Elementary fractal geometry. Networks and carpets involving irrational rotations
EElementary fractal geometry.Networks and carpets involving irrational rotations
Christoph Bandt and Dmitry MekhontsevMarch 20, 2020
Abstract
Self-similar sets with open set condition, the linear objects of fractal geometry, have beenconsidered mainly for crystallographic data. Here we introduce new symmetry classes in theplane, based on rotation by irrational angles. Examples without characteristic directions,with strong connectedness and small complexity were found in a computer-assisted search.They are surprising since the rotations are given by rational matrices, and the proof of theopen set condition usually requires integer data. We develop a classification of self-similarsets by symmetry class and algebraic numbers. Examples are given for various quadraticnumber fields. .
Self-similar sets and open set condition.
A self-similar set is a nonempty compactsubset A of R d which is the union of shrinked copies of itself. This is expressed by Hutchinson’sequation A = m (cid:91) k =1 f k ( A ) (1)Here F = { f , ..., f m } is a finite set of contractive similarity mappings, often called an iteratedfunction system, abbreviated IFS. Recall that a contractive similarity map from Euclidean R d to itself fulfils | f ( x ) − f ( y ) | = r f | x − y | where the constant r f < f. To keep things simple, we assume that all maps in F have the same factor r. For a given IFS F = { f , ..., f m } there is a unique self-similar set A which is often called the attractor of F. See [12, 13, 20] for details. Figure 1 shows standard examples with factor , , and (cid:112) / , respectively.The idea of equation (1) is that A subdivides into m pieces f k f k ( A ) of second level, andinto m n pieces f w ( A ) of level n, where w = k ...k n runs through all words of length n withletters from the alphabet K = { , ..., m } . Thus we have a homogeneous structure of little pieces,and we can define a uniform measure on A by assigning each pieces of level n the value m − n . In 1946, Moran [39] constructed the uniform measure on A as normalized Hausdorff measureof dimension α where mr α = 1 . He needed a condition, the so-called open set condition, or OSCfor short:There is an open U so that the image sets f ( U ) , ..., f m ( U ) are disjoint subsets of U. (2)1 a r X i v : . [ m a t h . M G ] M a r igure 1: Classical fractal shapes. The Sierpi´nski triangle, the Sierpi´nski carpet and a variationof the Gosper snowflake tile are standard examples of self-similar sets with open set condition.In Figure 1, U can be taken as an open triangle, a square, and the interior of the set A, respectively. It has turned out that without open set condition, the compact set A does nothave a nice local structure. The number of sister pieces in the vicinity of a piece of level n willtend to infinity with n if OSC does not hold [6, 45, 13]. However, self-similar sets play a similarpart in fractal geometry as lines in ordinary geometry. The concept of self-similarity is thatafter sufficient magnification, the view of A will repeat and should not become infinitely dense.That is why OSC is required. In the literature we find mainly examples like Figure 1 where theopen set can be easily constructed. The aim of this paper.
Sierpi´nski constructed his triangle and carpet more than 100years ago as topological spaces with curious properties [36]. After 1980, physicists took themas models of porous materials, and mathematicians developed an analysis of heat equation,Brownian motion, and eigenvalues of the Laplace operator on just these spaces. See the booksby Kigami and Strichartz [31, 51] for an introduction, and the literature cited there. However,both examples have special properties which are rarely met in nature: • They have a few characteristic directions. • They contain line segments. • Their holes have small perimeter compared to their area.Moreover, Sierpi´nski’s triangle and the related p.c.f. fractals [31, 51] have cutpoints (see Section4) and resemble networks more than porous materials. The vector space of harmonic functionson such spaces is finite-dimensional. The topology of the carpet is definitely more realistic andinteresting, but also hard to study.Nowadays, complex geometric structures are studied in nearly every active area of science:cell biology, the brain, soil and roots, foam, clouds, dust, nanostructures etc. Therefore we thinkthat fractal geometry should diversify its models. In this paper we show that this is possible,2ven within the framework of equation (1) with equal contraction factors in the plane. This isa beginning. We hope to study more general cases in a subsequent paper.Here we construct carpets based on non-crystallographic IFS data which • have no characteristic directions and an isotropic type of symmetry, • contain no line segments, and • have holes with large perimeter and rather complicated shape.In statistical physics, much more complicated random fractals were introduced by Sheffieldand Werner [47]. Their conformal loop ensemble is a probability space of fractal shapes which isinvariant not only under particular similitudes, but under arbitrary conformal maps. Instancesof the ensemble are hard to imagine and still harder to visualize. Concrete examples in thispaper can be seen as a step from Figure 1 to such abstract models.We shall focus on carpets with small complexity, for which there is a chance to developfractal analysis. In the next paragraphs, we sketch an approach to complexity of self-similarsets. We define the neighbor graph of an IFS, the automaton which generates the topology of A. The number of states of the automaton is taken as complexity.
Algebraic OSC.
There is an algebraic equivalent for the open set condition, formulatedexplicitly in terms of the IFS F [6, 13]. F generates a free semigroup F ∗ , and id is an isolated point in F ∗− F ∗ . (3)The first part of this condition says that two n th level pieces f w ( A ) and f v ( A ) will not coincide,for arbitrary n. This part is not essential for our concept of self-similarity, and a weak separationcondition WSC was defined by assuming only the second part of condition (3) [55, 41, 32, 16].However, in that case we would have to control pieces which appear once, twice, or n times, andwe want to keep things simple. The second part of (3) says that the maps h = f − w f v stay awayuniformly from the identity map, so that pieces f w ( A ) and f v ( A ) do not come arbitrary nearto the same position, for any level n. Like OSC, this is an accurate formulation of the vaguestatement that “the overlap of two different pieces is not too large”.Things become simpler when we go to a discrete setting. We assume that there is a matrix M associated with our iterated function system F on R d so that each f k in F has the form f k ( x ) = M − s k x + v k with a vector v k and an orthogonal matrix s k . (4)We can now assume that v is an integer vector and M, s k are integer matrices which commutewith each other. Then we have to check only finitely many maps h = f − w f v , and the secondpart of condition (3) will follow from the first one [6, 13, 55].Equation (4) is a strong assumption, however. To get similarity maps, M must fulfil M · M (cid:48) = r · I where I denotes the unity matrix. And even if we assume that M commutes only withthe whole group generated by the s k , as in Theorem 2 of [3], there are only a few choices ofinteger maps s k . On the positive side, however, we have the fact that all calculations are donein integer arithmetics and thus the check of the OSC is accurate. It was implemented in the3igure 2: An example with six pieces and considerable overlap for which the OSC is still true.In such cases, several magnifications are necessary to reveal the local structure. The data forthis IFS come from a crystallographic group with 60 o rotations.package IFStile [38] and has led to thousands of new examples with surprising properties. Afew of them, tightly related to the Sierpi´nski triangle in Figure 1, were discussed in [8].Another example is given in Figure 2. Since an open set in such examples consists of infinitelymany components (cf. [8]), the local structure of such examples cannot be seen from a globalpicture of the set A. Usually we need various magnifications of the set in order to understand thelocal structure. The number of pictures which we need can be seen as a measure of complexityof the self-similar set.
Neighbor complexity.
