Abstract
The fractal rep-tiles of the Euclidean plane considered in this article are examples of rep-tiles (tilings) with fractal boundaries. Several new examples of fractal rep-tiles are constructed using reflection transformations and integer matrices. A new class of foldable fractal rep-tiles based on general reflection mappings is introduced, and it is shown that these rep-tiles tile the plane using similitudes, including reflections, if the iterated function system (IFS) satisfies lattice tiling conditions. We prove the existence of foldable fractal 2-rep and 4-rep tiles that exhibit novel properties (chirality and aperiodicity) caused by reflection mappings. Fractal variations of foldable rep-tiles are also constructed. The fractal rep-tiles and the foldable rep-tiles presented here are in one-to-one correspondence with finite reflection groups, and this novel class of foldable rep-tiles can be lifted to construct new classes of fractal rep-tiles with roots in classical reflection groups. The images of rep-tiles are rendered using the random iteration algorithm, which is one of the popular iterative methods to generate self-similar fractals and tilings.
1 Introduction
A tiling is a covering of a given region using a given set of subsets (called “tiles”) without any overlap. Tilings can be seen in many fields, including arts, puzzles, textiles (e.g., on stitched, woven, or printed cloth), cultures and civilizations, and nature (e.g., a honeycomb). Tilings also have connections in pure mathematics in non-commutative geometry, dynamical systems, and operator -theory, and the mathematical theory of tilings has produced many interesting and challenging problems. We refer to the book by Grünbaum and Shephard [1], which is an excellent source of information on tilings and patterns. For example, a chessboard (Figure 1a) is an elementary self-similar tiling made up of congruent tiles (rep-tiles), each similar to the whole shape.
FIGURE 1
Fractal tilings of the plane refer to those having fractal boundaries. The Lévy dragon discovered by Paul Lévy (1938) is the classic example of a self-similar fractal tiling of the plane. Other examples include the twindragon and the Heighway dragon (Figure 1b). These tilings possess interesting properties such as symmetry and self-similarity. We refer to the articles [2–6], to the article by Husain et al. [7], and to the works by Bandt and Mekhontsev [8–10] for ongoing work and progress made to date. Many of these references have employed integer matrices extensively to generate fractal tilings because of their simple structure, integer determinants, algebraic eigenvalues, and so on.
In the Euclidean plane, topological and geometrical characteristics of fractal rep-tiles, including symmetry, interior points, boundaries, disk-likeness, connectedness, and self-similarity, have been studied by many researchers, and we refer to the articles [11–15] for details.
Fathauer [16] investigated fractal gaskets generated from self-replicating tiles whose outer and internal boundaries are themselves fractal. Hamiltonian cycles that traverse these gaskets and extend the constructions into three dimensions were studied by spatially separating successive generations and connecting them with polygonal faces. The resulting polyhedral forms highlight how planar fractal rep-tiles can be developed into architecturally evocative 3D structures.
Koss [17] presented visual and mathematical constructions of geometric series using fractal tilings. By iteratively scaling and arranging tiles, the article illustrates how geometric sums emerge naturally from self-similar patterns. The work emphasizes how fractal tilings provide intuitive, aesthetically compelling representations of convergence. For a summary of fractal rep-tiles with holes, we refer to [18], which presented families of planar fractal rep-tiles constructed through iterative methods using reflections and rotations.
Lai [19] proposed five geometric methods (including Escher-style rules and Conway’s criterion) to construct fractal tilings and rep-tiles. A concrete algorithm is presented based on expanding integer matrices and residue systems, showing how a self-replicating tile (an “ rep-tile”) can be generated via the affine iterated function system (IFS) using this framework.
In this article, reflection (mirror) rep-tiles of the Euclidean plane are presented using reflection isometries and integer matrices. Several examples of unfamiliar fractal rep-tiles and their fractal variations are constructed, which also include examples of previously known tilings and fractals. A novel class of fractal rep-tiles is introduced, namely, the “foldable rep-tiles” of the plane, which are essentially examples of finite reflection groups without fixed points (see Hoffman and Withers [27], Theorem 2.2). The foldable fractal rep-tiles are shown to tile the plane using similitudes containing at least one nontrivial reflection provided the IFS satisfies lattice tiling conditions. We prove new results on the existence of foldable fractal 2-rep and 4-rep tiles having novel properties, viz., chirality and aperiodicity coming from reflection mappings. These rep-tiles are topological disks, that is, orientable, closed, bounded, and connected sets. One may construct more examples of well-known fractals, new gaskets, and carpets by removing some of the functions in the IFS that have been used to create the rep-tiles.
