Hyperbolic Isogonal Tilings from Uniform Edge Colorings

Year: 2024 Authors: Agatha Kristel M. Abila; Ma. Louise Antonette N. De las Peñas; Mark D. Tomenes

Core claim

Uniform edge colorings of hyperbolic uniform tilings can systematically generate isogonal tilings whose edges may be replaced by symmetric motifs for artistic rendering.

Topics

hyperbolic tilings, isogonal tilings, uniform edge coloring, symmetry groups, mathematical art

Domains

tiling theory, hyperbolic geometry, group theory, symmetry, mathematical art, pattern design, textile motifs, indigenous design

Methods

uniform edge-coloring construction, subgroup orbit analysis, motif substitution, symmetry classification

Media

hyperbolic plane, colored edges, flowers and butterflies, Yakan textile motifs

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2024 Conference Proceedings

Hyperbolic Isogonal Tilings from Uniform Edge Colorings

Agatha Kristel M. Abila ${}^1$ , Ma. Louise Antonette N. De las Peñas ${}^2$ , Mark D. Tomenes ${}^3$

${}^1$ Ateneo de Manila University, Quezon City and Southern Luzon State University, Lucban, Quezon, Philippines; agatha.abila@student.ateneo.edu ${}^2$ Ateneo de Manila University, Quezon City, Philippines; mdelaspenas@ateneo.edu ${}^3$ Ateneo de Manila University, Quezon City, Philippines; mtomenes@ateneo.edu

Abstract

This paper presents the construction of an isogonal tiling from an edge coloring of a uniform tiling of the hyperbolic plane. In addition, it also discusses how the isogonal tiling can be transformed into a hyperbolic artwork by using motifs with appropriate symmetry properties as the edges of the tiling.

Introduction

In tiling theory, the study of tilings with vertex transitivity properties has always been of interest (see [3][5][8] and references herein). These tilings have been used for their applications in science, such as to model various chemical structures. For example, a patch of a two-coloring of the hexagonal tiling in Figure 1(a) can be used to model benzene [3]. In art, these tilings may also be used as a basis for constructing aesthetically pleasing colored patterns. The regular $(5^5)$ tiling of the hyperbolic plane $({\mathbb{H}}^2)$ with symmetry group *552 for instance, was used to render a hyperbolic pattern presented in Figure 1(b) using as motifs the textile designs from the Yakans, one of the indigenous communities in the Philippines [6].

(a)

(b) Figure 1: (a) A patch of a two-coloring of a hexagonal tiling used to model benzene; and (b) a hyperbolic pattern with motifs inspired by the indigenous designs of the Yakan tribe from Mindanao, Philippines.

In this paper, we discuss the construction of isogonal tilings of ${\mathbb{H}}^2$ . Isogonal tilings are tilings with vertices forming one orbit under the action of their respective symmetry groups [9]. This means that in an isogonal tiling, there is always a symmetry of the tiling that will send one vertex to another. In [8], a list of Euclidean isogonal tilings was derived using the concept of edge adjacency symbols. There is existing literature on hyperbolic uniform tilings [5][7] which are isogonal tilings consisting solely of regular polygons, but not much has been said on a systematic construction of isogonal tilings of ${\mathbb{H}}^2$ in general. In this work, an approach to construct an isogonal tiling of ${\mathbb{H}}^2$ from a uniform tiling will be presented, by considering its uniform edge coloring. It can be recalled that an edge coloring of a uniform tiling is uniform if for any two vertices of the tiling, there is a symmetry of the tiling that sends one vertex to the other and preserves the colors of the edges [9].

Abila, De Las Peñas, and Tomenes

Constructing Isogonal Tilings from Uniform Edge Colorings

We begin by first considering a uniform edge coloring of a uniform tiling. Let $\mathcal{T}$ be an uncolored uniform tiling with symmetry group $G$. Choose a subgroup $H$ of $G$ such that $H$ forms one orbit of vertices of $\mathcal{T}$. Suppose $H$ forms $n$ orbits of edges in $\mathcal{T}$, that is, if $E$ is the set consisting of edges of $\mathcal{T}$, $E$ can be partitioned as $E=H{e}_1\cup H{e}_2\cup \cdots \cup H{e}_n$ where we pick $e_1,e_2,\dots ,e_n$ as representatives from each orbit of edges. Assigning distinct colors to each $H{e}_j$, $j=1,\dots ,n$, we obtain an edge-$n$-coloring of $\mathcal{T}$. Since $H$ preserves the colors of the edges, the edge-$n$-coloring is uniform.

