A Novel Way to Generate Fractals

Transcription

1 A Novel Way to Generate Fractals Michelle Previte and Sean Yang. INTRODUCTION. For more than twenty years, fractals have intrigued mathematicians and nonmathematicians alike due to their inherent beauty and widespread appearance in nature and computer graphics. Intuitively, a fractal is a geometric object with intricate detail on an arbitrarily small scale and some measure of self-similarity. Formally, a fractal is a metric space with topological dimension strictly less than its Hausdorff dimension [4, p. ix]. For example, the Sierpiński triangle (Figure ) is a fractal with topological dimension and Hausdorff dimension ln 3/ ln. This means that the Sierpiński triangle is thinner than a solid triangle, but thicker than a line. Therefore, its Hausdorff dimension lies between the dimensions of a solid triangle and a line (two and one, respectively). Figure. The Sierpiński triangle. Figures and 3 illustrate the standard method for constructing the Sierpiński triangle. In Figure, we start with a solid equilateral triangle T, then delete the solid triangle from T whose vertices are the midpoints of the edges of T. The result is that T is reduced to three smaller solid equilateral triangles, T, T,andT 3.Next, we delete from each triangle T i (i =,, 3) the solid triangle whose vertices are the midpoints of the edges of T i. This further reduces T to the nine smaller solid equilateral triangles T 3,...,T 39 shown in Figure 3. We repeat the process for each triangle T T T T 3 Figure. The first step in constructing the Sierpiński triangle. January 008] A NOVEL WAY TO GENERATE FRACTALS 3

2 Figure 3. The triangles T 3,...,T 39, labeled arbitrarily as,...,9, respectively. T 3i (i =,...,9), deleting the solid triangle whose vertices are the midpoints of the edges of T 3i. Continuing this process indefinitely yields the Sierpiński triangle. In this article we introduce a new approach (see section 3) for constructing the Sierpiński triangle and many other fractals. This approach involves iterating a function R (called a replacement rule) that replaces certain vertices in a graph G with other graphs (called replacement graphs). The result is a sequence of graphs {R n (G)} that, after scaling, converges to a well-defined geometric object if the replacement rule R satisfies some very weak conditions. Such a limit object is a fractal if its topological dimension is less than its Hausdorff dimension. It turns out that these limits have topological dimension. In section 5, we illustrate a proof of this fact. Although the concept of Hausdorff dimension is rather technical, in the case of limits of vertex replacements it very often coincides with the (more accessible) box dimension. In section 7, we sketch the proof of the formula for the box dimension of a limit arising from a vertex replacement rule. Finally, in section 8 we give some examples of vertex replacement rules that yield fractals.. THE GROMOV-HAUSDORFF METRIC. In order to study the limits of sequences of scaled vertex replacements, we must understand in what sense these sequences converge. As mentioned earlier, when a replacement rule R satisfies some weak conditions, the resulting sequence of scaled graphs converges in what is called the Gromov-Hausdorff metric. To understand this metric, we begin by recalling the definition of Hausdorff distance. Interestingly, although this distance is commonly called the Hausdorff distance, Hausdorff himself credits Dimitrie Pompeiu for this definition, which is equivalent to the one Pompeiu penned in his Ph.D. thesis on continuous complex functions (see [] for the story). For any metric space Z we use dist Z to signify the metric on Z. For any set C contained in Z and for any ɛ>0weletc ɛ ={z Z : dist Z (z, C) <ɛ}. That is, C ɛ is a fattening up of C that comprises all points near (within distance ɛ) toc. The Hausdorff distance between two nonempty compact subsets A and B of a metric space Z is defined by dist Haus Z (A, B) = inf{ɛ >0 : A B ɛ and B A ɛ }. Recall that the diameter diam Z (C) of a subset C of a metric space Z is given by diam Z (C) = sup x,y C dist Z (x, y). In Figure 4 is a closed square A of diameter centered at the origin and a closed regular octagon B of diameter four centered at (3, 0). As A must be fattened by a collar of width at least four to encompass B and B can be fattened by a collar of thickness less than 4 to enclose A,dist Haus (A, B) = 4. R Now we are in a position to define the Gromov-Hausdorff distance (see [5]). Let S denote the collection of all isometry classes of compact metric spaces. The Gromov- Hausdorff distance dist GH S (X, Y ) between two compact metric spaces X and Y is 4 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