While classical constructions of self-similar rep-tiles generated by expanding integer matrices (see, e.g., [4, 6, 7, 12, 13, 15]) make use only of affine maps of the formour work extends this framework by allowing affine mapswhere each belongs to the symmetry group of the expanding matrix . This simple but powerful modification leads to new families of fractal -rep tiles that do not appear in the existing literature. For matrices with nontrivial symmetry groups (including reflections and rotations), our method yields explicit new rep-tiles with a nonempty interior. Figures throughout Section 4 illustrate these new examples arising from previously unexploited combinations of digit sets and symmetry elements. To our knowledge, these fractal rep-tiles have not been documented before, and their construction constitutes the main novelty of the present article.
Fractal rep-tiles provide systematically constructed 2D domains with controllable self-similar boundaries. Such domains are natural candidates for the study of spectral properties of irregular geometries. The rep-tile structure provides exact self-similarity, providing analytical and numerical investigation of eigenfrequencies, wave localization, and resonant modes on fractal boundaries from a physical point of view. The foldable 2-rep and 4-rep tiles constructed here exhibit aperiodicity induced by reflection operations. This parallels the aperiodicity observed in Penrose tilings [28, 29] and quasicrystalline structures [30, 31] but with the additional feature of fractal boundaries, opening new test geometries for studying wave diffraction, photonic band gaps, and self-similar material properties. The use of reflections produces geometric behaviors absent in classical rep-tiles, including chirality, symmetry breaking, and deterministic aperiodicity. Many of the rep-tiles constructed here appear to be new and present intriguing topological and geometric properties. Moreover, the construction naturally places foldable rep-tiles with finite reflection groups and affine Weyl groups, providing a potential algebraic framework for classifying and analyzing these objects.
Many software and algorithms are available for generating fractal tilings, and we particularly refer to the IFS Construction Kit [20] developed by Larry Riddle. The software offers improved graphics, several options for visualizations of fractals and tilings with in-depth functionalities, and an easy-to-use GUI. Most of the tilings in this article are generated using the IFS Construction Kit by employing the random iteration algorithm (one of the popular iterative methods based on suitably chosen probabilities). We have also used the deterministic algorithm to construct some tilings. Both algorithms are well known for constructing self-similar fractals and tilings by producing attractors of a given IFS [21] and have been implemented in almost all fractal-generating software.
The structure of the article is as follows. Section 2 introduces the mathematical framework for fractal rep-tiles. Section 3 constructs many examples of classical and new fractal rep-tiles using reflection mappings and integer matrices. Section 4 introduces the new class of foldable fractal rep-tiles, including examples of foldable rep-tiles generated by general reflections, and analyzes chirality, aperiodicity, and geometric consequences of reflections. Section 5 presents fractal variations and visual examples generated by the random iteration algorithm and outlines connections with finite reflection groups, including interpretations. Conclusions are provided in Section 6 along with potential applications in physical systems.
2 Some preliminaries
We now introduce some definitions and results that are needed in the subsequent sections. Let be a complete metric space. Let be the set of nonempty compact subsets of . As in [21], will be called the space of fractals. The space is a metric space with respect to the Hausdorff metric defined bywhere and [21].
Definition 2.1Let be a metric space. A function is called a contraction if there exists a constant such thatFor example, the map is a contraction on with the usual metric having the contraction constant .
Definition 2.2Let be a complete metric space. An iterated function system (IFS) on is a finite set of contraction mappings , having contraction constants , for . The number is called a contraction constant of the IFS.
[22] Let be a complete metric space, and let be an IFS with contractivity factor . Then the transformation (called the Hutchinson operator) defined byis a contraction on with contraction constant . Moreover, its unique fixed point (called the attractor), , satisfies
The attractor described in Theorem 2.1 is obtained by taking the limit of the sequence of forward iterates of ; that is, for any , where is the fold forward iterate of . Thus, the final image of the attractor is independent of the choice of initial set.
Definition 2.3Let be a subset of . A finite family of open sets is called a cover of ifIn this article, we are interested in fractal rep-tiles of the plane. Therefore, we shall assume unless otherwise stated.
Definition 2.4A countable collection of sets that covers the plane is called a tiling of the plane ifHere, denotes the interior of . Thus, in any tiling of the plane, the interior of a set does not intersect the interior of any other set; that is, the tiling has no overlaps.
Definition 2.5A map is called a similitude (or similarity transformation) if there exists a constant such thatHere, denotes the standard Euclidean norm. Every similitude can be written in the formwhere is an orthogonal transformation (rotation or reflection) and is a translation.For example, is a similitude in with scale factor 3 and reflection.
Definition 2.6A set is called a fractal rep-tile if there are sets congruent to , such that for andwhere is a similitude and boundary of ; that is, is a fractal (i.e., has a non-integer Hausdorff dimension).Thus, fractal rep-tiles are tiles with fractal boundaries. The sets are called fractiles. A very useful property of an IFS is a natural separation condition, called the open set condition (OSC).
