A single fractal pinwheel tile
Abstract
The pinwheel triangle of Conway and Radin is a standard example for tilings with selfsimilarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.
1 Selfsimilar tilings
A compact set with nonempty interior is called a replication tile with pieces, or reptile for short, if there exist a similarity map and isometries on Euclidean such that
(1) 
and any two different sets have no interior points in common. Figure 1 shows an example, others can be found in [17, Chapter 11], or in [14, 21, 29, 22]. A wellknown theorem of Hutchinson says that is determined by the data [8, 11].
The union of isometric copies of is an enlarged copy so the union of isometric copies of will be a still larger copy of Continuing this procedure and subdividing the larger copies, one can see that the whole is tiled by isometric copies of forming a selfsimilar tiling. There are different ways to form supertiles, by taking the small tile as first, second, or th piece of the larger tile, and one has to care a bit so that this socalled inflation process takes place around an interior point of
Our assumptions are quite restrictive. Among others, comparison of the volume in equation (1) shows that all eigenvalues of must have modulus There are more general concepts in the literature, allowing for affine maps for copies with different sizes, and for several types of tiles. Here we stick to the simplest case, and we take so that has ratio Moreover, we assume that and consider only tiles which are homeomorphic to a disk, bounded by a closed Jordan curve. Our point is that even under such restrictive assumptions new examples can be found.
To see why Figure 1 is important, we give a brief review of selfsimilar tilings. The main point is that the tiling property of is fulfilled only for very special choices of similarities and isometries The standard assumption is that the mappings generate a certain crystallographic group and the expanding map produces a subgroup
There are tilings which have as their symmetry group. This happens when is a complete residue system, that is [14], cf. [4, 5, 21]. This necessary and sufficient condition for crystallographic tilings is easy to check. For given data there exists only one tiling, and many ways to assemble the tiles into larger and larger supertiles. Examples include the plane regular tilings by squares and by equilateral triangles, with
A second class of tilings with and noncomplete residue system has been considered by many authors [19, 17, 29, 5, 31, 21, 3, 20]. We have more tilings, but less symmetric tilings, and fewer choices for forming supertiles than in the crystallographic case. The simplest case is on with where and [19]. There are two possible tilings with a fixed tile obtained by considering as left or as right part in a supertile. With the IFSTile program package [22], a lot of new cases in this class have been detected. Necessary and sufficient conditions for the second class were found only in the onedimensional case [20].
An extreme subclass of this class includes the chair and sphinx tilings [17, 29, 3]. These tilings are not periodic: no translation will transform them into themselves. We have a continuum of different tilings, which can be made a tiling space [30, 26], and the composition of supertiles is unique in each given tiling [31]. Such tilings are often associated with quasicrystals, although their Fourier spectra are not much different from those of periodic tilings [29, 3]. The ’typical quasicrystal’ Penrose, Robinson and Ammann tilings made from several types of tiles [17, Chapter 11] and are not considered here. They have symmetries of the Fourier spectrum which are forbidden for crystallographic pattern, and which have been found by physicists in quasicrystallic alloys [29, 3].