3 4 4 A B (5, 0) A 4 Figure 4. The sets A, B,andA 4. defined by dist GH S (X, Y ) = inf Z S I,J {ɛ >0 : dist Haus Z (I (X), J (Y )) < ɛ}, where I and J are isometric embeddings (i.e., distance preserving injective maps between two metric spaces) of X and Y into Z, respectively. Informally, when measuring the Gromov-Hausdorff distance between spaces X and Y, we place X and Y into some space in such a way that they are as close together as possible and then measure the resulting Hausdorff distance. We have already seen that the Hausdorff distance between the sets A and B appearing in Figure 4 is 4. However, in the Gromov-Hausdorff metric, we use isometric embeddings to move A and B as close together as possible. One can show that dist GH S (A, B) = cos(π/8) /(see Figure 5) Figure 5. The sets A and B after isometric embeddings have been applied and the set A cos(π/8) /. Interestingly, the set S of all isometry classes of compact metric spaces is itself a metric space under the Gromov-Hausdorff metric. Moreover, (S, dist GH S ) is a complete metric space. January 008] A NOVEL WAY TO GENERATE FRACTALS 5

4 3. VERTEX REPLACEMENT RULES. In this section we define and provide some basic examples of vertex replacements. Throughout this article we assume that all graphs are connected, locally finite, unit metric graphs (i.e., each graph is a metric space and every edge is isometric to the interval (0, )). Also, the distance between two points in a graph is measured by the shortest path in the graph between the two points. For example, in the graph shown in Figure 6 the distance between vertices a and b is 7. Furthermore, we think of each graph as a representative of an isometry class in (S, dist GH S ). v a b Figure 6. v v v 4 Figure 7. A graph H. 3 AgraphH with a designated set of vertices {v,...,v k } is called symmetric about {v,...,v k } if every permutation of {v,...,v k } can be realized by an isometry of H. The vertices in such a designated set are called boundary vertices of H, andtheset {v,...,v k } is denoted by H. The graph H shown in Figure 7 has several possible sets of boundary vertices. For example, H could consist of any pair of vertices, say v and v. Or perhaps H consists of only the single vertex v.however, H cannot be the set {v,v,v 3,v 4 },sinceh is not symmetric about this set. The boundary vertices of a graph H are determined by what we call a vertex replacement rule. A vertex replacement rule R consists of a finite set of finite graphs (called replacement graphs) H,...,H p, each with a specified set H i of boundary vertices, satisfying the requirement that H i = H j when i = j, where denotes the cardinality of a set. For example, we can define a vertex replacement rule R by the replacement graphs H and H depicted in Figure 8. The boundary vertices designated by R for each replacement graph are shown with unfilled-in circles (as opposed to dots). Note that each replacement graph is symmetric about its set of boundary vertices. H H Figure 8. A replacement rule R. Let G be a graph, and let R be a vertex replacement rule given by the replacement graphs H,...,H p. Recall that the degree of a vertex v in G, denoted deg(v), isthe number of edges in G adjacent to v.avertexv in a graph G is replaceable with respect to a replacement rule R if deg(v) = H i for some graph H i in R. The replacement rule R acts on G to produce a new graph R(G) by substituting for each replaceable vertex v in G a copy of the corresponding replacement graph H i in such a way that the deg(v) edges previously adjacent to v in G are now adjacent to the H i boundary 6 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

5 vertices of H i.since H i = H j when i = j, each replaceable vertex has a unique corresponding replacement graph. Also, since each replacement graph H i is symmetric about H i, it is irrelevant how the edges previously adjacent to v are attached to H i. Thus, vertex replacement is a well-defined procedure. x v w w v 3 w v 3 x x 3 Figure 9. A graph G. Let G be the graph in Figure 9. With respect to the replacement rule R in Figure 8, vertices w,w,andw 3 are replaceable with H, and vertices v,v,andv 3 are replaceable with H, but vertices x, x,andx 3 are not replaceable. Figure 0 shows R(G). R(G) Figure 0. Constructing the graph R(G). Figure. A graph G with no vertices of degree three. Note that it is not necessary for all members of a replacement rule to be used in forming R(G). For example, if the initial graph G is the graph in Figure and R is again the replacement rule in Figure 8, then forming R(G) requires two copies of H January 008] A NOVEL WAY TO GENERATE FRACTALS 7

