6 is correct or else every vertex of a tiling containing this figure is equivalent to one of vertices 1 to 5. n this case vertex z must be of type 4, and x

66 50 l Figure Figure Figure Figure 3.2: The three 4-isogonal, non vertex-homogeneous tilings used to show the existence of certain edge types in theorem 3.1.

67 51 Figure 3.3 (a) Figure 3.3 (b) Figure 3.3: Part (a) shows the configuration forced by the existence of anyone of the edge figures marked as bold edges. This cannot be embedded in any v-isogonal tiling for v < 5. Part (b) shows a 6-isogonal tiling which contains these edge figures.

68 52 cannot be (4 is adjacent to a vertex of type 1 along the edge of simple type 3.4, and x isn't). Vertex x must then be of type 3, hence is adjacent to a vertex of type 4; i.e. vertex y is of type 4. Since every vertex of type 4 is adjacent to another such vertex (i.e. 4 and z), w is also of type 4; but then w is not adjacent (along the hexagon) to a vertex of type 3, hence is not equivalent to vertex 4 and must be in a sixth orbit. Tables 2 and 3 contain some additional information about the 98 possible full edge types. Within a given edge figure we can identify various edges of several different simple edge types. For example, in figure 2.1a, the edge figure corresponding to the bold edge contains edges with simple types 3 2, 3.4, and 42. The numbers of different simple edge types which are contained in the edge figures become important in section 4.3, and these are listed in table 3.

69 53 Vertex Type ============= Minimum Edge Figure v =======-============== ======= / / " 3 " / " / / / " / " / / n / n / " / !~~!~_!!The 41 monogonal edge types which occur in edge-to-edge tilings by regular polygons.

70 54 Vertex Type ============= " Minimum Edge Figure v ====================== ======= / / / " " / / _.._ / / _._ / / :3 " / / " / / " / " / / " / " / /

71 55 Vertex 1 ======== Vertex ===~=========!~~2=~~~~2====== ====== Min. v 2-1 Simple Edge =~~2~ 2 Thm. 4.1 ===== Thm. 4.8 ===== 8 " " " * 6 " " ** " * " " " " " " " " " " " " ** * ** * ** * * * * * A * t t Table 3: The 57 non-monogonal edge types which occur in edge-to-edge tilings by regular polygons.

72 56 Simple Vertex Vertex Min. Edge Thm. Thm. 1 ======== 2 ======== v ----=~~~~-!~~~~==---= =~~~~ 4.1 ====== ===== 4.8 ===== B " " * " * * " * " " ** * " l * 2 " * " " ** " J " C '-3 4 * 2 2 " " * 2 2 " ** " * 2 " '-3 3 * * 2 " " " ** '* * " " ** " * " " * " " 4.3.6

73 57 Simple Vertex Vertex Min. Edge Thm. Thm. 1 ======== 2 ======== =====~~~~=~~~~~====== ====== v =~~g~~ 4.1 ===== 4.8 ===== ** " " " " ** * ** " ** " ** * " ' " ** ** * " " " " " ** " **

74 58 Section 3.2: Finiteness of Tilings with k Orbits of an Element. This section shows that there are only a finite number of tilings by regular polygons which have a fixed number of orbits of some class of elements. n fact, for the proof of the main result, the only facts needed are that there are a finite number of tiles, vertex figures, or edge figures possible in such cl tiling and that any element can be adjacent to only a finite number' of other elements of the same class. For simplicity, the theorem Ls stated and proved only for edge-to-edge tilings by regular polygons, and then a more general formulation (which can be established by the same proof) is stated as a corollary. This result answers two questions posed by Griinbaum and Shephard [1983] (see sec tion 1.2). To prove this result, we need some preparation. Assume T is a v-isogonal tiling. Given a representative set C of v vertices that induce a connected graph, we define the fundamental region R that they generate to be the set C together with all edges incident with at least one vertex of C. We call the original set C the center of R, and the vertices in R which are not in C are denoted by C'. An example of a fundamental region is the set of edges marked in figure 3.4 with bold edges. A tiling can have incongruent fundamental regions, and the centers of two such regions are marked by the circled vertices in figure 3.4. Also, the same fundamental region can generate more than one tiling -- for example, the two (33.42; 44) tilings of figure 4.2 have the same fundamental regions. However, the idea of the proof of theorem 3.3 is to show that there are at