Definition 2.7Let be an IFS. We say that the IFS fulfills the open set condition if there is an open set satisfyingThus, the OSC guarantees that the sets do not overlap. If OSC holds and , then is a tile. The mirror rep-tiles considered in this article will have some overlaps. However, using Theorem 2 from Falconer [23], it follows that the Hausdorff dimension of these rep-tiles with overlap (provided the fractiles do not overlap “too much”) is almost equal to the similarity dimension without overlap. The set of points for which this does not hold is essentially a set of Lebesgue measure zero. Hence, all rep-tiles presented in this article are examples of plane tilings with similarity dimension 2.
3 Reflection (mirror) rep-tiles of the plane
Recall that an IFS denotes an iterated function system (i.e., a finite family of contraction maps). We are going to construct fractal rep-tiles using integer matrices and reflection isometries with the help of Theorem 2 of Bandt [4].
Definition 3.1Let be a group, and let be a subgroup. For , the left coset of with representative isThe right coset is .Cosets partition the group into equal-sized disjoint subsets. For example, in and ,and the cosets , , and partition modulo 3.Suppose with (i.e., is an expanding matrix), and let be the group of lattice points in the plane. Then , and it is a subgroup of . We level any set of lattice points as a digit set. A digit set is called a residue system for if and there is exactly one point in from each coset of . In this case, the IFS defined bywith as translation vectors produces a fractal rep-tile [4]. DefineThen,where the sets in union on the right side are pairwise disjoint and, in this case, the vectors are said to constitute a complete residue system. The position of digits in the residue system dictates the location of the fractiles, and the fractiles may substantially change their shape by changing the residue system.
Definition 3.2Let be an IFS in , where each , each is either the identity or a reflection, and . The IFS satisfies the lattice tiling condition if the translation vectors generate a full-rank lattice ; that is,
Definition 3.3For an integer expanding matrix , the symmetry group of is the subgroup of the Euclidean isometry group consisting of all orthogonal matrices satisfying . Thus,that is, all orthogonal transformations (reflections or rotations) that commute with .An important result from [4] provides a general method to construct self-similar fractal rep-tiles using reflections and rotations involving integer matrices.
(Theorem 2, [4]). Let be an expanding matrix, and be a residue system for . Consider the IFS , where , belong to the symmetry group of the linear mapping . Then, the attractor of the IFS is self-similar with respect to and has a nonempty interior.
By choosing , where denotes the identity mapping, Theorem 3.1 reduces to Theorem 1 of [4], which is a particular case of constructing fractal rep-tiles using integer matrices without rotations and reflections and has been exploited by many authors [6, 7, 12, 13, 15].
Definition 3.4A reflection in a line is the transformation defined byIf the line is given by , and the point has image , then we have:
The reflection is given by
Proof. See [24] for a detailed proof.
Finally, we recall (for later use) that for a nonempty compact set , its Hausdorff dimension denoted by is defined via the standard Hausdorff measure (see [23] for more details on the Hausdorff measure). Throughout the remainder of the article, the self-similar geometry of the rep-tiles is produced by iterating contraction maps as defined in Definition 2.1, which provides a rigorous mathematical basis for the resulting fractal domains.
For simplicity, we shall construct reflection rep-tiles using the elementary reflections given in Table 1. However, the construction remains valid for more general reflections, and one can use Equation 3.1 to compute the corresponding matrix for any reflection mapping.
Notice that in Table 1, is the rotation about the origin of . However, we consider as a reflection because it can be written as a product (composition) of two reflections, and any product of reflections is an isometry (see Chapter five of [24]). Therefore, belong to the symmetry group and, hence, Theorem 3.1 is applicable for also.
To construct examples of reflection rep-tiles, we first consider reflections in lines through the origin. Later, we will allow for reflections along arbitrary lines, giving birth to foldable rep-tiles. In , we can always choose a basis such that the mappings in the IFS are similitudes if and only if the eigenvalues of are either complex or real but with equal modulus and independent eigenvectors [25]. This guarantees that the rep-tiles we generate are self-similar. The figures shown in this article are representative examples of the constructions rather than an exhaustive classification.
TABLE 1
| S. no. | Reflection | Description |
|---|---|---|
| 1 | Identity mapping (self-reflection) | |
| 2 | Reflection along the axis | |
| 3 | Reflection along the axis | |
| 4 | Reflection in the line | |
| 5 | Reflection in the line | |
| 6 | Rotation about the origin of |
Notations for reflections.
Example 3.1Let . The lattice generated by the columns of is shown in Figure 2A. We determine the digit set(’s are indicated by red bullets in Figure 2A) and the IFS mappingsHere, are reflections in the group of symmetries generated by . Different choices of give rise to the rep tiles shown in Figures 2b–f, which have been produced with the IFS Construction Kit [20] using the random iteration algorithm. The quality of images may be improved further by repeating the iterative method.A list of reflection isometries used in Figure 2 is given in Table 2. The rep-tiles shown in Figures 2b, d, e were also obtained by Bandt in [4, 26], respectively.Changing the digit set to (’s are shown by red bullets in Figure 3a). Using the reflection isometries as and , we obtain the rep-tiles shown in Figures 3b,c respectively, which may be considered as relatives of the twin/Lévy dragon family belonging to the group of symmetries of the same matrix .