6 and no copies of H since G contains exactly two degree two vertices and no degree three vertices. Notice that R(G) in Figure 0 contains vertices of degrees two and three. That is, R(G) has vertices that are replaceable with respect to R. Thus the replacement rule R may be iterated to create a sequence of graphs R n (G). Consider the degree three vertices of R(G) in Figure 0. Note that each one of these vertices was originally a degree two boundary vertex in a copy of either H or H. So it is important to keep in mind the fact that once a copy of a replacement graph H i is inserted into a graph G, an edge of G is attached to each boundary vertex and its degree increases. Therefore, to measure the effect of iterating a replacement rule R on an initial graph G, we extend the idea of a replaceable vertex to include the vertices of the replacement graphs themselves. However, one should not treat a replacement graph H i as an initial graph G, but always view it as having already replaced some vertex of a graph G. Thus, although a nonboundary vertex v in a replacement graph of a replacement rule R is replaceable with respect to R if deg(v) = H i for some graph H i in R, in the case of a boundary vertex v of a replacement graph in R,wesay v is replaceable with respect to R if deg(v) = H i for some graph H i in R. For example, in Figure 8 the boundary vertices of H are replaceable with H (each such vertex will have three edges adjacent after being inserted into a graph G), while the remaining vertices of H are replaceable with H. Likewise, the boundary vertices of H are replaceable with H, and the remaining vertices of H are replaceable with H. Recall that each edge in R n (G) has length one. Therefore, when each edge in R n (G) is scaled to have length /diam(r n (G)), we obtain a new graph (R n (G), ) that has diameter one. When each graph (R n (G), ) is viewed as a representative of an isometry class in (S, dist GH S ), then according to the main result in [8] thesequence {(R n (G), )} of (isometry classes of) scaled graphs converges in the Gromov- Hausdorff metric. For the replacement rule R and the initial graph G given in Figures 8 and 9, Figure shows (R (G), ), (R 3 (G), ), and a representative of the limiting isometry class of the scaled sequence {(R n (G), )}. We now introduce some notation for counting vertices that are replaceable with respect to a replacement rule R ={H,...,H p }.LetG be an arbitrary finite graph, let N i (G) denote the number of vertices in G replaceable with H i,andletn(g) be the total number of replaceable vertices in G. For a replacement graph H i we define N j (H i ) to be the number of vertices in H i replaceable with H j when one regards H i as a subset of R(G). Inotherwords,N j (H i ) is the number of vertices v in H i such that deg(v) = H j ifv is a boundary vertex or deg(v) = H j if v is not a boundary vertex. Certain matrices play a role in determining the growth of the complexity of (R n (G), ). One of these matrices is the replaceable vertex matrix of R. If R ={H,...,H p } is a replacement rule, the replaceable vertex matrix V (R) of R is given by V (R) = N (H ) N p (H )..... N (H p ) N p (H p ). For the replacement rule in Figure 8, the replaceable vertex matrix is [ ] c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

7 Figure. (R (G), ), (R 3 (G), ), and a representative of the limit of {(R n (G), )}. The other matrices that determine the growth of the complexity of (R n (G), ) are the path matrices of R. Path matrices of a replacement rule R are defined in terms of simple boundary connecting paths that reside in the replacement graphs of R. A path σ in a replacement graph is called a simple boundary connecting path if σ is a simple (i.e., non-self-intersecting) path with boundary vertices for endpoints and no boundary vertices on its interior. Let R ={H,...,H p } be a replacement rule, and let {σ,...,σ p } be a collection of simple boundary connecting paths such that σ i lies in H i for i =,...,p. Asin the case of the replacement graphs, we regard each simple boundary connecting path σ i H i as a subset of R(G). ThenweletN j (σ i ) denote the number of vertices on the path σ i replaceable by H j when one computes the degrees of the vertices on σ i in R(G). A matrix P of the type N (σ ) N p (σ ) P =..... N (σ p ) N p (σ p ) is called a path matrix for R. In general, a replacement rule has many path matrices. However, we will restrict attention to replacement rules that have unique path matrices. In this case, we denote the path matrix for the replacement rule R by P(R). For a simple path σ let L(σ ) denote the length of σ. A replacement rule R ={H,...,H p } is simple if, for i =,...,p and any pair of simple boundary connecting paths σ and σ in H i,itistrue that L(σ ) = L(σ ) and N j (σ ) = N j (σ ) for j =,...,p. Note that for a simple replacement rule R, the path matrix is unique. In the case where a replacement graph H i has only one boundary vertex (hence there is no path joining distinct boundary vertices), then we take the boundary connecting January 008] A NOVEL WAY TO GENERATE FRACTALS 9