75 y,...., ',,,....,...» -6. "...., >,.., >,, ',CJ',,,,... c,.,, <," " 59.,. -1.., Figure 3.4:,...,.., ~.!,.' 'c...,,, '...,,.. " ' '" 4 4 A 2-gonal, 3-isogonal (3.6; 3.6; ) tiling. Two different representative sets of vertices are marked with open circles, and the fundamental region R generated by one set is marked with bold edges. R includes all of the solid edges, and is the union of R and the fundamental regions generated by the five centers marked by inscribed triangles.,'.... Figure 3.5: An example of two different tilings which have the same fundamental regions and labeled closures with respect to vertices, edges, and tiles.

76 60 most a finite number of fundamental regions possible, and that each fundamental region can be extended to only a finite number of v-isogonal tilings. The second half of this proof uses the idea of a closure of a fundamental region. Let v be any vertex in ct. By the isogonality condition, v is contained in the center of at least one image of R under some symmetry of T, and this image is denoted by Rv' n figure 3.4 the centers of each such Rv are indicated by the five inscribed triangles. n a more general tiling, Rv may not be uniquely determined. A closure of R, denoted R, is the union of R and one Rv for every v in ct. n figure 3.4, R consists of all the solid edges. Again, R is not always uniquely dete!rmined. Finally, a labeled closure of R is a closure of R in which every vertex v' in C' is labeled by a pair (v, S) where v is the vertex in C to which v' is equivalent, and S is a symmetry of T that takes v to v' and R to Rv,1. For further examples of the ideas involved in the proof of lemma 3.2, we suggest that the reader consider Rand R in the two (3 3.42; 44) tilings of figure 4.2. Of course, knowing S tells us which vertex of C is equivalent to v', and knowing S for each vertex in C' also describes the structure of R, so there is a good deal of redundant information here. This definition, however, seems to make the proof follow more easily. Exactly how much information about Rand R is needed to give the result of lemma 3.2 is unclear, and this is discussed further in chapter 6.

77 61 Lemma 3.2: No two different tilings without singular points have the same fundamental region R and labeled closure R. Pf: We claim that a knowledge of Rand R, with R labeled, uniquely determines the tiling. Rather than comparing two different tilings, we imagine trying to construct a tiling T from Rand R, and show that this construction is unique. The idea of the construction is straight-forward: at any stage in extending T, there will be a vertex v' near the "border" of the construction with a label (v, S). v is a vertex in R which is equivalent to v' under the symmetry S, and S forces the existence of a copy Rv' of R which covers v'. Adding the tiles of Rv' to our construction of T will, in general, force the existence of a larger portion of T. Note that while the vertex v' may be contained in several different copies of R, the symmetry S specifies exactly one of these copies. Thus, while there may not be a unique extension of T which embeds v' in a copy of R, nevertheless the extension we will choose is forced, and hence the construction of T will be unique. We need to show that this kind of extension is always possible. and that it eventually forces all of the tiling. To make the idea of the "border" of the construction more precise, we let the distance from a vertex v' to C, the center of R, be the minimum distance from v' to some vertex of C. Every image of R under a symmetry a of T is embedded in the

78 62 same way as R in an image of R, and the image of a label (v,s) will be (v, o(s». For every vertex Vi at distance 1 from the center of R, Rv' is contained in R (R being the first step in our construction of T from R). and the symmetry which labels Vi.... extends Rv' uniquely to an RV. This gives a larger portion of the tiling which, for every vertex 00 at distance 2 from the center, contains 00, a fundamental region Roo' and a label for w. terating this process n times gives a portion of the tiling which includes the vertex figures and a copy of Roo for each vertex 00 at distance n+1 from the center. Since the graph of the tiling is connected, every vertex is at finite distance from the center, hence every vertex figure in the tiling is eventually forced. Lemmata similar to 3.2 hold for the edges and tiles. f we define a fundamental edge region [fundamental tile region] to be the union of a representative set of edges [tiles] that induce a connected graph [dual graph) together with all adjacent edges [tiles), and define the labeled closures in the natural way, then arguments identical to those used above show: Lemma 3.2': No two different tilings without singular points have the same fundamental edge region and labeled closure. Lemma 3.2": No two different tllings without singular points have the same fundamenta1 ti1e region and 1abe1ed c1osure.