As a rigorous concept of complexity, we take the number of neigh-bor maps. Consider the map h = f − w f v where w, v ∈ K n denote different words of length n from the alphabet K. It is an isometry. We call h a neighbor map or neighbor type if the corre-sponding pieces of A intersect: f w ( A ) ∩ f v ( A ) (cid:54) = ∅ . The number of neighbor types will be takenas the complexity of the IFS F. If the number is finite, we say that F is of finite type. As aconsequence, OSC is fulfilled: a special open set was constructed in [7].The neighbor map h = f − w f v represents the relative position of the intersecting pieces f w ( A ) and f v ( A ) . However, h does not map the pieces into each other. This is done by themap g = f v f − w . The neighbor map h always maps A to a potential neighbor h ( A ) in some‘superpiece’. Constructions with larger and larger pieces are familiar in the theory of self-similartilings, see for instance [26, 46, 2]. The maps h and g are conjugate: h = f − v gf v . But g dependson the size of the pieces while h is standardized and does not depend on the level n. Suppose the Sierpi´nski carpet in Figure 1 is realized in the complex plane, in a square withvertices 0 , , i, i. Then the neighbor maps will be translations h ( x ) = x + v with v ∈ {± , ± i } for neighbors with a common edge. For neighbors with a common vertex, we have the translationvectors v ∈ {± (1 + i ) , ± (1 − i ) } . So the Sierpi´nski carpet has 8 neighbor types. The square as aself-similar set with m = k pieces forming a k × k checkerboard pattern has the same neighbortypes. Thus to distinguish different k we need other measures of complexity. The Sierpi´nskitriangle has 6 neighbor types, two for each of its vertices. The tile in Figure 1 has 11 types and4he example in Figure 2 has 52.With the concept of neighbor type, we provide a quantitative version of the open set condi-tion. In fact we are not interested in the question “OSC or not OSC”. We rather want to findexamples of small complexity which we can understand. At present, an IFS with 1000 neighbortypes can hardly be distinguished from an IFS without OSC.Given an IFS (4) with integer vectors and matrices, the question whether there are atmost N neighbor types is decidable in finite time for every N. One of the authors has writtenthe program IFStile [38] which decides this question within milliseconds for N ≤ . Thealgorithm, discussed in Section 3, constructs the neighbor graph of F, an automaton whichdescribes geometric and topological properties of the set A. However, when we assume integer matrices, we are in the setting of crystallographic groups.In the plane, symmetries must be rotations by multiples of 30 degrees, or reflections, the axes ofwhich differ by angles of k ·
15 degrees. Besides Figure 1, this includes Figure 2 where we haveno reflection, only rotation by 60 o and 180 o . Nevertheless, this setting is rather restrictive.Figure 3: A carpet generated from an irrational rotation. It has only 16 neighbor types, muchless than Figure 2. There are no characteristic directions. See the text in Section 6.
Approaches to non-crystallographic patterns.
There are various ways to general-ize crystallographic finite type systems. One way is projection of integer data from a higher-dimensional space, known from quasiperiodic tilings used for modelling quasicrystals [46, 2].This is implemented in IFStile. Another possibility is to replace the equation (1) by a systemof equations for different types of sets, which is called a graph-directed construction [37]. Intheir study of self-similar tilings, Thurston, Kenyon, and Solomyak [53, 30, 50]. studied graph-directed systems without any symmetries. They assume that there are finitely many tiles up totranslation. This approach includes the case of rotations by rational angles - rational multiplesof 360 o . A self-similar set A with pieces rotated by multiples of 60 o , as Figure 2, can be generatedby a translationally finite graph-directed system of six sets without considering rotations.In this approach all neighbor maps are translations, and since all matrices are powers of abasic matrix M, their product is commutative. For self-similar tilings only few examples are5nown outside this setting [42, 15, 22, 10]. For self-similar fractals, however, we are going topresent plenty of examples which are not translationally finite.Figure 4: A carpet generated from an irrational rotation and a reflection. It has 13 neighbortypes and no characteristic directions. For details, see Section 6. Contents of the paper.
In the translationally finite case as well as in self-similar setswith crystallographic data, including those which are projected from higher-dimensional lattices,all motives appear in a finite number of directions. Here we construct fractals with a dense setof characteristic directions, in other words, with no characteristic directions at all. Figures 3and 4 give a first impression of our patterns. We replace integer data by rational data, whereno theorems guarantee the existence of finite type examples, except for trivial cases like Cantorsets. We performed an extensive computer search, checking some hundreds of millions of IFS,and present selected results.After stating our basic assumptions in Section 2, we introduce the main tool, the neighborgraph of an IFS, in Section 3. Its topological applications are listed in Section 4 while Section5 discusses an algebraic viewpoint and an open problem. The search leading to Figures 3 and4, based on Pythagorean triples, is described in Section 6. We briefly discuss the search on thehexagonal lattice in Section 7 and introduce a classification of planar self-similar sets in Section8. This leads to a very convenient concept of an algebraic planar IFS which saves all matrixcalculations. In Section 9, a family of IFS is given by a characteristic polynomial and some linearrelations between expansion maps and symmetries. It turns out that the polynomial describesan algebraic number field. In the final Section 10 we discuss the search for examples in differentquadratic number fields.
Motivation.
Beside the potential of isotropic fractals for modelling in science, there arevarious mathematical reasons for this research. One is pure curiosity: to see what is beyondcrystallographic symmetries. Another motivation is to show that our approach with symmetriesis much wider than the translationally finite setting. Moreover, fractals without characteristicdirections show some measure-theoretic uniformity. They have “statistical circular symmetry”,6s certain quasiperiodic tilings, and physical materials of this type show a diffraction spectrum ofrings [42, 22]. Their projections onto lines possess the same dimension for every direction whilein general we have an exceptional set of directions with smaller dimension. Further uniformityproperties were proved mainly by Shmerkin and coauthors, see [21, 28, 49, 48]. Here we constructconcrete examples of such sets which also have nice topological properties.
Basic assumptions.
We recall the basic equations (1) and (4): A = m (cid:91) k =1 f k ( A ) with f k ( x ) = M − h k ( x ) = M − ( s k x + v k ) . (5)In standard coordinates, the s k should be linear isometries, given by orthogonal matrices. Themap g ( x ) = M x should be an expanding similarity map, so that we can write equation (5) as g ( A ) = m (cid:91) k =1 h k ( A ) = m (cid:91) k =1 s k ( A ) + v k . (5a)Moreover, the maps must have a discrete structure to apply integer arithmetics. We assumethat with respect to a common base B = { b , b } M and the matrices of the s k contain rational numbers, and the v k are integer vectors. (6)The condition (5) or (5a) together with (6) is our basic assumption. Note that B need not be thestandard base. It is needed to determine the combinatorial structure of the IFS. Using rationalnumbers with a bounded denominator, this calculation will be accurate. The transformationfrom base B to the standard base, and the visualization of the set A in standard coordinates,are done by floating point arithmetics with numerical error. In the following sections, B will bethe standard base. In Sections 7-10 we shall consider other bases. An example with irrational rotation.