8 path σ i in H i to be the trivial path that consists solely of the boundary vertex of H i. Then L(σ i ) = 0andtherowN (σ i ),...,N p (σ i ) in a path matrix P is either a row of zeros (when the boundary vertex of H i is nonreplaceable) or a row in which all but the jth entry is a zero (when the boundary vertex of H i is replaceable with H j ). Not only will we focus entirely on simple replacement rules, which makes computations much easier, but we will also confine ourselves to primitive replacement rules so that Perron-Frobenius theory applies to the situation described in Theorem. If every path matrix P for a replacement rule R is primitive (i.e., there is a positive integer k such that P k has only positive entries), then R is called primitive. The replacement rule in Figure 8 is simple and primitive. If σ and σ are simple boundary connecting paths in H and H, respectively, then the path matrix is [ N (σ ) N (σ ) N (σ ) N (σ ) ] = [ Of course, not every replacement rule is simple and primitive. Figure 3 exibits an example [ ] of a replacement rule that is simple but not primitive, for it has a path matrix 0. The replacement rule in Figure 4 is primitive but not simple because there 0 are two simple [ boundary ] connecting paths in H that give rise to two different path matrices: comes from taking the upper route between the boundary vertices [ ] 0 3 of H and comes from taking the lower route. ]. H H Figure 3. A replacement rule that is simple but not primitive. H H Figure 4. A replacement rule that is primitive but not simple. The following result describes scenarios in which a sequence {(R n (G), )} of scaled replacement graphs is certain to converge: 0 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

9 Theorem. Let R be a vertex replacement rule, and let G be a finite graph with at least one replaceable vertex. When R consists of a single replacement graph H the scaled sequence {(R n (G), )} converges in the Gromov-Hausdorff metric if and only if either H contains exactly one replaceable vertex or every boundary connecting path in H contains at least two replaceable vertices. If R is given by more than one replacement graph, then the scaled sequence {(R n (G), )} converges in the Gromov- Hausdorff metric provided that R is simple and primitive. For the case in which R comprises one replacement graph, Theorem was established in [7]. The sufficient conditions for convergence for the case when R contains more than one replacement graph appear in [8]. It should be noted that one can easily construct replacement rules that are neither simple nor primitive yet that produce convergent sequences of scaled graphs. We will see later that the spectral radii of the replaceable vertex and path matrices of R determine the box dimension of the limit of {(R n (G), )}. We remind the reader that the spectral radius of a real or complex matrix A is the maximum of the moduli of the eigenvalues of A. The condition in Theorem that R be a primitive replacement rule is given so that we can apply the following theorem from Perron-Frobenius theory (see [, Theorem 4.]; we also use Theorem to prove the box dimension formula for limits of vertex replacements): Theorem. If A is a nonnegative primitive matrix of spectral radius r, then lim n (A/r) n is a positive matrix; that is, lim n (A/r) n is a matrix with all positive entries. 4. TOPOLOGICAL DIMENSION. We now define the topological dimension of a space X (see [6, sec. 50]). This definition, due to Lebesgue, was developed in the early 900s and was born out of a need to prove the obvious result that ordinary space is three-dimensional. See [3] for an interesting history of the controversy surrounding the concept of topological dimension. Roughly, the topological dimension of a topological space X measures the minimal amount of overlap (defined as order in what follows) of an open cover of X. A collection A of subsets of a topological space X is said to have order m + ifsome point of X lies in m + elementsofa, but no point of X lies in more than m + elements of A. Figure 5 depicts an open cover A of a closed disk X in R.The elements of A are rectangles, and each point of X lies within the number of elements 3 4 Figure 5. An order-four cover A of X. January 008] A NOVEL WAY TO GENERATE FRACTALS