FIGURE 2
TABLE 2
| S. no. | Reflections used | Reference figure |
|---|---|---|
| 1 | Figure 2b (twindragon) | |
| 2 | Figure 2c (Lévy dragon) | |
| 3 | Figure 2d (leaf rep-tile) | |
| 4 | Figure 2f | |
| 5 | Figure 2f |
Reflections used in the -rep tiles of Figure 2.
FIGURE 3
Example 3.2Consider the matrix . The lattice spanned by the columns of is shown in Figure 4a. From this, we choose translation vectors , , and . Therefore, the digit set is .Figures 4b, e show 3-rep tiles for various reflections and the digit set . Reflections used in these rep-tiles are given in Table 3.Setting the digit set asand choosing reflections and , we construct the 3-rep tiles shown in Figures 4g,h, respectively. Notice that the shape of the rep-tile changes drastically with the change in isometries, and most of these fractal rep-tiles are not seen or have very limited appearances in the literature to date.
FIGURE 4
TABLE 3
Reflections used in the 3-rep tiles of Figure 4.
Example 3.3Let . Then , and the set of lattice points by taking the integer span of columns of is shown in Figure 5a. From this, we set the digit set , where , , , and .Corresponding rep tiles are shown in Figures 5b–e. Identifying the set of lattice points as and iterating using IFS, we obtain the rep-tile of Figure 5f. The reflections used in Figure 5 are given in Table 4. We can construct more examples by changing the reflections and the vectors .
FIGURE 5
TABLE 4
Reflections used in the 4-rep tiles of Figure 5.
Example 3.4In this example, we construct rep-tiles for for the integer matrix and its transpose. We identify two digit sets and .Applying the IFS mappings, we acquire the rep-tiles shown in Figures 6a,b. Next, we take the matrix and identify two sets of lattice points as , and . The resulting 5-rep tiles are shown in Figures 6c,d, respectively. The reflections used in the construction of each rep-tile of Figure 6 are given in Table 5. Notice the artistic geometry of reflections exhibited by these examples.
FIGURE 6
4 Foldable rep-tiles
A geometric figure in is said to be foldable if it can be mapped into smaller copies of itself by reflections in hyperplanes. More precisely.
Definition 4.1Let be a compact connected subset of such that ; that is, is the closure of its interior. The set is called a foldable figure if there exists a set of hyperplanes that divide into finitely many congruent subfigures , each similar to , such that , for .We cite a result from [27], which states that foldable figures are convex polytopes, and they tile the Euclidean plane.
A foldable figure in is a convex polytope that tessellates by reflections in hyperplanes.
We state and prove the corresponding result for fractal rep-tiles in . For this, we define a foldable fractal rep-tile.
Definition 4.2
(Foldable fractal rep-tile). Let
be a compact, connected set such that
. We say
is a foldable fractal rep-tile of order
if there exists an iterated function system (IFS)
such that
Each is a similitude; that is, , where is the scaling factor, is either the identity or a reflection across a line in , and is a translation. At least one is a nontrivial reflection.
, with pairwise non-overlapping except possibly on their boundaries (satisfying the open set condition).
The boundary has a Hausdorff dimension .
Let be a foldable fractal rep-tile of order generated by the IFSsatisfying the open set condition, where at least one is a nontrivial reflection and where the boundary of satisfies . Then is not convex, and if the IFS satisfies the lattice tiling condition (Definition 3.2), then tiles by similitudes in the group generated by together with translations by the lattice.
Proof. Because is compact, connected, and , it has a nonempty interior. If were convex, its boundary would be a rectifiable Jordan curve, and henceHowever, by definition of a foldable fractal rep-tile, the boundary satisfies . This is impossible for a convex set and, therefore, cannot be convex.
The set is the unique attractor of the IFS and satisfies the self-similarity relationshipwith overlaps only on boundaries, due to the open set condition.
Let be the semigroup generated by the similitudes . For any word of length , the image is a contracted and possibly reflected copy of . The collectionforms a disjoint (up to boundaries) tiling of a supertile, which is a scaled copy of .
Because the IFS satisfies the lattice tiling condition, the translation vectors generate a full-rank lattice . Then the translated copiestile the entire plane without interior overlaps. Moreover, each supertile is itself tiled by the sets in . Combining these two tilings, we obtain a tiling of the plane by the family
The presence of a nontrivial reflection among the maps does not alter the lattice , which is generated purely by translation vectors. Instead, it enlarges the symmetry group of the tiling by introducing reflected orientations of the tile. Consequently, the tiling consists of both direct and reflected copies of , all positioned by translations in . Hence, under the stated lattice condition, tiles by similitudes in the group generated by the IFS together with the lattice of translations.