There is a third class of selfsimilar tilings, where either does not generate a crystallographic group, or is not a subset of In the language of Section 3 below, the neighbor maps are not contained in a crystallographic group. The first example was the pinwheel triangle of Conway and Radin where it is easy to verify that contains an irrational rotation. Using ergodicity, Radin [24] proved that the orientations of triangles in an infinite pinwheel triangle tiling are equidistributed on the circle. This implies that the tiling has a continuous spectrum. A physical material modelled by pinwheel triangles would have an extraordinary diffraction pattern consisting of circles, like a disordered system, cf. [2, 23]. At the same time, the pinwheel triangle tilings have a finite set of matching rules, similar to a crystal, as proved in [25]. This apparent contradiction motivated the work of the mathematical physicist Radin. For mathematical work on tiling spaces and spectra of tilings see [30, 26].
Various modifications of the pinwheel triangle have been presented [27, 13]. They use triangles, and most of them have a larger number of tiles. There is also a fractal pinwheel version [12] which uses 13 different types of tiles.
Figure 1 shows an unexpected single fractal tile with irrational rotations between neighbor tiles, denoted as ’fractal pinwheel’. Irrational rotations imply statistical circular symmetry of orientations [13, Section 6], and continuous diffraction and dynamical spectrum [2, 23, 30]. The assumption of GoodmanStrauss [16] are also fulfilled, so there exists a finite set of matching rules. Straight line boundaries are not necessary for such tiles.
In the following section the reptile in Figure 1 is defined, and geometrical properties are studied. While the tiling property is obvious for a triangle like Radin’s pinwheel, considerable effort is needed for the proof in the fractal case where the tile is defined only implicitly by contraction maps. Theorem 4 in Section 3 uses the technique of neighbor maps. Actually, the tile was found by a computer search with IFSTile [22] which analyzed neighbor graphs of many random parameter sets. Since then, a search of parameter sets with IFSTile [22] has not provided further pinwheels. In Section 4 we study a second reptile structure on the fractal pinwheel which is quite different.
After this paper was completed, we found that both of our tiles were already 2012 presented by Ventrella in his inspiring book [33, p.8586] on planefilling curves. Ventrella used Lsystems and gave no rigorous argument for the planefilling property. He noted that these two curves do not fit into a square grid and show an extraordinary ’mixture of 90 and 45degree angles’. Our careful analysis of irrational angles will provide mathematical clarity.
2 Definition and geometry of the fractal tile
The two outer pieces 1 and 5 of Figure 1 have different orientation from the whole figure while the three middle pieces have the same orientation. The apparent vertices of the tiles are on a lattice. We take part 1 as our basic tile and choose coordinates so that has vertices and Then is a triangle with fractal boundaries and vertices and For an affine mapping the coefficients are given as
(2)  
(3) 
Of course, other descriptions are possible. Choosing part 4 as basic tile and point as origin of a coordinate system with we would get the expanding matrix as Thus the expansion map is the same as for the pinwheel triangle in [24, 25, 12]. The are quite different: in Figure 1, piece is connected only with pieces by a fractal edge, while in the pinwheel triangle three pieces are pairwise connected by edges or ’halfedges’.
Proposition 1 (Convex hull and diameter of fractal pinwheel)
The convex hull of the fractal in Figure 1 is the hexagon with The diameter of is and the diameter of is
Proof. We denote the convex hexagon by The selfsimilar set contains the points and which are fixed points of mappings and see (4) below. Thus contains and So the hexagon is a subset of the convex hull of On the other hand, it is easy to check that and This implies that (the Hutchinson operator for fulfils its iteration yields a decreasing sequence converging to , see [8]). Thus is the convex hull of It has three sides of length 1, two of length and a short side of length
Since the diameter of a polygon is its longest side or diagonal, the diameter of and of is the length of Since has factor the diameter of is
Two Jordan arcs and with a common endpoint will be said to form a Jordan angle of size if a rotation by with center transforms into The fractal boundary curves between and are called sides of the triangle
Proposition 2 (The fractal pinwheel as a triangle)