10 of A shown in the figure. As every point in X is covered by four or fewer elements of A, A has order four. To define topological dimension, we will need to talk about refinements of open covers. Given a collection A of subsets of X, a collection B is said to refine A (or to be a refinement of A) if each element B of B is contained in at least one element of A. Figure 6 shows a refinement B of the open cover A depicted in Figure 5 that contains every element of A except the one in the upper right-hand corner of Figure 5. This element has been replaced with three smaller open sets (outlined in bold in Figure 6). Again, each point of X lies within the number of elements of B shown in the figure. Note that B has order three, for each point of X lies within at most three elements of B Figure 6. An order-three refinement B of A. We are now in a position to give the formal definition of topological dimension. A topological space X has topological dimension m if m is the smallest integer such that each open covering A of X has the following property: there is an open covering B of X that refines A and has order at most m +. The closed disk X in Figures 5 and 6 has topological dimension two since every open covering of X can be refined to an open covering of order at most three. For a given graph G let (R n (G), ) again signify the graph R n (G) scaled to have diameter. As mentioned in the introduction, when a sequence {(R n (G), )} of isometry classes of scaled vertex replacements converges in the Gromov-Hausdorff metric, then each metric space X in the limiting isometry class has topological dimension one. For the sake of brevity, we will no longer refer to isometry classes and simply say that the sequence of graphs {(R n (G), )} converges to the metric space X. SinceX is a compact metric space, we can use a Lebesgue number of an open covering A of X to construct a refinement of A (see [6, Lemma 7.5]). Theorem 3. If A is an open covering of a compact metric space X, then there is a positive number δ with the following property: for each subset of X of diameter less than δ there exists a member of A that contains it. Any such δ is called a Lesbegue number for A. 5. TOPOLOGICAL DIMENSION AND LIMITS OF VERTEX REPLACE- MENTS. We now apply the preceding concepts to convince the reader that the limit of a sequence of scaled vertex replacements has topological dimension one. c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

11 Theorem 4. If G is a finite graph and if graphs H,...,H p constitute a simple, primitive vertex replacement rule R, then the sequence {(R n (G), )} converges in the Gromov-Hausdorff metric to a metric space X that has topological dimension one. In lieu of a proof (which can be found in []), we illustrate the key ideas with the following example. Let R be the replacement rule from Figure 8, and let G be the initial graph in Figure 9. As we have already seen, the sequence {(R n (G), )} of graphs converges in the Gromov-Hausdorff metric to the space X pictured in Figure. Let A be an arbitrary open covering of X. By Theorem 3, the covering A has a Lebesgue number δ. Consider the order-two open covering B depicted schematically in Figure 7. In order to describe this covering, we first introduce a little more notation. Let H i be a replacement graph in the replacement rule R, andletv i beavertexina graph G replaceable with H i.thenweviewr(v i ) as a copy of H i in R(G) and R n (v i ) = R n (H i ) as a subset of R n (G). R ( G) Figure 7. Some of the elements in B. Now to obtain B, we begin with the set V of replaceable vertices in R(G). Since {(R n (G), )} converges to X, for each vertex v in V the sequence {(R n (v), )} converges to a subset Y v of X. The arrows in Figure 7 indicate the limits in X of a few vertices in V. The hexagonally shaped sets in Figure 7 represent the interiors Y v of the sets Y v for v in V. The circular sets in the figure represent open balls B p in X centered at the points p in the union of boundaries v V Y v and of a radius chosen so that B p B q = for distinct points p and q of v V Y v. If the diameters of the sets in B are less than δ, thenb is a refinement of A, and we are done. Otherwise, we construct an order-two open covering B of X like B, except that we use the set V of replaceable vertices in R (G) (see Figure 8). Notice that the diameters of the sets in B are less than the diameters of the sets in B.However, if these diameters are still not less than δ, we can create another order-two open cover B 3 in the same fashion, now using the set V 3 of replaceable vertices in R 3 (G). Again, the diameters of the sets in this cover are smaller than the diameters of the sets in B. In general, we construct an open cover B m using the set V m of replaceable vertices in R m (G). Form large enough B m will have sets whose diameters in X are less than δ. Inotherwords,B m is an order-two refinement of A. Hence X has topological dimension. January 008] A NOVEL WAY TO GENERATE FRACTALS 3