Now, to construct an isogonal tiling ${\mathcal{T}}^{\ast }$ from the uniform edge coloring of $\mathcal{T}$, a new set of edges $e_1^{\ast },e_2^{\ast },\dots ,e_n^{\ast }$ is introduced to replace $e_1,e_2,\dots ,e_n$ respectively. There are four possibilities for $e_j^{\ast }$ ($j=1,\dots ,n$) depending on its finite group $F$ of symmetries in the group $G$. The group $F$ is either of type $C_n$ (cyclic group of order $n$) or $D_n$ (dihedral group of order $2n$). Figure 2(a)-2(d) shows the four possibilities for $e_j^{\ast }$ with various symmetry types. Figure 2(a) type $C_1$: $F$ is generated by the identity isometry, Figure 2(b) type $C_2$: $F$ is generated by a $180^{\circ }$ rotation with center $C$, Figure 2(c) type $D_1$: generated by a reflection with axis $l$, and Figure 2(d) type $D_2$: generated by two reflections with axes perpendicular to each other, one of which passes through the edge. It is important to note that we can replace the edge $e_j$ with one of the four possibilities of $e_j^{\ast }$, provided the symmetries of $e_j$ in $H$ is contained in $F$.

Figure 2: Edges with finite symmetry group types (a) $C_1$; (b) $C_2$; (c) $D_1$; and (d) $D_2$.

The tiling ${\mathcal{T}}^{\ast }$ is formed by applying $H$ to the newly constructed edges $e_1^{\ast },e_2^{\ast },\dots ,e_n^{\ast }$. That is, ${\mathcal{T}}^{\ast }=H{e}_1^{\ast }\cup H{e}_2^{\ast }\cup \cdots \cup H{e}_n^{\ast }$. The tiling ${\mathcal{T}}^{\ast }$ is a tiling whose vertices are the vertices of $\mathcal{T}$, the edges are the union of orbits of the $e_j^{\ast \ast }$s under $H$ and the tiles are the regions bounded by these edges. Its symmetry group is $H$. Since $H$ is transitive on the vertices of $\mathcal{T}$, it follows that ${\mathcal{T}}^{\ast }$ is an isogonal tiling.

To illustrate the construction, consider the uniform tiling $\mathcal{T}:=(4^6)$ of ${\mathbb{H}}^2$, a tiling with 6 regular 4-gons incident to each vertex. Its symmetry group is $G=⟨P,Q,R⟩\cong \ast 642$ generated by reflections $P,Q$ and $R$ with axes shown in Figure 3(a). We consider a subgroup $H=⟨P,R,QPQ,QRQRPQRQ⟩\cong 2^{\ast }222$ of $G$ generated by reflections $P,R$, and $QPQ$, and the $180^{\circ }$ rotation $QRQRPQRQ$ with axes and center, respectively, shown in Figure 3(b), where $H$ forms one orbit of vertices of $\mathcal{T}$. Moreover, $H$ forms four orbits of edges of $\mathcal{T}$ namely $H{e}_1,H{e}_2,H{e}_3$ and $H{e}_4$. Assigning the colors red, blue, green, and orange, respectively to $H{e}_1,H{e}_2,H{e}_3$ and $H{e}_4$, we obtain the uniform edge coloring in Figure 3(b).

We now construct an isogonal tiling from this uniform edge coloring of $\mathcal{T}$. To do this, we introduce a new set of edges $e_1^{\ast },e_2^{\ast },\dots ,e_4^{\ast }$. We explain the construction in detail. We use the straight edge $e_1^{\ast }$ of symmetry group in $G$ generated by reflections $P$ and $R$. It can be checked that the symmetry group of $e_1$ in $H$ is also generated by $P$ and $R$. The edge $e_2^{\ast }$ may be an edge of symmetry group in $G$ generated by the reflection $QPQ$ or generated by perpendicular reflections $QPQ$ and $QRQ$. Note that the symmetry group of $e_2$ in $H$ is generated by $QPQ$. We choose $e_2^{\ast }$ of symmetry group generated by $QPQ$. The edge $e_3^{\ast }$ may be an edge of symmetry group in $G$ generated by the $180^{\circ }$ rotation $QRQRPQRQ$ or generated by the $180^{\circ }$ rotation $QRQRPQRQ$ and reflection $QRQPQRQ$. Observe that the symmetry group of $e_3$ in $H$ is generated by $QRQRPQRQ$. We choose $e_3^{\ast }$ of symmetry group generated by $QRQRPQRQ$. Lastly, we use a straight edge $e_4^{\ast }$ of symmetry group in $G$ generated by perpendicular reflections $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$. Observe that the symmetry group of $e_4$ in $H$ is also generated by $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$. We now form the isogonal tiling ${\mathcal{T}}_1^{\ast }$ by forming the union of the orbits of these new edges under the subgroup $H$. That is, ${\mathcal{T}}_1^{\ast }=H{e}_1^{\ast }\cup H{e}_2^{\ast }\cup H{e}_3^{\ast }\cup H{e}_4^{\ast }$ (see Figure 3(c)).