Example 4.1(Foldable fractal rep-tiles).We now construct some examples of foldable fractal rep-tiles for the simple mapping . The resulting 4-rep tiles are shown in Figures 7b–e.Figure 7a is obtained by taking , reflection in the line , reflection in the line , and reflection in the line . In Figure 7b, , reflection in the line , reflection in the line , and reflection in the line . Figure 7c is obtained with , reflection in the line , reflection in the line , and reflection in the line . Finally, in Figure 7d, we have taken , reflection in the line , reflection in the line , and reflection in the line . All these rep-tiles are examples of generalized foldable figures studied by Hoffman and Whithers [27], and the rep-tile in Figure 7a was also studied by Bandt in [4].We now state and prove two novel results about the existence of foldable rep-tiles having properties that have not often been mentioned in the literature to date. In what follows, the numerical boundary dimension values are conjectural estimates rather than rigorously derived values.
FIGURE 7
Definition 4.3A set is chiral if for every orientation-reversing isometry of , . Equivalently, is chiral if it is not congruent to any of its reflections.
(Chiral foldable fractal 2-rep tile). There exists a foldable fractal 2-rep tile defined by an IFS with two mappings, one including a reflection, such that is chiral (not congruent to its mirror image), and its boundary is a fractal having a Hausdorff dimension .
Proof. Define an IFS in the complex plane
by the set of mappings
where
, and
is the complex conjugate (reflection across the real axis) of
. Let
be the attractor of this IFS. We show that
is a chiral foldable fractal 2-rep tile. This will be done in four steps: (1) existence of the attractor, (2) 2-rep tile property, (3) boundary dimension, and (4) chirality.
- 1.
Step 1 (existence): Note that the mappings and are contractive similitudes, each having the contraction ratio . For , is a reflection (isometry), and the scaling by ensures contractivity. By Hutchinson’s theorem, the IFS has a unique compact attractor such that
Moreover, the open set condition (OSC) holds. To see this, consider an open set
containing
. Because
scales by
and
reflects, scales, and translates by 1, the sets
and
are disjoint except possibly on their boundaries. Numerical iterations suggest that
has measure zero (points of overlap are sparse), satisfying OSC. Thus,
is a well-defined fractal attractor.
- 2.
Step 2 (2-rep tile property): Because , therefore is tiled by two copies, where is a scaled copy of (by factor ), and is a reflected, scaled, and translated copy of . Because and are similitudes, therefore and are similar to . The OSC ensures minimal overlap (of measure zero). Hence, is tiled by two similar copies, confirming the 2-rep property.
- 3.
Step 3 (boundary structure and dimension). The similarity dimension of the attractor is determined by the equation
which for
yields
. Because the open set condition holds, it follows that
. In particular,
has a positive Lebesgue measure and a nonempty interior.
We now consider the boundary . Becauseand the interiors of and are disjoint, the boundary satisfiesThus, the boundary is contained in the forward invariant set generated by the overlap set and its images under the IFS.
The set has an empty interior and a Lebesgue measure of zero, and its iterates under the maps and form a compact, self-similar subset of with a strictly smaller Hausdorff dimension than . However, due to the presence of a reflection in , this boundary-generating dynamic is not purely self-similar but involves orientation reversal, which prevents a direct application of standard dimension formulas.
We do not provide a rigorous formula for
, but numerical iterations indicate that its boundary exhibits a self-similar branching structure similar to the classic Lévy dragon with the same contraction ratio
. Motivated by this similarity, we conjecture that
understanding this as an approximate, non-rigorous estimate.
- 4.
Step 4 (chirality): Finally, we investigate whether is chiral by examining its structure and the IFS defining its mirror image. Suppose it is possible that is achiral. Then, there exists an isometry mapping from to . The fixed point of is . For , the fixed point solves , yielding , which lies on the real axis. The fixed points and are not symmetric across the real axis, which implies asymmetry in the attractor. Now, the mirror image is the set of complex conjugates of points in , and the IFS for is
In the mirror IFS,
scales and reflects, and
scales and translates without reflection. This confirms that
and
are not isometric. Indeed, it is easy to check that the translation
breaks symmetry across the real axis because the points like
map to
. Iterations show asymmetric clustering, so
. No isometry can map
to
, confirming that
is chiral.
Remark 4.1A rigorous determination of the Hausdorff dimension of would require a detailed analysis of the boundary overlap structure, potentially via a graph-directed iterated function system or the study of neighbor maps associated with the tiling. Such an analysis lies beyond the scope of the present work and is left for future investigation.The term “foldable” in the proof of Theorem 4.3 refers to the reflection in , allowing to be constructed via a reflection-based IFS, akin to folding across the real axis. Unlike the achiral Lévy dragon (defined by rotations and translations), the reflection and asymmetric translation in induce chirality. This is a novel property of foldable fractal 2-rep tiles, as most known examples (e.g., the Lévy dragon) are achiral due to symmetric mappings.