12 R ( G ) Figure 8. Some of the first type of elements in B. 6. BOX DIMENSION. Although to determine whether a compact set is a fractal, one must compute its Hausdorff dimension, it is the box dimension that is usually determined experimentally. This is because the box and Hausdorff dimensions very often agree and the box dimension is easier to compute. In both the box and Hausdorff dimensions, we measure how the number of boxes in a cover of the set scales as the size of the largest box is taken to zero. However, in the Hausdorff case, the number of boxes is minimized by allowing different box sizes. It is this minimization that gives the Hausdorff dimension its theoretical advantage because it excludes pathologies that may arise in the limit of smaller boxes and countably many isolated points. An excellent introduction to the box dimension can be found at Scott Sutherland s website []. The remainder of this section is an adaptation of what is there. The box and Hausdorff dimensions detect fractals by relating a set s diameter with its length (or area or volume). For example, if we try to cover the unit square with little squares of side length ɛ, we need /ɛ squares. However, if we try to cover a line segment of length, we will need only /ɛ squares. If we think of the square and the segment as sitting in space (as in Figure 9) and try to cover them with little cubes having sides of Figure 9. Covering a curve, a surface, and a solid cube with cubes having sides of length ɛ (used with permission []). 4 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

13 length ɛ, we get the same answer. Then if we use the little cubes to cover a cube, we will need exactly /ɛ 3 of them. Notice that the exponent here is the same as the dimension of the thing that we are trying to cover. This is no coincidence. The box dimension (or box-counting dimension) of a compact set X in Euclidean n-space R n is defined as follows: For a compact subset X of R n and ɛ>0let Cov(X,ɛ) be the minimum number of closed n-dimensional cubes having sides of length ɛ that are needed to cover X. Thebox dimension dim Box (X) of X is the unique real number d (if such exists) such that there is some positive constant C so that lim sup ɛ 0 + Cov(X,ɛ) /ɛ d = C. () From equation (), we have ( ) ln Cov(X,ɛ) lim sup( ln ɛ) d = ln C. () ɛ 0 + ln ɛ Since ln ɛ as ɛ 0 +, this implies that so lim sup ɛ 0 + ln Cov(X,ɛ) ln ɛ d = lim sup ɛ 0 + d = 0, ln Cov(X,ɛ). (3) ln ɛ We remark that for the sets X that we consider, the limit supremum in (3) is in fact a limit. (There are, however, cases in which the limit supremum above does not equal the corresponding limit infimum, but that issue will not concern us here.) Also, when computing box dimension, we need not use boxes if some other shape is more convenient. If we cover a plane set X, say, with closed disks of diameter ɛ (as in section 7), open disks of diameter ɛ, or even stars of diameter ɛ, we get the same answer. 7. BOX DIMENSION AND LIMITS OF VERTEX REPLACEMENTS. We now present a formula for the box dimensions of limits of vertex replacements. Theorem 5. If R is a simple primitive replacement rule, G is a graph with at least one replaceable vertex, and the metric space X is the limit of the sequence {(R n (G), )} in the Gromov-Hausdorff metric, then dim Box (X) = ln r ln ρ, where r and ρ are the unique spectral radii of the replaceable vertex matrix of R and the path matrix of R, respectively. Here, we simply sketch the proof that dim Box (X) ln r/ ln ρ for the replacement rule R from Figure 8 and the initial graph G in Figure 9. The key ingredients in the general proof are illustrated in this example. The interested reader can consult [9] for a complete proof. January 008] A NOVEL WAY TO GENERATE FRACTALS 5

14 Assume that ɛ is some arbitrary small positive number. As stated earlier, we need not cover X with boxes if some other shape is more convenient. In this case, we use disks. Thus Cov(X,ɛ) denotes the minimum number of disks of diameter ɛ required to cover X. Recall that the path matrix of R is P(R) = radius ρ = 3. Fix m so that [ ], which has spectral N(G)ρ = <ɛ m+ 6 3m+ 6 3 = m N(G)ρ. (4) m The idea of the proof is to create an ɛ/-separated set S n in the graph (R n+m (G), ) that for large n yields a corresponding ɛ/4-separated set S in X. Thus, any element of a cover of X by ɛ/8 disks can contain at most one element of S,soCov(X,ɛ/8) S. Since the number of elements in S can be expressed in terms of the replaceable vertex matrix V (R) of R, the result follows from a simple computation. Before constructing an ɛ/-separated set, recall that if v i is a vertex that is replaceable with the replacement graph H i,thenr(v i ) is a copy of H i in R(G) and R n (v i ) = R n (H i ) is a subset of R n (G). Furthermore, we define R n (v i ) to be the set of all vertices w in R n (v i ) that are adjacent to one of the deg(v i ) edges outside of R n (v i ) that were adjacent to v i. Thus R n (v i ) is the set of vertices through which a path in R n (G) passes when entering or exiting R n (v i ). Note that R n (v i ) = H i. For example, if R is the replacement rule given in Figure 8 and w 3 is as in Figure 9, then Figure 0 depicts the two vertices in each of R(w 3 ), R (w 3 ),and R 3 (w 3 ) with unfilled-in circles. We now construct an ɛ/-separated set S n in (R n+m (G), ). LetV be the set of all replaceable vertices in R m (G). For each vertex w in V and each natural number n we R(w 3 ) R (w 3 ) R 3 (w 3 ) Figure 0. 6 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