79 63 Throughout this chapter, we have been assuming that our ti1ings contain no singu1ar points, and these 1ast three 1emmata are a11 false if we allow such points. Figure 3.5 shows two tilings which have the same fundamental regions and labeled closures with respect to vertices, edges, and tiles. We are now ready to move on to the main result of this section. Theorem 3.3: Let S be the set of all edge-to-edge ti1ings by regu1ar polygons and let k be a fixed integer. Then the number of k-isogonal tilings in S is finite; the number of k- isotoxal tilings in S is finite; and the number of k-isohedral tilings in S is finite. Pf: The proofs of the three parts of this theorem are essentially the same, so we give only the proof for the k-isogonal tilings, using 1emma 3.2. For this theorem (tillngs by regular polygons), as opposed to corollary 3.4 (more general ti1ings). the finiteness of the k-isotoxal and k-isohedral tilings also follows from the finiteness of the k-isogonal tilings via theorem 3.6. Fix k. We claim there are a finite number of possible fundamental regions with center of size k. This fol10ws since there are a finite number of connected graphs G on k vertices; each vertex in G can be occupied by at most 1 of 15 vertex figures; and each edge of G can correspond to at most 1 of 6 edges in the vertex figure of either endpoint.

80 64 Fix a fundamental region R with center C and let C be the set of vertices in R but not in C. For each vertex v' in C' there are k choices for which vertex v in C is equiva~ent to Vi. Vi is adjacent to C, and there are at most valence(v') (~6) choices for what edge in the vertex figure of Vi joins it to C, and at most 1 symmetry of T which fixes that edge and its endpoints. Thus there are at most 12 symmetries of T which take v to v'. Hence, for a fixed v, this gives at most 12 choices for Rv" and for the ~abel attached to Vi. Since C is finite, there are thus a finite number of choices for the labeled closure of R, and each of these choices extends to at most one tiling. Corollary 3.4: Let k be a fixed number. Then: (1) Given a finite set of vertex figures, each of finite valence, then the number of k-isogonal ti~ings that contain on~y these vertex figures is finite; (2) Given a finite set of edge figures, with each incident vertex having finite va~ence, then the number of k-isotoxa~ ti~ings that contain only these edge figures is finite. (3) Given a finite set of ti~es, each with a finite number of edges, then the number of k-isohedral tilings that contain only these tiles is finite. Remark: n (3), we emphasize that the "edges" of the tiles must be edges in the tiling. This is equivalent to saying that for each tile

81 65 we know what points along its boundary will be vertices of the tiling. f we were to try to omit this restriction we would find, for example. that there are infinitely many isohedral tilings by congruent squares, i.e. the standard (44) tiling by squares with rows slid along each other by a constant a.

82 66 Section 3.3: Bounds on the Number of Orbits of an E~ement. This section considers improvements in the bounds of theorem 2.3 under the added assumption that the tiling is edge-to-edge and all tiles are regu~ar polygons. Specifica~~y, bounds are estab~ished on the number of orbits of one c~ass of elements in the tiling as a function of the number of orbits of another c~ass of e~ements. To take advantage of the fact that equality can ho~d in theorem 2.3 (1), on~y if the graph of the ti~ing is bipartite, we first classify these graphs. Lemma 3.5: Let T be an edge-to-edge ti~ing by regu~ar po~ygons whose graph is bipartite. Then T is one of the four isogonal tilings (4 4 ); (6 3 ); (4.8 2 ); or (4.6.12); hence v = 1 < e. Pf: A ti~ing whose graph is bipartite contains no triang~es, and the four vertex types mentioned in the lemma are the only such vertex types in table 1. No non-monogonal edge exists which uses only these four vertices (see table 3), hence the tiling must be monogonal. t is well known (see section 4.1), that this implies the tiling is isogonal. Grunbaum and Shephard [1977a] ask if for (v, e, t)-ti~ings by regular polygons there are functions t(v) and vet) such that for a fixed t, v ~ vet) and for a fixed v, t~ t(v). We answer these questions, and the four re~ated questions invo~ving the parameter e, in theorem 3.6.