Example 4.2(Chiral 2-rep fractal tile).
Consider the IFS in the complex plane
consisting of the mappings.
, where .
, where is the complex conjugate (reflection across the real axis).
Definition 4.4A tiling of the plane is periodic if there exist two linearly independent translation vectors such that translating the entire tiling by or leaves unchanged. A tiling that is not periodic is non-periodic. A tile (or rep-tile) is called “aperiodic” if it admits at least one tiling of the plane, but every tiling it admits is non-periodic.The following theorem guarantees the existence of an aperiodic, foldable 4-rep tile. Strong aperiodicity in the sense of Definition 4.4 is not established in this article, and Theorem 4.4 establishes only the existence of at least one non-periodic tiling arising from the hierarchical inflation construction.
There exists an iterated function system on consisting of similitudeswhere at least one is a reflection composed with a rotation, such that the attractor is a foldable fractal 4-rep tile and admits a non-periodic tiling of .
Proof. We give a constructive proof in several steps and provide rigorous arguments for each claim.
LetFix with for sufficiently small and assume . Definewhere denotes rotation by angle , and is reflection across the -axis. SetEach is a similitude with contraction ratio .
By Hutchinson’s theorem, there exists a unique nonempty compact set such that
For , the maps reduce to the standard quarter-square subdivision of the unit square. Let . Then and for . Because the maps depend continuously on , there exists such that for all , the images remain pairwise disjoint subsets of . Hence, the open set condition holds.
Each map scales area by a factor of and, therefore,Under the open set condition and this equality, standard results in self-similar measure theory imply that the self-similar (Hutchinson) measure coincides with Lebesgue measure restricted to . In particular, , so . Thus is a fractal 4-rep tile.
Define the inflation map . From the self-similarity of ,Hence, decomposes into four copies of , possibly rotated or reflected. Iterating this subdivision yields a nested hierarchy of supertiles whose union produces a tiling of . The open set condition guarantees that interiors do not overlap, so is well defined.
Consider the -fold composition of . Its linear part is . After inflating by the factor , the corresponding tile appears as a copy of rotated by angle . Because is irrational, the setis infinite. Thus, the tiling contains tiles in infinitely many distinct orientations.
Assume for contradiction that is periodic. Then there exists a nonzero translation lattice such that for all . Let be a fundamental domain of . Because has finite area and has positive area, only finitely many tiles intersect , so only finitely many tile orientations occur in and hence in . This contradicts the existence of infinitely many distinct orientations. Therefore, is non-periodic.
The map is orientation-reversing. Hence, one subtile of is obtained from by an orientation-reversing isometry and, therefore, is foldable. This completes the proof.
Remark 4.2In contrast to classic periodic 4-rep tiles (such as trapezoids), the present construction shows that the inclusion of orientation-reversing similitudes is compatible with the existence of non-periodic tilings. Thus, foldability does not preclude non-periodic behavior within the class of 4-rep tiles.
Remark 4.3The rep-tile constructed in Theorem 4.4 is intrinsically chiral due to the presence of reflections composed with rotations in its defining iterated function system, resulting in a global lack of mirror symmetry of the fractal domain. Such chiral fractal “drums” are of interest in mathematical physics, as the absence of reflection symmetry may lead to more spectral asymmetries, modified eigenvalue degeneracies, and altered nodal structures compared with non-chiral fractal domains. When combined with hierarchical self-similarity and non-periodicity, this geometric chirality can also influence wave propagation, scattering, and mode localization in acoustic or electromagnetic settings, making these rep-tile domains natural model geometries for studying wave phenomena in aperiodic and self-similar media.
5 Variations
Because our IFS satisfies OSC with , therefore the OSC holds for any subset of the IFS as well. Hence, all the rep-tiles presented in Sections 3, 4 can be reduced into fractal carpets and gaskets by removing some of the mappings from the IFS. Figures 8a–d display some self-similar fractals variations of the rep-tiles of Figure 5 obtained on removing , , , and from the corresponding IFSs of Figures 5b–e respectively. Notice that these variations are relatives of the Sierpinski gasket.
FIGURE 8
The variation in Figure 8e is obtained by removing the mapping from the IFS that corresponds to Figure 6c, while the fractal in Figure 8f is the result of deleting the mapping from the foldable rep-tile in Figure 7a. The fractal in Figure 8f is not seen anywhere, and by deleting mappings from all other examples presented in the article, many appealing and new fractals can be constructed.
A brief note about the geometry and abstraction of foldable rep-tiles follows. The foldable rep-tiles presented here can be regarded as elements of discrete reflection groups closely related to Coxeter groups and, in particular cases, to affine Weyl groups. This connection arises because foldable rep-tiles are constructed via reflections along hyperplanes, analogous to how affine Weyl groups are generated by reflections through the hyperplanes orthogonal to the simple roots of a root system.