Before we discuss the difficulties with the base B, we consider examples where the standard base can be taken as B. Let M = (cid:18) − (cid:19) and s = 15 (cid:18) −
33 4 (cid:19) . (7)The map s is a rotation with irrational angle, as shown in proposition 6 below. We wonderwhether an IFS with rotation s between the pieces can have a nice connected attractor. A tile,as on the right of Figure 1, seems not possible. Thus we would like to have at least somethinglike the Sierpi´nski carpet.Let us note that a fractal tiling with irrational rotations was found in [10] using a reflectionand the fact that M involves an irrational rotation. This was a rare and special example. Herewe prescribe the irrational rotation directly as a symmetry. The matrix M was chosen becauseit has determinant 5. With two or three pieces, we have few degrees of freedom, and only a7mall number of IFS with OSC, most of which are well known [24, 40]. With 8 or 9 pieces,we already have a huge choice of parameters s k and v k which we cannot control even with acomputer (see [9] for discussion of a similar case). A maximal number of five pieces is betterto handle. Since we find no tiles, we shall be content with fractals with m = 4 pieces and nicetopological structure.Figure 5: A carpet with simple structure involving an irrational rotation, and a close-up.Figure 5 shows the simplest result. In various runs of our experiments it was always obtainedearly. The pieces are well-connected: they intersect in Cantor sets which all have Hausdorffdimension 0.4307, as shown below. Most importantly, there are only five neighbor types. Thusthe complexity of this example is not larger as for the square or Sierpi´nski triangle. Thiswill be shown in the next section. First we complete the data of the IFS, listing the maps h k ( x ) = s k ( x ) + v k for k = 1 , ..., m.h = − s ( x − (cid:0) (cid:1) ) , h = − x − (cid:0) (cid:1) , h = x , h = − x + (cid:0) (cid:1) . (8)Thus the irrational rotation acts only between the pieces A and A while A and A are parallel,and A is rotated by 180 o with respect to them. Definition.
We shall now determine the combinatorial structure of our example IFS, theso-called neighbor graph. For the case of pieces of equal size, this object has been defined invarious papers, including [1, 4, 11, 14, 17, 18, 27, 35, 43, 44, 54]. We introduce the conceptbriefly and refer to the literature for details. The IFStile package determines neighbor graphsalso for IFS with different contraction ratios and for graph-directed constructions.The neighbor graph is a directed edge-labelled graph G = ( V, E ) which can have multipleedges and loops. The vertex set V consists of all neighbor maps h = f − w f v , where A u = f u ( A )and A v = f v ( A ) are two pieces of A of the same level which intersect each other. Different pairs( w, v ) of words on 1 , ..., m can belong to the same neighbor map.8ow if A w and A v intersect, there are intersecting subpieces A wk , A vj which must belong tosome neighbor map h (cid:48) which is another vertex. For each such pair of subpieces there is an edgefrom h to h (cid:48) labelled by k, j, indicating that h (cid:48) = f − k hf j . Moreover, we have initial edges from h = id to h = f − k f j , labelled by k, j, for each pair of first level pieces A k , A j which intersect.We consider id as the root vertex of G. But id is not a neighbor map and must not be reachedby an edge when the OSC is fulfilled. So we shall not draw id as a vertex. We just indicate theinitial edges.If a finite neighbor graph G can be constructed for an IFS, we say that the IFS has finitetype. We consider only finite type IFS. In the computer treatment, an IFS is discarded if thecorresponding neighbor graph requires more than a prescribed number N of vertices. In ourcontext, we take N = 100 or even smaller.For our example, we first explain how a human can geometrically construct the neighborgraph. Then we describe how a computer does the job. n n n n n Figure 6: The neighbor graph of Figure 5 has a very simple structure.
Intuitive construction.
The first level intersections all involve A . Initial edges labelled1 , , , , , , and 3 , n = f − f = − s − x + (cid:0) (cid:1) , n = f − f = h = s ( − x + (cid:0) (cid:1) ) ,n = f − f = h = − x − (cid:0) (cid:1) , n = f − f = h = − x + (cid:0) (cid:1) . In this first stage, f − k f j = h − k h j since g cancels out, and h = id allows to calculate directlyfrom (8). Moreover, n and n are point reflections, they are self-inverse. Thus f − f = n and f − f = n . Instead of drawing new edges leading to the vertices n and n , we just write asecond label 2 , , A ∩ A = A ∩ A , and that the neighboring position betweenthese subpieces is the same as between A and A . This can be confirmed by the equation f − n f = n = f − n f . Thus we draw an edge labelled 4 , n to n , and an edge withlabel 1 , n to n . A determine the boundary sets.Next, we see that A ∩ A = A ∩ A , and the neighbor type of these subpieces is just n . So we draw an edge with label 2 , n to n . Moreover, the neighbor type n divides intotwo subtypes: A ∩ A = ( A ∩ A ) ∪ ( A ∩ A ) . The corresponding neighbor map n = f − n f = − x − (cid:0) (cid:1) agrees with f − n f . So we draw an edge with two labels 2 , , n to n . Thesubdivision of neighbor types n to n is completed. We still have to study the subdivision ofthe new type n by considering the third level pieces of A, Fortunately, it can be seen that n describes just the intersection of the subpieces with index 1 on both sides, and represents type n on the next level. Thus only one more edge from n to n with label 1 , The computer algorithm.
A computer easily generates lots of neighbor maps h wv = f − w f v by repeatedly applying the recursive formula h (cid:48) = f − k hf j with k, j ∈ { , ..., m } . Theproblem is to decide which of these actually fulfil f w ( A ) ∩ f v ( A ) (cid:54) = ∅ . Proposition 1
For closed sets
A, B let d ( A, B ) = inf {| x − y | | x ∈ A, y ∈ B } . Suppose thatan isometry h fulfils d ( h ( A ) , A ) ≥ ε > . Then d ( h (cid:48) ( A ) , A ) ≥ ε/r for each h (cid:48) = f − k hf j with k, j ∈ { , ..., m } . Proof.
We have d ( h ( f j ( A )) , f k ( A )) ≥ d ( h ( A ) , A ) ≥ ε. Applying the similarity map f − k with factor 1 /r to both sets, we conclude d ( f − k hf j ( A ) , A ) ≥ ε/r. (cid:3) The proposition says that when h ( A ) does not intersect A for some generated map h, then h (cid:48) ( A ) will not intersect A for all its successor maps h (cid:48) in forthcoming levels. Moreover, the size10f the translation of h (cid:48) will grow exponentially with the level. So even for small ε it will becomeobvious after few recursion steps that we do not have a neighbor map.The algorithm is now clear. We stop the recursive calculation as soon as we recognize that h (cid:48) is not a neighbor map. Afterwards, we repeatedly remove from our recursive tree of mapsall vertices h without successors. Then we are left with all maps which lead to a cycle inthe neighbor graph, and this is exactly the graph G. On the other hand, when the recursivecalculation produces too many (say 10 ) isometries for which we cannot decide whether theyare neighbor maps, we give up and say that the IFS seems not to be of small finite type.Now we provide a simple general criterion to decide when an isometry is not a neighbormap. The IFStile package uses more complicated IFS-specific estimates which reduce the effortof recursive calculations. Proposition 2
Let ˜ x denote the center of gravity of A, determined from the IFS by the equation ˜ x = m (cid:80) mk =1 f k (˜ x ) . Moreover, let δ = max mk =1 | f k (˜ x ) − ˜ x | . Then A is contained in the ball B with radius δ/ (1 − r ) around ˜ x. When an isometry h fulfils | h (˜ x ) − ˜ x | > δ/ (1 − r ) then h ( A ) and A are disjoint.Proof. Note that ˜ x and δ can be easily calculated from the IFS data. For each j, k thedistance of x = f j f k (˜ x ) to f j (˜ x ) is at most rδ. So | x − ˜ x | ≤ (1 + r ) δ. For a word w = k ...k n and x = f w (˜ x ) we get | x − ˜ x | ≤ (1 + r + ... + r n − ) δ. Since these points, with arbitrary n and w, are dense in A, we obtain the estimate | y − ˜ x | ≤ δ/ (1 − r ) for all y ∈ A. This proves the firstassertion. If | h (˜ x ) − ˜ x | is greater than twice the radius of a ball B with centre ˜ x, then B and h ( B ) must be disjoint. (cid:3) Topology-generating automaton.