83 67 Theorem 3.6: Let T be an edge-to-edge (v, e, t)-tiling by regular polygons. Then: (1) v < e (2) e < 5v (3) e < 6t - 2 and and and t~ e + 1; t < 4v + 1; v < 6t - 2. Pf: (1) From theorem 2.3, t~ e+1; and v ~ e except possibly if the graph of T is bipartite. Lemma 3.5 shows that v < e for these tilings also. (2) By lemma 2.1 there is a representative set V of v vertices whose induced subgraph is connected. Every edge and tile in T is equivalent to an edge or tile incident with a vertex in V, so e and t are bounded by the number of edges and tiles incident to V. From table 1, valence(v) < 6 for each vertex v in V. Also, if every vertex of V has valence 6, then every vertex is of type 3 6 (see table 1) and the tiling is the isogonal (3 6 ) tiling (see section 4.1) which satisfies (2). Thus, w.l.o.g., there is some v in V with valence(v) ~ 5. Hence the number of edges and the number of tiles incident with V is at most 6v-1, even counting multiplicities for tiles or edges incident with more than one vertex of V. Since the graph induced by V is connected, it contains a spanning tree with v-1 edges, each of which has been counted twice, and each of these edges is incident with two tiles that have been counted twice. Thus we have over counted the edges by at least v-1 and the tiles by at least 2(v-1), proving (2).

84 68 (3) Let D be a representative set of t tiles whose induced dual graph is connected. Since D may contain 12-gons, imitating the argument of (2) does not give strong results. Let tn be the number of n-gons in D. As remarked earlier, the only tiling with an octagon is the isogonal (4.8 2 ) tiling, which satisfies (3). We may thus assume tn - 0 except for n = 3, 4, 6 or 12. Since no vertex is incident with three 12-gons, at least six edges of each 12-gon are incident with an m-gon where m ~ 6, and are thus equivalent to an edge of such an m-gon in D. Thus, the number of inequivalent edges incident to a tile in D is at most: 3t3 + 4t4 + 6t6 + 6t'2 < 6t so e < 6t. f e ~ 6t-l, then t3 and t4 = O. Lemma 3.5 shows this happens only if T is the isogonal (6 3 ) tiling (where e = 1), hence e < 6t - 2. By table 1, every vertex of T is incident with an m-gon where m~ 6, so to count the inequivalent vertices, we need only count the vertices incident with such an m-gon in D. As with the edges, we find that v ~ 6t, with v > 6t-1 only if t3 and t4-0 (in which case v = 1), hence v < 6t - 2. One immediate corollary to this theorem is the proof of a conjectured enumeration of the 2-isotoxal tilings by Grunbaum and Shephard [1977a] (see section 1.2). Theorem 4.8 will prove a stronger version of corollary 3.7.

85 ---, 69 Corollary 3.7: The only isotoxal edge-to-edge tilings by regular polygons are the four isogonal tilings: (36); (44); (63); and ( ). The only 2-isotoxal edge-to-edge tilings by regular polygons are the four isogonal tilings: (3.122); (4.82); ( ); and ( ). Pf: By theorem 3.6, isotoxal tilings are isogonal, and 2-isotoxal tilings are either isogonal or 2-isogonal. These tilings have been classified (Kepler [1619] and Krotenheerdt [1969], see figures 4.1 and 4.2), and inspection of these tilings yields the desired result. We have no reason to believe that the bounds of theorem 3.6 are sharp. except for (1) with small values of e. The isogonal ( ) tiling is a (1, 1, 2)-tiling which achieves equality in both parts of (1). There are also tilings where v. e for e ~ 8; and one tiling where t e+ 1 = 3. neql1alities (2) and (3) can probably be improved, and possibly (1) can be improved for larger values of e. Theorem 3.8, however, gives some limits on possible improvements. To establish these limits. we need to construct various classes of tilings. One way to do this is to combine "strips" of tiles to cover the plane. This technique will be used here and again in chapter 4, so it is helpful to be precise as to what we mean by this. A strip is a set of polygons which lie between two parallel lines and cover the area between those lines; the lines are assumed to be close enough that polygons meet both lines. There are three kinds of