For example, the 2-rep tile generated by reflections along two orthogonal lines in corresponds naturally to the simple roots of the root system, where denotes the rank-1 root system consisting of two points on the real line. The associated reflections generate the finite Weyl group of . While a rigorous one-to-one correspondence between all foldable rep-tiles and affine Weyl groups has not been established, this example illustrates a potential connection and opens a new avenue for further investigation. In this sense, foldable rep-tiles can be viewed as fractal elements of discrete reflection groups, linking classical geometric constructions with the abstract algebraic framework of Coxeter and Weyl groups.
6 Conclusion
Fractals and self-similar structures play an essential role in both mathematics and physics. In this article, we investigated new families of planar fractal rep-tiles generated by integer expansion matrices and reflection mappings, including a novel class of foldable fractal rep-tiles. We proved the existence of foldable fractal 2-rep and 4-rep tiles with fractal boundaries that tile the plane by similitudes involving nontrivial reflections and exhibit reflection-induced chirality and aperiodicity. These constructions establish new connections between fractal tiling geometry and finite reflection groups and are relevant to physical models involving self-similar and aperiodic domains.
The foldable rep-tiles are shown to tile the Euclidean plane by taking reflections (nontrivial) along hyperplanes provided the IFS satisfies lattice tiling conditions. We proved the existence and rigorously characterized foldable fractal 2-rep and 4-rep tiles, which exhibit novel properties (chirality and aperiodicity) driven by reflection mappings, fractal boundaries, and plane-tiling capability. Many reflection rep-tiles and foldable rep-tiles discovered in the article are new and seem to have interesting topological (and possibly algebraic) properties with roots in finite reflection groups and affine Weyl groups. All images of rep-tiles and fractals are obtained using the IFS Construction Kit [20]. More intriguing rep-tiles can be constructed by changing integer matrices, digit sets, and reflection mappings.
Future work will focus on developing the algebraic structures underlying foldable rep-tiles, particularly their relationship to reflection groups. We also plan to study the spectral, dynamical, and topological properties of these tiles, such as orientation, boundary behavior, connectedness, and the eigenvalue spectrum of the Laplacian on foldable fractal domains.
Beyond the mathematical interest, the rep-tiles introduced here connect to several areas of physics. Fractal boundaries and self-similar domains are relevant to spectral geometry, where questions related to vibrating fractal drums [32–34] or the spectral sensitivity to boundary irregularity can be studied using systematically constructed rep-tile domains. The aperiodicity generated by reflections parallels quasicrystalline structures and may find applications in modeling wave propagation, photonic band structures, and acoustic resonances in hierarchical media. Because rep-tiles can produce reproducible fractal interfaces, they also offer natural platforms for investigating transport, scattering, and diffusion in self-similar environments. These investigations may yield insights of interest both to mathematics and to physics, especially in fields involving self-similar or aperiodic geometries.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
MS: Writing – original draft, Investigation, Validation, Supervision, Writing – review and editing. AH: Visualization, Writing – review and editing, Software, Conceptualization, Writing – original draft, Methodology, Investigation. PK: Methodology, Writing – review and editing, Software, Conceptualization.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
GrünbaumBShephardGC. Tilings and patterns. New York, NY: Freeman (1987).
2.
VinceA. Replicating tessellations, SIAM J. Discret. Math (1993) 6:501–21. 10.1137/0406040
3.
AkiyamaS. Symbolic dynamical system and number theoretical tilings: selected papers on analysis and related topics. Amer Math Soc Transl Ser (2008) 223:97–113.
4.
BandtC. Self-similar sets 5. Integer matrices and fractal tilings. Proc Amer Math Soc (1991) 112(2):549–62. 10.1090/s0002-9939-1991-1036982-1
5.
DengQRLauK. Connectedness of a class of planar self-affine tiles. J Math Anal Appl (2011) 380(2):493–500. 10.1016/j.jmaa.2011.03.043
6.
GröchenigKHaasA. Self-similar lattice tilings. J Fourier Anal Appl (1994) 1:131–70. 10.1007/s00041-001-4007-6
7.
HusainAKarthikGMeghamMAshishS. Fractal rep-tiles of and using integer matrices. Fractals (2021) 29:2150027. 10.1142/S0218348X21500274
8.
BandtCMekhontsevD. Elementary fractal geometry. New relatives of the sierpinski gasket. Chaos: Interdiscip J Nonlinear Sci (2018) 28(6):63–104. 10.1063/1.5023890
9.
BandtCMekhontsevD. Elementary fractal geometry. In: Networks and carpets involving irrational rotations (2020).
10.
BandtCMekhontsevDTetenovA. A single fractal pinwheel tile. Proc Amer Math Soc (2018) 146:1271–85. 10.1090/proc/13774
11.