The neighbor graph is an automaton which generatesthe topology of A. All topological properties of A are encoded in the neighbor graph. In theargument above, we pretended that we can see that the pieces A and A intersect. This wasnot true. Actually, pictures did repeatedly lead us to wrong conclusions. Only the calculationof the neighbor graph can verify that A and A have common points.The neighbor graph is empty if the pieces A k = f k ( A ) , k = 1 , ..., m are pairwise disjoint.Otherwise, the neighbor graph tells us which pieces intersect. Consider any infinite path ofedges, starting with an initial edge, that is, starting from the root h = id. Paths are directedby definition, and if they are infinite, they must contain directed cycles. If the path has thelabels k , j ; k , j ; ... then the two sequences k k ... and j j ... describe the same point, whichis determined by the decreasing sequence of compact sets A k ∩ A j , A k k ∩ A j j and so on (cf.[12, 13, 20]). In our example, infinite paths are all given by the cycle between n and n whichwe have to run through infinitely many times. Actually, it consists of two cycles since from n to n we have two edges with labels 2 , , , respectively. Thus the addresses 3 u u u ... and 4 v v v ... describe the same point whenever either ( u i , v i ) = (2 ,
4) or ( u i , v i ) = (4 ,
2) foreach i = 1 , , .... This means that A and A have a whole Cantor set in common. And the sameholds for A and A , and A and A , if we consider paths starting in n and n . On the other11and, A and A have no common points since 1 , , Connectedness properties.
Recall that the neighbor graph G is constructed so thatthere is an outgoing edge from each vertex, and hence an infinite directed path starting in eachvertex. Thus there is an initial edge with label k, j if and only if A k ∩ A j (cid:54) = ∅ . So from G we candefine the connectedness graph G c of A, with vertex set { , ..., m } and undirected edges between k, j whenever A k intersects A j . For our example, the connectedness graph is a 3-star, or letterY, with central vertex 3. The following is known. The first assertion is a classical theorem ofHata, proved by inductively constructing chains of intersecting pieces.
Proposition 3 (i) The attractor A is connected if and only if G c is connected.(ii) Suppose A is connected. Then A contains a closed Jordan curve if and if and only if either G c is connected, or two pieces A j , A k intersect in more than one point.(iii) Let h = f − k f j be the vertex of the neighbor graph G which corresponds to D = A k ∩ A j (cid:54) = ∅ . Let C denote the set of all directed cycles which can be reached by a path from vertex h. The intersection D is a singleton if and only if C consists of one element C and there isonly one path from h to C . The intersection is finite if there is no path between C and C for any C , C ∈ C . The set D is uncountable if there exist C , C ∈ C and paths from C to C and back. Otherwise D is countably infinite. The proof is simple but there are various details, cf. [11]. The graph in Figure 6, for instance,contains two cycles from n through n to n . The two cycles can be reached from each othersince they start in the same point - the connecting path is empty. Similarly, care has to be takenwhen points belong to three or more pieces.According to (iii), we have different levels of connectedness of A which we can read fromthe neighbor graph: ordinary connectedness in the sense of Hata, connectedness by single-pointintersections which means that A is a dendrite, connectedness by finite intersections (called p.c.f.self-similar sets in [31]), connectedness by infinite, at most countably infinite or uncountableintersections. All such properties will be determined in IFStile, and can be used to characterizeand order the exmples.Some related properties still require research, for instance the study of connectedness com-ponents of A when A is not connected. For distinguishing Cantor set intersections A k ∩ A j from intervals, one needs to consider neighbor graphs for intersections of three or more pieces[4, 14, 54]. This is important for tilings, especially in three dimensions. In the example above,all three-piece intersections are empty. Cutpoints and first-level intersections.
Consider the fixed point y = 0 of f in theabove example. It does not belong to two pieces since the address 3 = 333 ... - even the word33 - is not labelling a path in the neighbor graph. Nevertheless, y is an important point for thetopology of A. It is a cutpoint of degree 3: A \ { y } consists of three connected components. Thiscan be easily seen: removal of A from A results in three different pieces, and removal of A from A results in three larger pieces, and so on. Thus A involves a rather simple tree structure,12espite containing Cantor set intersections and closed Jordan curves. We call such a space aweb.We do not go further into detail. We just want to give an impression of the potential anduniversal role of the neighbor graph. For the computer search, certain simple parameters areuseful even if they do not define topological properties of A. The number FLI of first-levelintersections is just the number of initial edge labels k, j with k < j.
This is an invariant of theIFS (cf. [8]). In our example FLI = 3 . When we look for connected examples, and do not findthem - which was the case in several experiments related to this paper - then FLI will give usan idea on how far we are from connectedness.
Hausdorff dimension of the boundary.
Apart from connectedness properties, let usremind the fact that the neighbor graph verifies that our IFS is of finite type and fulfils theOSC. In particular, the Hausdorff dimension of A equals [12, 13, 20] α = log m − log r = 2 log m log det M . (9)Moreover, A is a very nice type of self-similar set, completely described by an automaton. It isa computable self-similar set.Now we show that the boundary dimensions of A can also be read from the neighbor graph.This was found for the twindragon by Gilbert [25], for the L´evy dragon by [19, 52] and forself-affine tiles by [1, 17, 33] and others. To each neighbor type n k there corresponds a boundaryset B k = A ∩ n k ( A ) . The first parts of the labels of the edges are used to determine a set ofequations for the boundary sets: B = f ( B ) = f ( B ) , B = f ( B ) = f ( B ) , B = f ( B ) ∪ f ( B ) . The boundary sets form a graph-directed system, which inherits the OSC from A. Therefore itis possible to determine the Hausdorff dimensions of all boundary sets. In our case they all havethe same dimension, and we see that B is even a self-similar set: B = f f ( B ) ∪ f f ( B ) . The 4 mappings f k have the factor r = 1 / √ . So A has dimension α = log 4log √ ≈ . , and all the boundary sets have dimension β = log 2log 5 = α ≈ . . Note that with f , the irrational rotation is involved in all boundary sets.All the mentioned properties and parameters are automatically determined by IFStile, andcan be used to identify non-isomorphic examples, or to select specimen with prescribed topo-logical properties from a computer-generated collection, cf. [8, Section VI].13 Algebraic aspects of the neighbor graph
An open question.
The neighbor graph completely describes the topology of A, but notthe Hausdorff dimension of A. It is well-known that the dimension of a Koch curve can bechanged without changing the topology, just by varying the angle of the maps [12]. So thequestion is which properties should be added to the neighbor graph in order to completelycharacterize the IFS, up to isomorphy. Isomorphy involves change of the coordinate system andpermutation of the maps, see [8]. The question seems difficult. We shall discuss it for our simpleexample.
Generating relations.
The neighbor maps generate a group of isometries. Wheneverwe extend the self-similar construction of a connected attractor A to the outside, by formingsupertiles f − k ( A ) , f − k f − k ( A ) , ..., any isometry between two ‘tiles’ of such a pattern will belongto that group. In algebra, groups are often defined by a system of generators and generatingrelations. It turns out that the neighbor graph provides such relations between the neighbormaps. In this way it gives algebraic information on the IFS beyond the topology of A. Usuallythe equations are highly nonlinear. For our simple example we can separate and solve them,however.