The fractal triangle has two congruent sides The fractal curves and on the long side are also congruent to the short sides.

The short sides are symmetric with reflection at the perpendicular bisectors of The long side is symmetric with respect to a rotation around its midpoint

The triangle has Jordan angle at and irrational angles at and at

In dimension all sides have positive and finite Hausdorff measure Taking this as length measure, Pythagoras’ theorem holds for the triangle.

The area of is
First part of proof. Properties a) and b) seem obvious from Figure 1, and a proof is given in Section 3. c) immediately follows from a). The Euclidean triangle is a Radin pinwheel and has angle at The size of the Jordan angle can be checked by noting that where is outside Figure 1. Irrational angles are the basis for concluding that the tilings are statistically circular symmetric [25, 13].
Concerning d), we shall prove in Section 3 that the sides of form a graphdirected system of selfsimilar sets. Calculation of the Hausdorff dimension of such fractal boundaries is standard, starting with classical work of Gilbert in the 1980s [15, 10, 32, 18, 7]. The Hausdorff measure of dimension of each side is positive and finite. Assuming this fact, we can prove d) by the following argument.
Hausdorff measure is invariant under isometry, and Hausdorff measure of an image under a similitude with factor decreases by the factor Here the factor of is and we let If we put then
Now cancels out, and is the positive solution of the quadratic equation. Thus which means that Pythagoras’ theorem is true for our triangle with fractal side lengths. Under contractions like where Euclidean distances shrink by the lengths of fractal boundaries shrink faster, by The dimension of boundaries is as stated in d).
It is a wellknown open problem whether there exists a single puzzle tile such that the plane can be tiled with isometric copies of but it cannot be tiled periodically [17, Chapter 11]. Could Figure 1 be such an aperiodic tile? Unfortunately not. If we add to three copies obtained by successive rotation around we get a fractal square which tiles the plane like an ordinary square. This directly follows from a) and b). Another periodic tile is the fractal rectangle the union of parts 2 and 3 in Figure 1. If we consider the square lattice as a checkerboard, and we put a copy of the rectangle on each white square, and a copy rotated around on each black square, we have a tiling, due to Proposition 2 a) and b). Part of such checkerboard pattern can be seen in Figure 2. As a consequence, the fractal rectangle must have area one, and the area of the tile is This proves e).
3 The neighbor graph
Applying to both sides of (1), we represent as a selfsimilar set or IFS attractor
(4) 
For our example, let us list the expressions of to be used below.
(5) 
A basic theorem of Hutchinson says that there is exactly one compact solution of (4) for given contractions [8, 11]. To get good topological and geometric properties, however, we have to assume that the do not overlap too much. Usually one assumes the open set condition: there is an open set so that the are disjoint subsets of Such a set is hard to determine, so one requires that , or better is of finite type. This means that the neighbor graph, defined below, is finite. The topology of is determined by a finite automaton which is obtained from the by an algorithm. We shall calculate the automaton for our example.
The technique of neighbor graphs is now well established. The beginnings go back to Gilbert in the eighties, see [15] and references in [4]. In [6], neighbor maps were introduced as where for some The open set condition was shown to be equivalent to the fact that neighbor maps cannot converge to the identity map. For tiles neighbor maps are exactly the isometries between neighboring pieces in any selfsimilar tiling made up of copies of cf. [5]. We shall consider only proper neighbors, for which Here and Thus a neighbor map transforms to an isometric copy which intersects and is a tile in some patch obtained by inflation of around one of its pieces
A neighbor map describes a potential boundary set Actually, the study of tile boundaries, especially for the Levy curve [10, 32] was a key motivation for developing the method of neighbor graphs [18, 28, 7, 1]. The neighbor graph yields a system of set equations for the boundary of similar to (4). In the program IFSTile [22], a fast and very general algorithm was implemented to determine neighbor graphs and dimensions of boundary tiles.
The vertices of the neighbor graph are the neighbor maps An arrow with label is drawn from vertex to vertex if for two marks We keep only those arrows which correspond to proper neighbors, that is, See [7] for details. The identity map is the root vertex of the graph, with loops labelled It is not drawn in Figure 4, and arrows from have no intial vertex.
If is a finite graph, the reptile or IFS attractor generated by resp. is called finite type. If there are no incoming edges to the root vertex then the open set condition is fulfilled, which together with the condition that all have similarity ratio implies the tiling property [6, 5].
In a plane tiling, we can have two different kinds of neighbors: point neighbors which have a single intersection point, and edge neighbors which have uncountably many points in common. Other kinds of neighbors can occur [6] but not in the case of our example. Moreover, the intersection of two edge neighbors is always homeomorphic to an interval, as will be proved now.
Theorem 3 (Neighbor graph and boundary of fractal pinwheel)
Let denote the fractal pinwheel, with mappings defined by (4), (2) and (3).

There are exactly 11 edge neighbors illustrated in Figure 3, 69 point neighbors and no other neighbors. Thus is finite type, has nonempty interior and is a reptile.

Two of the maps for edge neighbors are irrational rotations. So there is a continuum of different tilings. They are not lattice tilings, and have statistical circular symmetry.

Edges are of two types: rotation on one hand, glide reflections and irrational rotations on the other. All subedges of an edge at any level have the same type as the original edge.