Hyperbolic Isogonal Tilings from Uniform Edge Colorings

Another isogonal tiling may be formed by choosing a different set of edges for $e_1^{\ast },e_2^{\ast },e_3^{\ast }$ , and $e_4^{\ast }$ . The isogonal tiling ${\mathcal{T}}_2^{\ast }$ in Figure 3(d) is formed by considering $H{e}_1^{\ast }\cup H{e}_2^{\ast }\cup H{e}_3^{\ast }\cup H{e}_4^{\ast }$ where the symmetry group of $e_1^{\ast }$ in $G$ is generated by the reflections $P$ and $R$ , the symmetry group of $e_2^{\ast }$ in $G$ is generated by the reflection $QPQ$ , the symmetry group of $e_3^{\ast }$ in $G$ is generated by the $180^{\circ }$ rotation $QRQRPQRQ$ and reflection $QRQPQRQ$ , and the symmetry group of $e_4^{\ast }$ in $G$ is generated by the perpendicular reflections $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$ .

(a)

(b)

(c) Figure 3: (a) The hyperbolic tiling $\mathcal{T}$ with symmetry group $G$ generated by reflections $P,Q,R$ ; (b) a uniform edge-4-coloring of $\mathcal{T}$ , generators $P,R,QPQ,QRQRPQRQ$ of $H$ and axes of reflections $QRQ$ , $QRQPQRQ$ , $(QR{)}^{2}QPQ(RQ{)}^2$ ; and (c-d) isogonal tilings ${\mathcal{T}}_1^{\ast }$ and ${\mathcal{T}}_2^{\ast }$ with symmetry group $H$ .

(d)

Constructing Patterns from Isogonal Tilings

An isogonal tiling may be transformed to arrive at a pattern with varying motifs, having the same symmetry properties as the tiling. To illustrate this, consider the hyperbolic isogonal tiling ${\mathcal{T}}_1^{\ast }$ shown in Figure 3(c). We consider the symmetry group in $G$ of each edge $e_j^{\ast },j=1,\dots ,4$ in ${\mathcal{T}}_1^{\ast }$ and construct a motif with the same symmetry group in $G$ as that of $e_j^{\ast }$ . For example, the edge $e_1^{\ast }$ of symmetry group in $G$ generated by two reflections $P$ and $R$ is replaced by a blue butterfly of symmetry group in $G$ also generated by $P$ and $R$ . The rest of the edges are replaced as follows: The edge $e_2^{\ast }$ of symmetry group in $G$ generated by the reflection $QPQ$ by a green dragonfly of symmetry group in $G$ generated by the reflection $QPQ$ ; the edge $e_3^{\ast }$ of symmetry group in $G$ generated by the $180^{\circ }$ rotation $QRQRPQRQ$ by a yellow flower of symmetry group generated by $QRQRPQRQ$ ; and the edge $e_4^{\ast }$ of symmetry group in $G$ generated by perpendicular reflections $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$ by the pink butterfly of symmetry group in $G$ generated by perpendicular reflections $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$ . The resulting pattern is shown in Figure 4(a) with symmetry group $2^{\ast }222$ and having four different classes of motifs corresponding to the four orbits of edges in the original tiling ${\mathcal{T}}_1^{\ast }$ .