Finding IFS data from a neighbor graph.
Let us assume that we have an IFS givenby g, h , ..., h which produces the neighbor graph of Figure 6. For simplicity we assume h = id which can be arranged by passing to the isomorphic IFS ˜ g = h − g, ˜ h j = h − h j . Then the initialedges 3,1, 3,2 and 3,4 imply that h = n , h = n , and h = n . If two initial edges with label k, j and reversed label j, k lead to a vertex h, then the map h is self-inverse. This also holds for paths with reversed labels starting from the root. Thus n , n , and n are self-inverse. Now when we restrict ourselves to IFS without reflections,the only self-inverse isometries in the plane are point reflections. Thus we can conclude that h ( z ) = w − x, h ( x ) = w − x, and n ( x ) = w − x, for some vectors w k . In order to simplify equations and avoid confusion with the IFS data above, we now work inthe complex plane, writing z instead of x. Our aim is to determine unknown complex numbers λ, t, and v from the neighbor graph. They correspond to M, − s, and s (cid:0) (cid:1) in the example. The w k are now also considered unknown complex numbers, but we use the same letter as for vectorsabove, and keep the notation g, h k , f k for the unknown maps. We have proved the first part of(i) in the following proposition. Proposition 4
Suppose an IFS is defined in the complex plane by g ( A ) = (cid:83) k =1 h k ( A ) with g ( z ) = λz, h ( z ) = tz + v, h ( z ) = z, with λ, t, v ∈ C and | t | = 1 . If the neighbor graph of thisIFS is the graph in Figure 6 then(i) h ( z ) = w − z and h ( z ) = w − z where w , w ∈ C fulfil w = (2 − λ ) w . (ii) v = tw . (iii) t (4 − λ ) = λ ( λ − λ + 3) . roof. For (i) we derive the relation between w and w from the neighbor graph. Notethat f k = λ − h k . The edge from n to n generates the relation f − n f = n . We multiply by f from the left and use h = n , h = n . Thus h λ − h = λ − h h , or w − ( λ − ( w − z ) = λ − ( w − ( w − z )) . Ordering terms, we obtain (i).The edge from n = h to n = h implies h f = f h , or t ( λ − ( w − z )+ v ) = λ − ( t ( w − z )+ v ) . Thus tw + λv = tw + v. Together with the relation in (i), this gives (ii).To verify (iii), consider the edge from n to n . It generates the relation f − n f = n , or h λ − h = λ − h n . Calculation gives w = w − λw + w = ( λ − λ + 3) w . The edge from n to n says n f = f n , or w − λ − ( tz + v ) = λ − ( t ( w − z ) + v ) . Thus λw = tw + 2 v = t ( w + 2 w ) = t (4 − λ ) w . Comparing the two expressions for w we get (iii). (cid:3) The parameter w was not determined. Since the assumption h = id did not involvecoordinate axes, we are still free to choose the coordinate system. We can set w = 1 , or w = − i if we like to obtain equation (8) exactly. Thus we have almost determined the wholeIFS from the neighbor graph. We got a rational function t = r ( λ ) = λ ( λ − λ + 3) / (4 − λ ) . For λ = 2 − i, the value t = r (2 − i ) = (2 − i )( − i ) / (2 + i ) = − (4 + 3 i ) / s in (7).Since t varies on the unit circle, it seems better to consider λ as a function of t, that is, of therotation angle of h . The derivative of r is r (cid:48) ( λ ) = (3( λ − + r ( λ )) / (4 − λ ) . Since r (cid:48) (2 − i ) =( − i + t ) / (2 + i ) (cid:54) = 0 , the inverse function theorem applies: there is a holomorphic function λ = r − ( t ) defined in a neighborhood of t . Moreover, in a sufficiently small neighborhood of t , the maps f − k f j which correspond to non-intersecting pieces at ( t , λ ) will still correspondto non-intersecting pieces at ( t, λ = r − ( t )) since only a finite number of such pairs need to beconsidered. And the above calculations can be reversed so that the parameters ( t, λ = r − ( t ))provide IFS with the neighbor graph of Figure 6. This proves Proposition 5
In a small neighborhood of our example IFS parameters λ = 2 − i, t = − (4 + 3 i ) / together with w = 1 , v = t, w = 2 − λ, there is a one-dimensional parametricfamily of IFS which all have the same neighbor graph as the example. In this neighborhoodand family, each IFS is uniquely determined by the neighbor graph and | λ | which respresentsHausdorff dimension of the attractor. The last assertion was checked only by a Matlab calculation which shows that | λ | is decreasingaolmost linearly on the curve λ = r − ( t ) for 30 o ≤ angle t ≤ o while at t we have the anglearctan ≈ . o . Since (iii) is a cubic equation, there are two other solutions λ , λ which fulfil r ( λ j ) = t . One of them has modulus smaller than one, and for the other one IFStile could not calculatea neighbor graph, so it certainly will not generate Figure 6. Thus it seems even globally truethat for this special neighbor graph, an associated IFS is uniquely determined by its Hausdorffdimension. 15
Examples from Pythagorean triples
Pythagorean triples and irrational rotation.
A rotation of the plane by an angle α iscalled rational if α = kn · π for integers k, n, and irrational otherwise. Proposition 6
A rotation s has a matrix of the form (cid:0) u − vv u (cid:1) with u + v = 1 . If u, v are rationalthen the rotation angle is irrational or a multiple of o . Proof.
If there are integers k, n with α = kn · π then z = u + iv is an n -th root of unity in C . That is, z is a root of the n -th cyclotomic polynomial which is irreducible. Thus except for n = 1 , , and 4, z is not in the rational field Q ( i ) , and u or v must be irrational. (cid:3) The rational points ( u = a/c, v = b/c ) on the unit sphere correspond to the Pythagoreantriples ( a, b, c ) of integers with a + b = c . They can be generated by a well-known formula ofEuclid, see Section 10. The rotation s in (7) came from the basic triple (3 , , . The two nexttriples are (5 , ,
13) and (8 , , . Our question here is whether the corresponding irrational rotations, combined with appro-priate expansions g ( x ) = M x, do generate self-similar sets with strong connectedness properties.Figure 8: A pattern of dimension 1.69 generated from (3 , ,
5) together with 90 o rotations. Suchpatterns are fairly easy to generate. This one has only 9 neighbor types. The challenge.
We briefly explain the goal of our experiments. For crystallographic data,each integer expansion matrix M leads to IFS with OSC and m = det M. This is proved by takingso-called complete residue systems as h , ..., h m [3], and it is well-known that the attractors aretiles. By dropping one or more of the mappings we can then easily create carpets, as in Figure1. For irrational data, however, only very few tiles are known [42, 15, 22, 10], and we found nofurther tiles in our experiments.It is not difficult to find totally disconnected, Cantor-type sets A with all kinds of irrationalIFS. But whenever two pieces have a point in common, this creates an equation for the mappingsof the IFS. In general the equation is given as a limit when the level of pieces tends to infinity.16owever, if we assume finite type, edges of the neighbor graph define equations for finite com-positions of the f k , as we have seen above. In case of complex linear functions f k ( z ) = t k z + v k we directly get polynomial equations in z. When we require the pieces to have only one point with eventually periodic addresses incommon (so-called p.c.f. fractals, like the Sierpi´nski triangle [31, 51]), we can still establishsuch equations by hand and try to solve them. However, we do not know any mathematicalarguments which would lead to the construction of the example above, even though there isonly one double cycle in the neighbor graph. It seems a mystery that two pieces differing byan irrational rotation intersect in a similar Cantor set as pieces which just differ by a pointreflection.