is homeomorphic to a disk, bounded by a closed Jordan curve of dimension This boundary set is the union of intersections of with its three neighbors by rational rotation.
Proof. First we sketch our proof of the difficult part a) by calculation of all neighbor maps with computer. This was done independently by two authors with different software. We build the graph recursively, calculating all possible for all previously constructed maps We want to neglect if but this cannot be checked directly. However, by Proposition 1, the diameter of is smaller than whenever intersects Since the origin belongs to this implies that
We determine the graph of neighbor maps with an orthogonal matrix for which is fulfilled. This graph turns out to be finite, with 955 vertices. Then we take the subgraph of all vertices which lie on cycles of the large graph. This is our graph of proper neighbors with only 81 vertices including the root. Point neighbor maps have the property that for each only one path of length starts at vertex They are easily singled out by checking powers of the adjacency matrix of There were 69 point neighbors, 11 remaining neighbor maps and the identity, which proves a). On a PC, all this is done in less than 2 seconds.
Now we give a computerfree proof of the theorem, except for the number of point neighbors. As explained below, the 11 edge neighbors in Figure 3 can be found by inspection of the second subdivision of Figure 2, and confirming by calculation. A check of the next subdivision, or of Figure 3, then verifies that no other edge neighbors exist, and there are no incoming arrows to the root vertex in the neighbor graph.
Note that (4) implies and thus for So the successors of the root vertex include the rational rotations a rotation around
So far we have studied the maps between pieces 2 and 3, and 3 and 4 in Figure 1. Now we consider their subpiece neighbors in Figures 2 or 3. We see that subpieces 24 and 34 have the same relative position as pieces 2 and 3, which is algebraically verified by the equation and results in a loop from vertex to itself with label Subpieces 41 and 35 also intersect, and correspond to the neighbor map which results in arrows from vertex to with label and from to with label Checking two other pairs of subpieces of 2 and 3, and one remaining pair of subpieces of 3 and 4 in Figure 2, we obtain the graph in Figure 4. This argument proves that there are no other arrows starting in (which the computer checked algebraically). It is enough to take only the first label of any arrow from a vertex to a vertex since the second label is the same as the first label of the arrow from to Drawing arrows from the root without an initial vertex, we obtain a reduced form of the graph [7] on the left of Figure 4.
rational rotations  glide reflections, irrational rotations  
initial vertex  id  id  p  p  r  r  id  id  s  s  s  a  a  a  t  b  b  b 
terminal vertex  p  r  p  r  s  s  a  t  b  
first label  2  3  4  5  5  3  1  4  3  5  5  1  1  2  1  4  4  5 
second label  3  4  4  1  1  5  2  5  1  1  2  1  4  4  5  3  5  5 
To get all edge neighbors, we still have to consider the boundary between pieces 1 and 2, or 4 and 5. There we get the glide reflection and its inverse seen in the second row of Figure 3. The subpieces 15 and 21 lead to the glide reflection and its inverse Subpieces 13 and 21 yield the neighbor map which is an irrational rotation by around see Section 2 and Figure 3. Subpieces 15 and 22 yield the neighbor map which is an irrational rotation by around
We found edge neighbors in the second subdivision for which the neighbor map is an irrational rotation! This shows the noncrystallographic character of our fractal tile. This property implies that there is a continuum of different tilings and that for each tiling, the orientations of tiles, defined as angles, are dense in Their distribution within a large circle of radius around 0 converges to the uniform distribution on when runs to infinity. This is called ’statistical circular symmetry’ [13]. The Fourier spectrum, important from the physicists viewpoint, is also symmetric under rotations. This was shown in [25, 23, 13] which completes the proof of b).
To get the complete graph of edge neighbors, we still have to study the subpieces of neighbors in Figure 3. They all represent neighbor maps of the second and third row of Figure 3, providing arrows in leading to previous vertices. Instead of drawing this part of which is not planar, we list the arrows in Table 1. To each arrow in the table, except there is another arrow which is not listed for brevity. This proves a) when we neglect point neighbors. Assertion c) follows since the right part of Table 1 contains no arrows leading to rational rotations. The graph of edge neighbors, without root, splits into two components.
Can we really neglect point neighbors? Yes, we can. The proof of d) will be done only with the graph of rational rotations in Figure 4 which was derived by simple calculation. d) implies that the three sides of the triangle studied in Section 2 are really Jordan curves, and thus the angles are correctly defined. Together with the list of edge neighbors and the Jordan curve theorem, this implies that any nonedge neighbor can intersect only in one of the vertices or Moreover, since subtiles meet at such a point with their vertices, only finitely many angles are possible. This shows that beside edge neighbors, only finitely many point neighbors exist, and shows the finite type property of (If we are satisfied with the open set condition for instead of finite type, the finite number of angles will not be needed.)
Now let us prove d). Abstract methods as in [1, 7] are not needed since our case is rather simple. Consider with three edge neighbors defined by The subpiece 33 in the middle of Figure 2 has this structure. Let and denote the corresponding boundary sets of The reduced form of the neighbor graph for this configuration in Figure 4 yields the equation system
(6) 