Similarly, Figure 4(b) and Figure 4(c) show patterns corresponding to the isogonal tiling ${\mathcal{T}}_2^{\ast }$ . The motifs used in Figure 4(b) are inspired by the fabric designs of the Northern Kankana-ey from Northern Luzon in the Philippines [2]. The motifs used to replace $e_1^{\ast }$ , of symmetry group in $G$ generated by two reflections $P$ and $R$ , and $e_3^{\ast }$ of symmetry group in $G$ generated by the $180^{\circ }$ rotation $QRQRPQRQ$ and reflection $(QR{)}^{3}Q$ , are known as matmata which represents rice grains and the eyes, as they admire rice as an all-seeing god that gives their body the nourishment that it needs. The motifs used to replace $e_2^{\ast }$ , of symmetry group in $G$ generated by the reflection $QPQ$ , and $e_4^{\ast }$ of symmetry group in $G$ generated by the perpendicular reflections $R$ and $(QR{)}^{2}QPQ(RQ{)}^2$ , are called tiktiko, which shows distinguishing zigzag designs that symbolize the mountains and forests where their rice fields are located. These two types of motifs imply wealth and abundance for the Northern Kankana-ey.

Moreover, Figure 4(c) shows a hyperbolic pattern with motifs that are inspired by the indigenous designs of the Yakan tribe in the Philippines [4]. The designs of the Yakans usually include colorful motifs to showcase bravery in battle, joy in birth, and marriage rituals. The motifs used to replace $e_1^{\ast },e_2^{\ast },e_3^{\ast }$ and $e_4^{\ast }$

Abila, De Las Peñas, and Tomenes

consist of colorful diamonds which is a traditional motif used in Yakan textiles. These motifs also show symmetries which satisfy the symmetry group condition mentioned in the previous section.

(a)

(b)

(c) Figure 4: Hyperbolic isogonal tiling using different motifs as edges: (a) flowers and butterflies; motifs from indigenous designs of the (b) Northern Kankana-ey; and (c) Yakan tribe in the Philippines.

Conclusion

The connection of uniform edge colorings and isogonal tilings may lead to the construction of other types of tilings such as $k$-isogonal tilings from edge colorings of $k$-uniform tilings of the Euclidean plane, hyperbolic plane and 2-sphere. The problem of how to efficiently construct uniform edge-$n$-colorings has been addressed in response to the problem posed by Grünbaum and Shephard in [9] and is discussed in detail in [1]. Consequently, various aesthetically pleasing patterns may also arise depending on the underlying isogonal or $k$-isogonal tiling and the set of motifs that will be used.

Acknowledgment

The authors would like to thank the Symmetries in Algebra and Geometry (SAGE) Laboratory, Department of Mathematics, Ateneo de Manila University where this research was carried out. Agatha Kristel Abila would like to express her gratitude to the Department of Science and Technology (DOST) through the Accelerated Science and Technology Human Resource Development Program (ASTHRDP) for the scholarship grant for PhD studies.

References

[1] A.K.M. Abila. “Uniform Edge-$n$-Colorings of Uniform Tilings.” PhD Dissertation. Ateneo de Manila University, Philippines, 2024. [2] N. Baylas, T. Rapanut, and M.L.A.N. De Las Peñas. “Weaving Symmetry of the Philippine Northern Kankana-ey.” Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, 2012, p.267-274. https://archive.bridgesmathart.org/2012/bridges2012-267.html [3] M.L.A.N. De Las Peñas and A. Basilio. “Vertex Colorings of Archimedean Tilings with Applications in Molecular Symmetry.” Loyola Schools Review, 2004. [4] M.L.A.N. De Las Peñas, A. Garciano, D.M. Versoza, and E. Taganap. “Crystallographic Patterns in Philippine Indigenous Textiles.” J. Appl. Cryst, 51, 456-469, 2008. [5] M.L.A.N. De Las Peñas, G.R. Laigo, and E.D. Provido. “Hyperbolic Semi-Regular Tilings and their Symmetry Properties.” Bridges Donostia: Mathematics, Music, Art, Architecture, Culture, 2007, p.361-368. http://archive.bridgesmathart.org/2007/bridges2007-361.html [6] M.L.A.N. De Las Peñas. “Symmetrical Patterns: Studies and Discoveries.” Mathematical Society of the Philippines Annual Convention Pre-conference Public Lectures, Ateneo de Manila University, June 1-4, 2023. [7] D. Dunham. “Hyperbolic Symmetry.” Comp. and Maths. with Appls. Vol 12B, p. 139-153, 1986. [8] B. Grünbaum and G. Shephard. “The Ninety-one Types of Isogonal Tilings in the Plane.” Trans. Amer. Math. Soc. 242, p.335-353, 1978. [9] B. Grünbaum and G. Shephard. Tilings and Patterns, W.H.Freeman and Company, New York, 1987.