Results of the computer experiments.
While a computer search with rational rotationsgives thousands of examples within a minute, many of them with high complexity [8], theinteractive search for examples with irrational IFS is more difficult. First we get only Cantorsets without any intersections. Then few examples may have pieces with intersections, creatinga non-trivial neighbor graph, but A will still remain a Cantor set. Then by modifying the bestdatasets, we get some fractals which are connected, or have at least connected subsets. Finally,deleting all bad examples and modifying only the best ones, we get carpets with uncountableintersections in most of our experiments. As a rule, they have small complexity, between 10 and40 neighbor types. The number of such carpets varied between 1 and 100, for different g and s. Two results were given in Figures 3 and 4 in Section 1. They are from the same family asFigure 5, with g and s defined in (7) and m = 4 . Hence they have the same Hausdorff dimension α ≈ . . We shall not further specify the IFS data since all .png files produced by IFStilecontain their data. They can be opened in IFStile, allowing visual study of details. The IFSdata, including code of the neighbor graph, can be found under ’View-Console’. A list containingall our figures will be available on the web page [38].Most of our examples with irrational rotations contained cutpoints like Figure 5. We thoughtthat such cutpoints exist for mathematical reasons. Figure 3, with 16 neighbor types andboundary dimension 0.24, seems to reject this conjecture. It has a single point intersection oftwo pieces. However, this is not a global cutpoint. It seems that there are only local cutpointswhich separate a small neighborhood.If a reflection is allowed as additional symmetry, we get lots of carpets like Figure 4. Theyhave no cutpoints, neither global nor local ones. The influence of the reflection is visible in thelocal structure. In contrast to Sierpi´nski’s triangle and carpet, our examples do not contain linesegments. It would be interesting to know how much fractal analysis on these spaces differs fromthe classical carpet.Any additional symmetry will increase the number of patterns. Instead of a reflection, arotation by 90 o can be taken. A similar effect arises when we take two Pythagorean rotationswith angles adding to 90 o . We tried this for the expansion g = 5 s − (cid:0) −
33 1 (cid:1) whichhas determinant 10. A triangular tile with m = 10 and a dense set of characteristic directionsis known [23]. It uses a reflection, as in [42, 10]. For the Pythagorean rotation together with90 o rotations we found plenty of patterns with m = 7 like Figure 8, with m = 8 like Figure9, and various carpets with m = 9 and Hausdorff dimension 1.91. The influence of the 90 o rotations often dominated the effect of the irrational rotation. We found a few similar figures17igure 9: A carpet generated from (3 , ,
5) together with 90 o rotations. The expansion hasdeterminant 10, and there are m = 8 pieces. The Hausdorff dimension is 1.81, the intersectionsof pieces have dimension 1.25, and there are 13 neighbor types.with other Pythagorean triples. Most examples had rather small complexity, in contrast torational examples like Figure 2. If we want to include 60 o rotations, we have to distinguish the standard base and the base B = { b , b } mentioned in Section 2 for which the matrices of g and s k have rational entries. Acounterclockwise rotation s by 60 o has matrix (cid:0) −
11 1 (cid:1) with respect to the vectors b = (cid:0) (cid:1) , b = (cid:0) / √ / (cid:1) . When we work with such rotations, as in Figure 2, we determine the neighbor graph ofthe IFS with respect to the base B, using accurate integer calculation. For visualization on anumerical level we transform back to standard coordinates.As an example, we take g = 2 s + 1 as expansion and t = (3 s + 5) / g ( x ) = 2 s ( x ) + x is the similarity map transforming b into 2 b + b = (cid:0) √ (cid:1) , with determinant 7. By expressing g and t as rational linear combinationsof s and 1 , we guarantee that their matrices with respect to base B are rational, and that themappings commute. Moreover, we save a lot of matrix calculations.Figure 10 shows the most interesting carpet with m = 6 pieces resulting from a search with g, s, and t. While in most of our examples, only one piece involves the irrational rotation (relativeto the other ones), here we have three pieces with irrational rotation and three without. Similarto Figure 3, there seems to be no global cutpoint although local cutpoints are apparent. Theclose-up indicates both the symmetry of order 6 and the non-crystallographic character.We still have to prove that t is an irrational rotation. We use the fact that u = , v = isa rational solution of u + 3 v = 1 , and the following statement. Proposition 7
Let s be the counterclockwise o rotation, and let t = as + c with rational a, c. Let u = c + a/ , v = a/ . Then t is a rotation if and only if u + 3 v = 1 . The rotation angle isirrational or a multiple of o . roof. Since s has the matrix (cid:0) −√ √ (cid:1) with respect to the standard base, t has the matrix (cid:0) u −√ vv √ u (cid:1) . For a rotation, the determinant is 1. Suppose the rotation angle is rational and nota multiple of 60 o . Then the corresponding root of unity is the root of a cyclotomic polynomialwhich is irreducible over the field Q ( √− . This contradicts the assumption that t is a rationallinear combination of s and 1 . (cid:3) There are a number of similar patterns in this family. As reflection for the base B we canadd the exchange matrix (cid:0) (cid:1) which exchanges b with b . This will create a large family ofcarpets, related to the Gosper flake in Figure 1. Some of them have 100 neighbor types, butmost have smaller complexity.Figure 10: A carpet generated from irrational rotation by arctan together with 60 o rotations.The expansion has determinant 7, so for m = 6 pieces we have dimension 1.84. There are 18neighbor types. Principles.
Now we shall try to get some order into our zoo of fractal examples. As inbiology and crystallography, we have to divide them into species and families. This will be donein three steps.1. We fix the lattice, which corresponds to the species. We can choose the square latticewith the basic 90 o rotation, or the hexagonal lattice with its 60 o rotation. Other choicesare given below. Algebraically, the lattice is induced by a number field which specifiesalgebraic numbers like i or √ g ( x ) = M x or g ( z ) = λz. The number λ is taken as analgebraic integer of norm greater one in our number field, sometimes as an algebraic19ational. Then √ det M or | λ | is the contraction factor r of the IFS. Since we require OSC,the number of maps in the plane is bounded by mr ≤ , or m ≤ det M.
3. The choice of lattice and expansion, together with a finite selection of rotations s, deter-mines the family of fractals. In the last step, concrete instances of this family are producedby choosing particular s k , v k for k = 1 , ..., m such that the OSC is fulfilled.It makes sense to fix only a maximum number of maps and let m vary within the family. Oftenthe IFS with different numbers of maps and their neighbor graphs are related. Moreover, theactual symmetry class of an example can only be determined after the s k are chosen. Usually weare not using all symmetries which are offered. Then we follow the convention used in classifyingcrystallographic patterns [26]. The symmetry type of an example is determined by the group ofmaps generated from the actual neighbor maps. Examples.