15 choose a path γ n in R n (w), viewed as a subgraph of R n+m (G), that realizes the distance between distinct points of R n (w).lets n be the subset of R n+m (G) comprising all midpoints w n of these paths γ n for all w in V and natural numbers n. Ifm = 0, then V ={a,..., f } is in (G, ), and Figure illustrates one possible choice for the sets S ={a,..., f } in (R(G), ) and S ={a,..., f } in (R (G), ). Notice that as n increases the minimum distance between any pair of points in S n does not shrink toward 0. In fact, it can be shown that this distance is at least ɛ/ for all large n. a b c f e b a e e d c d c d (G, ) (R(G), ) (R (G), ) f Figure. The elements of V, S,andS. a b f Since {(R n (G), )} converges to X in the Gromov-Hausdorff metric, we can choose n sufficiently large so that dist GH S (X,(Rn+m (G), )) < ɛ/8. Then for each w n in S n we can choose an associated point x(w n ) in X and create an ɛ/4-separated set S in X. Figure shows S when m = 0andn =. Accordingly, any element of a cover of X by disks of radius ɛ/8 can contain at most one element of S. The number of disks in such a cover is at least S = V =[N (G), N (G)] V(R) m [, ] T, [ ] where V (R) = is the replaceable vertex matrix of R. Hence, 3 3 Cov(X,ɛ/8) [3, 3] V (R) m [, ] T and ln (Cov(X,ɛ/8)) ln(8/ɛ) ln ( [3, 3] V (R) m [, ] T ) ln ( 8N(G)3 m+). x ( a ) x ( b ) x ( f ) x ( e ) x ( c ) x ( d ) Figure. An ɛ/4-separated set S in X. January 008] A NOVEL WAY TO GENERATE FRACTALS 7

16 In view of inequality (4), m as ɛ 0. Since the spectral radius of V (R) is 5, Theorem asserts that lim m (V (R)/5) m is a positive matrix. Thus, dim Box (X) = lim sup ɛ 0 ln(cov(x,ɛ)) ln(ɛ) ln ( [3, 3] (V (R)/5) m [, ] ) T + ln (5 m ) lim sup m ln (8N(G)3) + ln (3 m ) We conclude that dim Box (X) ln r/ ln ρ = ln 5/ ln 3. = ln 5 ln EXAMPLES. We now give some examples of replacement rules that generate fractals. As before, we depict the boundary vertices of the replacement graphs with unfilled-in circles. Recall that a fractal is a metric space with topological dimension strictly less than its Hausdorff dimension. In each of the examples, the Hausdorff dimension of the fractal coincides with its box dimension. In fact, it was shown in [9] that the Hausdorff and box dimensions of a limit of vertex replacements agree whenever the boundary vertices of the replacement graphs are replaceable. As a result, it is very easy to construct lots of examples of replacement rules that yield fractals. Example [ ]. (Figure 3) In this example, [ the replacement ] rule R has path matrix 4 and replaceable vertex matrix. A few iterations of the replacement rule are shown in Figure 4. The sequence {(R n (G), )} converges in the Gromov- Hausdorff metric to the set we call Peter s Cross that appears in Figure 5. The cross has topological dimension one and box (and Hausdorff) dimension ln(3+ 7) ln. ln 3 H H G Figure 3. A replacement rule R ={H, H } and an initial graph G. Figure 4. (R(G), ), (R (G), ),and(r 3 (G), ). Figure 5. Peter s Cross. [ ] 3 Example. (Figure 6) The replacement rule R here has path matrix and [ ] 4 3 replaceable vertex matrix. Starting with the initial graph G in Figure 6, a 4 8 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

17 H H Figure 6. A replacement rule R ={H, H } and an initial graph G. G Figure 7. (R(G), ), (R (G), ),and(r 3 (G), ). Figure 8. A doily. few iterations of the replacement are shown in Figure 7. The sequence {(R n (G), )} converges to the metric space having topological dimension one and box dimension ln(5+ 57) ln ln 4 that is seen in Figure 8. Example 3. (Figure 9) This example must be [ embedded ] in at least three dimensions. The replacement rule has path matrix and replaceable vertex ma- H H G Figure 9. A replacement rule R ={H, H } and an initial graph G. January 008] A NOVEL WAY TO GENERATE FRACTALS 9