The sets form a socalled graphdirected construction. The crucial point is that the convex hulls and provide the open set condition for this system. Similar to Proposition 1, these are the quadrilaterals with and with and and with and The open set condition says that the interiors of the quadrilaterals contain disjoint unions of their images defined in (6):
(7) 
This is verified by simply calculating images of vertices with (5). We get chains where each quadrilateral has one vertex in common with its predecessor and successor. Moreover, each of the points belongs to two components of the boundary (Details: addresses of points of a boundary set are given by the paths starting in the corresponding vertex of the graph in Figure 4. Paths with start in both and So the point with address fixed point of belongs to Similarly, fixed point of belongs to Since and paths labelled start in respectively, )
The intersection points of consecutive small quadrilaterals belong to since they are images of such intersection points on previous levels, for example Iterating the graphdirected construction on quadrilaterals we obtain longer chains of smaller quadrilaterals with vertices in This is a classical ’Koch curve’ construction. In the limit we have three Jordan arcs which form the closed Jordan curve By definition,
We show that has no points in the exterior region of The neighbors contain the closed Jordan arcs and which have similar neighborhoods of quadrilaterals as Comparing slopes of lines, we see that the convex hull of given as the outer boundary in Figure 4, is within the interior region of the union (for the neighbors, use inner sides of quadrilaterals as bound). Thus each point of exterior to must belong to one neighbor. So by definition it belongs to Thus such exterior points cannot exist.
We note that the fractal arc is invariant under the reflection at the line To see this, we check that for each point connecting two quadrilaterals in the approximating chain of on some level , the reflected point will also be on two quadrilaterals, at least on level For and this can be seen in Figure 4. By induction we prove that all vertices of the quadrilaterals within lie on They form a reflection symmetric set.
As a consequence, the neighbor map describes the same boundary set as On one hand implies that contains On the other hand, points in fulfil and do their images under so that cannot contain other points of
Now and all boundaries between the pieces form a network of Jordan curves which belong to because the edge neighbor maps within are and We can apply the to the union of all these Jordan curves and get a more dense network of Jordan curves bounding the second level pieces and forming a subset of The diameter of holes within this network is at most Iterating further, the diameter of holes tends to zero. Thus the closed set contains the whole interior region of
Once we know that is homeomorphic to a disk, we immediately have the open set condition and the tiling property. For topological reasons, disklike neighbors can only meet in a Jordan arc or in a single point. So all remaining neighbors of are point neighbors. As mentioned above, this implies the finite type property. Since we had a graphdirected system (6) with open set condition (7), the calculation of the dimension in Section 2 is justified. The computerfree proof of the theorem is finished.
Completion of proof of Proposition 2. We know that is homeomorphic to a disk, and the neighbors and intersect in the fractal arcs and respectively. This immediately implies that the long side is invariant under rotation and that is mapped by rotation onto Reflectioninvariance of was shown above. The congruence of with the fractal arc and of with is given by the neighbor maps and see Figure 3. This proves the corresponding statements a)–c) for The argument of d) was justified above, and e) is based on the neighbor map and the reflection invariance of Everything is proved.
4 The second inflation structure
The union of pieces 2 and 3 in Figure 1 has the symmetry group of a rectangle. It is mapped to itself by the rotation and also by reflection at the line This kind of symmetry is rare in fractal tiles. We apply the reflection to and obtaining new maps
(8) 
Since there is a new reptile with maps and As a set, this reptile coincides with but the subdivision is different, as shown in Figure 5. This leads to other tilings.
Reflection of a rectangle consisting of two pieces in a selfsimilar triangle was Radin’s trick to come from a crystallographic tile to the noncrystallographic pinwheel triangle. A similar trick was used by Conway and Radin [9] to obtain threedimensional quaquaversal tilings from crystallographic ones. In our case, however the reflection of pieces 2,3 leads from one noncrystallographic tile to another noncrystallographic tile.
The second subdivision of this fractal structure is shown in Figure 5. Four of the mappings are orientationreversing. All three vertices of the fractal triangle are fixed points of corresponding contraction maps, resulting in a smaller number of point neighbors. The graph of edge neighbors is planar, as shown in Figure 7. This second similarity structure has quite different matching rules than the first. The irrational rotations in the last row of Figure 3 do not appear as neighbors maps. Instead, we have the glide reflections shown in Figure 6. An irrational rotation occurs between point neighbors, as for the pinwheel triangle: has the form with in both structures. Part a) of the following statement is proved like Theorem 4. c) follows from the graph in Figure 7.
Proposition 4 (Neighbor graph of the second fractal pinwheel structure)
The second fractal pinwheel structure on has the following properties.

Point neighbor maps include irrational rotations. So there are no lattice tilings, and we have statistical circular symmetry.

Edges come with two types of maps: rational rotations on one hand, glide reflections on the other. All subedges of an edge of first type are again of first type. Edges of second type contain a dense set of subedges of first type.
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