In Figure 1, the Sierpi´nski triangle was drawn in a symmetric way. When wemake use of this symmetry, or include 120 o rotations in our IFS, we are in the hexagonal lattice,or the number field Q ( √− , and have a symmetry group of three rotations. However, usuallywe generate a symmetric fractal in its most simple form. That is, f k ( x ) = ( x + c k ) / c k is the fixed point of f k . Then the symmetry group is trivial, s k = id for all k, and all choicesof non-collinear points c k are isomorphic. This symmetry type consists of a single isomorphyclass for m = 3 . For m = 2 we would add intervals. We work in Q , no extension of the rationalnumbers is required. When we go to m = 4 , we would obtain the parallelogram, the triangle,and a lot of more fragmented tiles [3, 38].When we stay with m = 3 and allow s k ( x ) = ± x, we obtain three other modified Sierpi´nskigaskets which are well known. Again, the choice of the c k is irrelevant. We stay in Q , and thefamily remains very small.In this paper, we shall always include s ( x ) = − x in the group of symmetries, since we donot want to distinguish too many symmetry types.In [8] we studied the Sierpi´nski triangle in a square lattice and its modifications obtainedby chosing rotations s ( z ) = i n z with n ∈ { , , , } . This is a huge family. We used the squarelattice Q ( i ) , and the symmetry group of most examples was the rotational group of the square.For m = 4 , we get tiles like square, right-angled isosceles triangle and the aperiodic chair [26].Of course, one gets an even larger family if reflections are added as symmetries.The tile on the right of Figure 1 obviously has 120 o and 60 o rotations as neighbor maps, sowe work in the hexagonal lattice, or Q ( √− . The symmetry group is a rotation group, evenwhen we define the s k with reflections in order to keep the positive determinant of g. The familyis large. For m = 7 it contains plenty of tiles [38]. One example for m = 6 is Figure 2.The new constructions in this paper have infinite symmetry groups. In Figures 3 and 5 thegroup is cyclic, with a single generator. A second generator is chosen as a reflection in Figure4, and as a rotation in Figures 8 and 9. The number field is Q ( i ) , the lattice is formed by theGaussian integers. For Figure 10, with an irrational rotation plus 60 o rotation, we have thehexagonal lattice of Eisenstein integers.A large part of the literature on fractal tilings [5, 17, 18, 29, 30, 33, 44, 53] is concerned withself-affine tiles where neighbor maps are translations and symmetries play no part. However,20here are also crystallographic fractal tiles where we have a crystallographic group which acts onthe tiling [24, 34, 35]. Only very few fractal tiles have infinite symmetry groups [42, 15, 22, 10].This can be explained by a result on algebraic expansion constants which goes back to Thurston[29, 30, 53].In the study of Ngai, Sirvent, Veerman and Wang [40] on fractal tiles with m = 2 pieces,symmetries play an important rˆole. They consider rational rotations and no reflection. Theyshow that there are only six cases: twindragon, L´evy curve, Heighway dragon, triangle, rectangleand tame twindragon. The first four cases are realized in Q ( i ) with expansion constant λ = 1+ i. The hexagonal lattice does not contain a lattice point with norm m = 2 , so it can be used onlyfor tiles with 3 pieces. This means that rectangle and tame twindragon are based on otherlattices which we consider below. Parameter-free description of mappings.
In this paper, a lattice was defined by abase B = { b , b } for which matrices M, s k have rational entries, and vectors v k have integercoordinates. How do we find B ? Let us first reformulate condition (6) for a rotational symmetryand expansion. Proposition 8
In the plane, let s be a rotation, and g an expanding similarity map with positivedeterminant. Assume that s and g have a rational matrix with respect to the same base B. (i) The characteristic polynomial of s is p s ( z ) = z + az + 1 , where a is rational and | a | ≤ . (ii) There are rational numbers b, c such that g ( x ) = bs ( x ) + cx for all x. Proof.
The determinant of a rotation is 1, and the trace is rational by assumption. Theeigenvalues of s are − a/ ± (cid:112) a / − . For | a | > , the eigenvalues are different reals so that s cannot be a rotation.For (ii) we consider similitudes with positive determinant in standard coordinates. Theyhave matrices of the type (cid:0) u − vv u (cid:1) and thus form a two-dimensional vector space. The identitymap 1 together with s forms a base of this space. So the map g has the form g = b · s + c · b, c. This equation is true for standard coordinates as well as for coordinateswith respect to any other base B. Taking a base for which both maps have a rational matrix,we see that b and c must be rational, by studying first values off the diagonal and then on thediagonal. (cid:3) The proposition says that any triple a, b, c of rational parameters defines a family of fractals.Not all triples lead to interesting examples. Our approach first selects a basic symmetry s andthen an expansion g which fits the symmetry s. One can also first define the expansion and thenadd symmetries in the form s = bg + c. However, ordinary integer expansions, like g ( x ) = 2 x, will fit any symmetry group. Moreover, the coefficients b, c for g = bs + c are often integers whilefor s = bg + c they are always rational. This led us to choose the symmetry type first. Togetherwith a given rotation s, all powers s q or s − q will be admitted as symmetry maps s k in (5a). Inreality, the integer q will of course be small. 21 ompanion matrix and exchange matrix. The definition of a fractal family by onecharacteristic polynomial p s ( z ) and a polynomial equation g = (cid:80) nk =0 b k s k was implemented inIFStile for a higher-dimensional setting. The user has only to think about the master equations,while all matrix calculations are done by the computer. In the case when the polynomial p s ( z ) isirreducible, the canonical base B is given by the successive images b k = s k − ( b ) , k = 2 , ..., n ofthe first base vector b . In other words, we work with the companion matrix of s. An advantageis that the transform to standard coordinates can be done by fast numerical procedures.Here we work with quadratic polynomials p s ( z ) = z + az + 1 which are irreducible for | a | < a = ± b = (cid:0) (cid:1) , b = s ( b ) . For this base, s has the companion matrix M s = (cid:0) − − a (cid:1) . Because of themaster equation g = bs + c, the matrix of g in this base is M g = bM s + cI where I denotes theunity matrix. Thus det M = b + c − abc and trace M = 2 c − ab . (10)So far we discussed only rotations and expansions with positive determinant. What aboutreflections? It turns out that we need only add a single reflection r to our symmetry group.Further reflections are produced from composition of r with rotations. When working with thecompanion base, a canonical choice for r is the exchange matrix M r = (cid:0) (cid:1) . In other words, r ( b ) = b and r ( b ) = b . Other definitions of a basic reflection r are possible. For Q ( i ) , ordinaryconjugation is the canonical choice.With a reflection r, we can also describe IFS with expansion maps with negative determi-nant. Note that the definition of an attractor by an IFS g, h , .., h m by (5a) is by no meansunique. For any isometry t, we can pass to the IFS tg, th , ..., th m which has the same attractor A. Moreover, we can change the coordinate system and permute the maps of the IFS. The recog-nition of isomorphic IFS representations is still a problem of the IFStile software, in particularfor symmetric examples as square and cube. See [8, Section VI] for a brief discussion of thisproblem. We try to choose the simplest IFS for a given attractor, but the computer will notalways do.Finally, we mention that in the two-dimensional case we have little problems with commuta-tivity of matrix multiplication: s commutes with g, and different rotations commute with eachother. For a reflection r and a rotation s we have sr = rs − .
10 Constructions in quadratic number fields
Quadratic number fields.