18 Figure 30. (R(G), ), (R (G), ),and(r 3 (G), ). Figure 3. AmodifiedSierpiński tetrahedron. [ ] trix. Figure 30 indicates a few iterations of the replacement, and the limit 6 4 space of the sequence {(R n (G), )} is exibited in Figure 3. Its box dimension is ln(3 + 7)/ ln 3. Example 4. (Figure 3) This example illustrates a replacement [ ] rule R with a nonpositive path matrix. This R has path matrix P(R) = and replaceable vertex 0 [ ] 0 matrix. Although P(R) is not positive, it is primitive since P(R) 6 4 is positive. Consequently, R is a primitive replacement rule. For any initial graph G the limit of the sequence {(R n (G), )} has topological dimension one and box dimension ln 6/ ln 3. Figure 33 shows a few iterations of the replacement for the initial graph G given in Figure 3. H G Figure 3. A replacement rule R ={H, H } and an initial graph G. H Example 5. (Figure 34) Here is a replacement rule that has only one replacement graph and yields the Sierpiński triangle when starting with the initial graph G pictured. 30 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5

19 Figure 33. (R(G), ), (R (G), ),and(r 3 (G), ). H Figure 34. A replacement rule R ={H} and an initial graph G. G Figure 35. (R (G), ), (R 3 (G), ),and(r 4 (G), ). In this case, the replaceable vertex matrix is [3] and the path matrix is []. Figure 35 shows (R (G), ), (R 3 (G), ),and(r 4 (G), ). Example 6. (Figure 36) We conclude with an example of a fractal that has integral box dimension. The Sierpiński tetrahedron (Figure 37), which has topological dimension one and box dimension two, is the limit space of {(R n (G), )},whereg and R ={H} are given in Figure 36. H G Figure 36. H and G. Figure 37. The Sierpiński tetrahedron. January 008] A NOVEL WAY TO GENERATE FRACTALS 3

20 REFERENCES. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, New York T. Birsan and D. Tiba, One hundred years since the introduction of the set distance by Dimitrie Pompeiu, in System Modeling and Optimization, IFIP, vol. 99, F. Ceragioli, A. Dontchev, H. Furuta, K. Marti, L. Pandolfi, eds., Springer, Boston, 006, pp T. Crilly with D. Johnson, The emergence of topological dimension theory, in History of Topology,I.M. James, ed., North-Holland, Amsterdam, 999, pp K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, M. Gromov, Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits), IHES 53 (98) J. Munkres, Topology, nd ed., Prentice Hall, Upper Saddle River, NJ, J. Previte, Graph substitutions, Ergodic Theory Dynam. Systems 8 (998) J. Previte, M. Previte, and M. Vanderschoot, Limits of vertex replacement rules, Rocky Mountain Journal of Mathematics 37 (007) M. Previte, The dimensions of limits of vertex replacements, Illinois J. Math. 5 (007) M. Previte and M. Vanderschoot, The topological dimension of limits of graph substitutions, Forum Math. 5 (003) M. Previte and S.-H. Yang, The topological dimension of limits of vertex replacements, Topology Appl. 53 (006) S. Sutherland, Fractal dimension, available at Fractal_Dimension.html. MICHELLE PREVITE received her B.S. from Westmont College in Santa Barbara, California and her Ph.D. from the University of Maryland at College Park. She is currently an associate professor of mathematics at Penn State Erie, The Behrend College. Aside from her fabulous job, she enjoys singing old hymns, taking long walks with her husband, playing with her three little children, and cooking for friends. School of Science, Penn State Erie, The Behrend College, Erie, PA 6563 MichellePrevite@psu.edu SEAN YANG recieved his B.S. in electrical engineering with minors in mathematics and computer engineering from Penn State Behrend in May of 005. Currently he is working at Lockheed Martin in Manassas, Virginia, as a systems engineer. In the future, Sean plans to pursue a master s degree in which he can specialize in digital signal processing. His interests include surfing online, baseball, hiking, reading, and of course, solving challenging math problems. 04 Rifle Road, Bristow, VA 036 sean.yang@lmco.com 3 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 5