LuoJRaoHTanB. Topological structure of self-similar sets. Fractals (2002) 10:223–7. 10.1142/s0218348x0200104x
12.
KiratILauKS. On the connectedness of self-affine tiles. J Lond Math Soc (2000) 62:291–304. 10.1112/s002461070000106x
13.
LeungKSLuoJJWangL. Connectedness of a class of self-affine carpets. Fractals (2020) 28(4):2050065. 10.1142/s0218348x20500656
14.
AkiyamaSThuswaldnerJ. On the topological structure of fractal tilings generated by quadratic number systems. Comput Math Appl (2005) 49:1439–85. 10.1016/j.camwa.2004.09.008
15.
BandtCWangY. Disk-like self-affine tiles. Discrete Computat Geom (2001) 26:591–601. 10.1007/s00454-001-0034-y
16.
FathauerRW. Fractal gaskets: rep-tiles, hamiltonian cycles, and spatial development. In: Bridges Finland conference proceedings (2016). p. 217–24.
17.
KossL. Fractal tiling illustrations of geometric series. In: Proceedings of bridges 2015: mathematics, music, art, architecture, culture (2015). p. 423–6.
18.
HusainA. Fractal rep-tiles of the plane with holes using reflections and rotations. In: Computing and simulation for engineers. Boca Raton: Taylor & Francis (2022).
19.
LaiP-J. How to make fractal tiling and fractal rep-tiles. Fractals (2009) 17/04:493–504. 10.1142/S0218348X09004533
20.
RiddleL, IFS construction kit, Available online at: https://larryriddle.agnesscott.org/ifskit/index.htm (Accessed February 21, 2026), (2019).
21.
BarnsleyM. Fractals everywhere. 2nd ed.Academic Press (1993).
22.
HutchinsonJE. Fractals and self-similarity. Indiana Univ Mathematics J (1981) 30(5):713–47. 10.1512/iumj.1981.30.30055
23.
FalconerKJ. The hausdorff dimension of some fractals and attractors of overlapping construction. J Statist Phys (1987) 47:123–32. 10.1007/bf01009037
24.
MartinGE. Transformation geometry: an introduction to symmetry. New York, NY, USA: Springer (1982).
25.
KenyonRSolomyakB. On the characterization of expansion maps for self-affine tilings. Discrete and Comput Geometry (2010) 43(3):577–93. 10.1007/s00454-009-9199-6
26.
BandtC. Self-similar sets 1. Topological markov chains and mixed self-similar sets. Math Nachr (1989) 142:107–23. 10.1002/mana.19891420107
27.
HoffmanMEWithersWD. Generalized chebyshev polynomials associated with affine weyl groups. Trans Amer Math Soc (1988) 308:91–104. 10.1090/s0002-9947-1988-0946432-3
28.
PenroseR. The role of aesthetics in pure and applied mathematical research. Bull Inst Mathematics Its Appl (1974) 10:266–71.
29.
PenroseR. Pentaplexity: a class of non-periodic tilings of the plane. The Math Intelligencer (1979) 2:32–7. 10.1007/bf03024384
30.
ShechtmanDBlechIGratiasDCahnJW. Metallic phase with long-range orientational order and no translational symmetry. Phys Rev Lett (1984) 53:1951–3. 10.1103/physrevlett.53.1951
31.
LevineDSteinhardtPJ. Quasicrystals: a new class of ordered structures. Phys Rev Lett (1984) 53:2477–80. 10.1103/physrevlett.53.2477
32.
KacM. Can one hear the shape of a drum?The Am Math Monthly (1966) 1–23. 10.1080/00029890.1966.11970915
33.
SapovalBFilocheMKaramanosKBrizziR. Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer. Eur Phys J B (1999) 9:739–53. 10.1007/s100510050819
34.
FilocheMSapovalB. Can one hear the shape of an electrode? II. Theoretical study of the Laplacian transfer. Eur Phys J B (1999) 9:755–63. 10.1007/s100510050820
Summary
Keywords
algorithm, foldable rep-tiles, fractal rep-tiles, integer matrices, iterated function system, iterative methods, reflections
Citation
Sajid M, Husain A and Kumar P (2026) Fractal rep-tiles of the plane via reflections and integer matrices. Front. Phys. 14:1699796. doi: 10.3389/fphy.2026.1699796
Received
05 September 2025
Revised
16 January 2026
Accepted
20 January 2026
Published
12 March 2026
Volume
14 - 2026
Edited by
Zbigniew R. Struzik, The University of Tokyo, Japan
Reviewed by
Alex Hansen, NTNU, Norway
Iden Rainal Ihsan, Universitas Samudra, Indonesia
Updates
Copyright
© 2026 Sajid, Husain and Kumar.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Akhlaq Husain, ahusain10@jmi.ac.in
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.