Every irreducible polynomial p with rational coefficients givesrise to a field extension of Q . In particular, p ( z ) = z + d with a positive square-free integer d generates the field Q ( √− d ) = { c + c · i √ d | c , c ∈ Q } . (11)The integer d is square-free if it is not divisible by a square of an integer n > . For otherintegers d such an n would be included in the coefficient c . Any quadratic polynomial withrational coefficients and complex roots will generate one of these fields, as can be easily checkedby solving the quadratic equation. 22igure 11: A carpet generated from three irrational rotations with a = , , and an expansion g of determinant 8. It has dimension 1.87 and 19 neighbor types. The number field is Q ( √− , as for the tame twindragon.Here we are interested only in polynomials p s ( z ) = z + az + 1 of rotations with a positiverational number a < . The number − a gives the same quadratic field, and is included in any IFSwith s since the map − x is always adjoined to the symmetry group. We take positive integers u < w with a = 2 u/w. Then z = u + i √ w − u w = u + v √− dw where w − u = dv . The number d is taken as square-free part of w − u . Table 1 shows allparameters a with denominator w ≤ d ≤ . We have characterized therotations which belong to quadratic number fields:
Proposition 9
Consider the polynomial p s ( z ) = z + az + 1 with rational a ∈ (0 , . Let a =2 u/w where u, w are positive integers.(i) p s generates the quadratic number field Q ( √− d ) where d is the square-free part of w − u . (ii) If a is different from 0 and 1, the rotation angle α is irrational and fulfils cos α = − a/ . (iii) The parameter a corresponds to the integer triple ( u, v, w ) which solves the equation u + dv = w . For every d, these generalized Pythagorean triples are generated by Euclid’sformula u = n − dm , v = 2 mn , w = n + dm with integers m, n such that n > m √ d. Proof.
The first assertion was proved above, the second follows from the fact that thecyclotomic polynomials which generate rational rotations have integer coefficients. The rotation23ngle is defined by z = cos α + i sin α. The quadratic Diophantine equation was derived above.It is equivalent to ( w − u )( w + u ) = v d. Now take relatively prime integers m, n with w + uv = nm = vdw − u and thus w − uv = mdn . Sum and difference of both equations yields wv = n + dm mn and uv = n − dm mn , respectively. Since m, n are relatively prime and d is square-free, this implies (iii). (cid:3) d a d a d a , ,
85 10 67 19 952 23 ,
149 11 53 , ,
79 21 453 1 , , ,
137 13 127 23 116 ,
785 43 , ,
29 14 109 31 1586 25 ,
107 15 12 , ,
118 33 877 32 , ,
98 17 169 35 13 , Table 1: The first positive square-free integers d, and corresponding fractions a with denominatorup to 9 which generate the field Q ( √− d ) . Results of the computer experiments.
We studied rational numbers a with smalldenominator and | a | < , and the rotation with polynomial p s ( z ) = z + az + 1 . The expansionwas defined as g = bs + c with rational numbers. The best results were obtained when g represents an algebraic integer in Q ( √− d ) . That is, determinant and trace of M in (10) mustbe integers. Recall that an element of the field (11) is an algebraic integer if either c , c areintegers, or d = 3mod4 and c = d / , c = d / d , d . For a = , we get the field Q ( √−
7) which corresponds to the tame twindragon [3, 40]. Thetwindragon is obtained for g = 2 s + 1 but does not use the symmetry s. With g = 3 s + 1 , wegot an example with m = 4 which is extremely similar to Figure 5. It has 6 neighbor types andfulfils β = α/ g has determinant 11/2 and trace5/2 and thus is not an algebraic integer. 24or the expansion g = 4 s + 1 with determinant 11, we got many nice carpets with m = 9 andsmall complexity. Adding reflections, we got an even greater variety of shapes. One examplehad dimension α = 1 .
83 and almost the same boundary dimension β = 1 . , similar to the L´evycurve [19, 52]. Another interesting expansion was g = 2 s − g = 2 s − m = 7 pieces.The Hausdorff dimension is 1.87, and there are no cutpoints whatsoever. Only two pieces areparallel and two pieces related by point reflection. All other pairs of pieces differ by differentirrational rotations. From all our IFSs, the isotropic character is most obvious here. Neverthelesswe have only 19 neighbors. There were two variations of this example, one with 16 neighborsand cutpoints, another one with 21 neighbors and still better geometry. On the whole a = 3 / Q ( √−
7) is a rich source of irrational examples.For a = 1 / , which corresponds to Q ( √− , we got also rich families. The simplest expan-sion is g = 2 s − m = 5 . Similar to crystallographic IFS like Figure 2, the complexity can be large, and thepieces of A may cross each other which indicates that the OSC can only be fulfilled with com-plicated open sets [8]. An example with only 12 neighbors is shown in Figure 12. The rotationwith a = 7 / a = 1 / , which corresponds to Q ( √− . The rectangle as a self-similar set with two pieces, used as standard size for writing paper,belongs to the field Q ( √− . An appropriate rotation is given by a = 2 / . The expansion g = 2 s − m = 4 similar to Figure 5, but with smaller dimension 1.5and larger number 12 of neighbors. The reason might be that g with determinant 19/3 is not analgebraic integer. Another expansion is g = s − m = 8 . Still these examples have global cutpoints, like Figure5, even if reflections are used. A third choice was g = 3 s + 1 with determinant 8. The searchgave a single example with m = 7 and Cantor sets as boundaries. It is exceptional since it hasonly 6 neighbor types. But like Figure 5, it has global cutpoints and involves a tree structure.Altogether, the field Q ( √−
2) seems not to lead to nice carpets for m ≤ . We performed similar experiments for d = 5 , , , , ,
91 and got similar results as for25 = 2 , at least as good as Figure 5. For d = 6 there were less examples. Summary.
Including irrational rotations into IFS is not easy. No tiles could be constructedthis way. Nevertheless, we found connected self-similar sets with pieces intersecting in uncount-able sets in all quadratic fields which we studied. Various examples have the topology of theSierpi´nski carpet, almost the same Hausdorff dimension, and a much more interesting isotropicgeometry. For the Sierpi´nski triangle, with finite intersections of pieces and dimension between1.5 and 1.6, there are lots of variations in all quadratic fields considered. In general, the com-plexity of irrational examples is smaller than that of IFS with crystallographic data. Some ofthe examples with small number of neighbors seem to be unique and worth of further study.
Some open problems.
This is a beginning. We discussed self-similar sets in the planewith equal contraction factors, between 4 and 14 pieces, and data from quadratic number fields.Graph-directed constructions and projection schemes for arbitrary number fields are more ex-citing. For three-dimensional self-similar sets a general framework does not yet exist, despitemuch work, including [4, 14, 15, 18, 23, 30, 38, 54].In Section 1 we claimed that our examples do not contain line segments. How can thisbe proved? If there is a line segment L in A, it will intersect a small piece A w in an interiorsubsegment L , and meet two subpieces A wj and A wk . Then L (cid:48) = f − w ( L ∩ A w ) is a segmentwhich connects two boundary points of A in different pieces A j , A k . In all our figures, visualinspection shows that such L (cid:48) cannot exist. It would be better to prove this by a characterizationof all IFS which lead to segments in A. It should also be possible to characterize attractors with cutpoints and give a fast algorithmfor detecting Cantor set attractors, which may have very large neighbor graphs [8]. The holesof all irrational examples obviously look more complicated than the holes of Sierpi´nski’s spaces.However, calculation of the Hausdorff dimension of the topological boundary of such holes isstill an open problem.
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Proc. Amer. Math. Soc. ,124:3529–3539, 1996.Christoph BandtInstitute of Mathematics, University of Greifswald, 17487 Greifswald, Germany. [email protected]
Dmitry MekhontsevSobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090 Novosibirsk Russia [email protected]@gmail.com