86 70 strips: strips of squares; strips of triangles; and strips of hexagons. These strips are all demonstrated in figure 3.8. A "strip of hexagons" also includes triangles (although only the hexagons meet both lines bounding the strip) and the hexagons all meet at vertices of type Theorem 3.8: For each of the following equations there are infinitely many edge-to-edge (v, e, t)-tilings by regular polygons which satisfy that equality: (1) v e - 1; t e - 1; (2) e = 3v - 2; t = 2v; (3 ) e = 3t - 9; v 2t - 1. Pf: Figure 3.6 is an example of a strip of squares separated by k strips of triangles (here, k=4). This family of tilings has parameters (v, e, t) = ( Lk;2J ' k+2, k+1 ) where LxJ denotes the greatest integer function. This class of tilings satisfies t = e-1, for any e > 3 and also satisfies t - 2v when k is odd. Figure 3.7 is an example of a column of triangles separated by k columns of hexagons (in figure 3.7, k=4). This family has parameters (v, e, t) = ( k+1, k+2, Lk;3 J ). These tilings satisfy v e-1, for any e ~ 3, and also satisfy v = 2t-1 when k " is even. Figure 3.8 may be modified by using 2k strips of squares (in the figure, k=2), to give a family of ( 2k+4, 3k+6, k+5 )-

87 71 Figure 3.6: A (3.6.5)-tiling which demonstrates a class of Lk+Z 2-' ( L.J' k+2, k+l)-tilings. Representative sets of edges and vertices are marked. Fig. 3.7: A (5,6,3)-tiling which demonstrates a class of ( k+l, k+2, ~~3J )-things. Representative sets of vertices and edges are marked. Fig. 3.8: A (8,lZ,7)-tiling which demonstrates a class of ( 2k+4, 3k+6, k+5)-tilings. Representative sets of vertices and edges are marked.

88 72 tilings. which satisfies e = 3t-9. f we remove the strips of triangles (and translate the squares over to meet with the hexagons), and use 2k+1 strips of squares, the result is a class of ( k+2, 3k+4, 2k+4 )-tilings, which satisfies e = 3v-2. With til.ings by arbitrary tiles (theorem 2.3). we showed the existence of a wide. continuous range of realizable pairs of parameters. We can also show the existence of continuous ranges of realizable parameters in the case of tilings by regular polygons. Unlike theorem 2.3. however. these ranges cover most, but not all, of the values within the extremes of theorem 3.8. Theorem 3.9: For edge-to-edge tilings by regular polygons: (1) Any pair of parameters (v, e) is realizable if v + 1 < e < 2v (2) Any pair of parameters (v, t) is realizable if Lt;1J.s v < 2t-1 ; (3) Any pair of parameters (e, t) is realizable if t + < e < 2t and t > 2 or t + < e < 3t Pf: (1) f we broaden the class of tilings referred to in figure 3.7 to allow j columns of triangles and k columns of hexagons, the resulting tilings have parameters (v, e, t) ( l ;+211J ~J + k, j + k + 1, l~j ).

89 73 (This class of examples is used again for theorem 4.7, and two examples with j > are shown in figure 4.9). When j = 1, e = v+1; and when k = 1 and j is odd, e = 2v. t is readily verified that any intermediate values of e and v can be obtained. n this class of examples, we must have v > 2; but corollary 3.7 gives four tilings with v = 1 and e = 2. (2) We look again at the class of examples of (1). When k = 1, v = t; and when j = 1 and k is even, v 2t-1. The range is complete between these extremes, so this gives tilings with all values (v, t) satisfying t ~ v ~ 2t-1. We now generalize the class of ex~mples of figure 3.6 so as to alternate j strips of squares with k strips of triangles. The resulting tilings have parameters (v, e, t) = i l~j l.i:!:...l2 +L~J +1, j+k+1 J + k i. When k = 1 and j is even, v = t; and when j - 1, v = t;]. The range between these extremes is again complete. n these classes of examples, t~ 2, but the three isohedral tilings (tilings #1, 2, and 3 in figure 4.1) satisfy t = 1 and v = 1. (3) The second class of examples from (2) (strips of squares and triangles) gives tilings with any pair (e, t) satisfying t+1 < e < 2t (with t * 1). To conclude the argument, we need only consider the range 2t+1 < e < 3t-19, which is empty unless t > 20. Hence we need not concern ourselves with