An Approach in Coloring Semi-Regular Tilings on the Hyperbolic Plane

Year: 2005 Authors: Ma. Louise Antonette N. De Las Peñas; Glenn R. Laigo; René P. Felix

Core claim

Perfect colorings of the hyperbolic semi-regular tilings 8·8·5 and 4·10·8 can be systematically obtained from subgroups of their symmetry group.

Topics

perfect colorings, semi-regular tilings, hyperbolic plane, subgroup structure, symmetry groups

Domains

group theory, hyperbolic geometry, tiling theory, coset actions, pattern coloring, geometric design, visual symmetry

Methods

subgroup enumeration, coset coloring, GAP computation, symmetry analysis

Media

hyperbolic tessellations, tile diagrams, computer-generated colorings

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.

An Approach in Coloring Semi-Regular Tilings on the Hyperbolic Plane

Ma. Louise Antonette N. De Las Peñas, mlp@mathsci.math.admu.edu.ph Glenn R. Laigo, glaigo@yahoo.com Math Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines

René P. Felix, rene@math01.upd.edu.ph Math Department, University of the Philippines, Diliman, Quezon City, Philippines

Abstract

A coloring of a semi-regular tiling is perfect if every symmetry of the tiling permutes the colors of the tiling. In this paper, an approach to the construction of perfect colorings of semi-regular tilings on the hyperbolic plane is presented.

1. Introduction

In [3], a method for coloring symmetrical patterns was presented where a fundamental domain of the pattern is assigned exactly one color. In this paper, we present a general framework for coloring planar patterns where a fundamental domain of the pattern may be assigned more than one color. We apply the framework to construct perfect colorings of semi-regular tilings on the hyperbolic plane. We will use the subgroup structure of the symmetry group of the tiling to systematically construct the colorings.

An edge-to-edge tiling is a plane tiling where the corners and sides of the polygonal tiles form all the vertices and edges of the tiling and vice versa. A vertex of an edge-to-edge tiling is said to be of type $p_1\cdot p_2\cdot \dots \cdot p_q$ if the polygons about this vertex in cyclic order are a $p_1$-gon, a $p_2$-gon, …, and a $p_q$-gon. An edge-to-edge tiling having regular polygons as its tiles with vertices all of the same type, and where the symmetries of the tiling act transitively on the vertices is called semi-regular. We denote the semi-regular tiling as $p_1\cdot p_2\cdot \dots \cdot p_q$ depending on its vertex type $p_1\cdot p_2\cdot \dots \cdot p_q$. If the polygons in the tiling are of the same type, particularly a $p$-gon meeting $q$ at a vertex, we denote the tiling as $p^q$.

In this paper, we will present an approach to color perfectly the $8\cdot 8\cdot 5$ and $4\cdot 10\cdot 8$ hyperbolic semi-regular tilings.

2. General Framework for Coloring Planar Patterns

The following general framework for coloring planar patterns shall be used to obtain colorings of semi-regular tilings.

Let $X$ be the set of tiles in the tiling to be assigned colors; $G$ be the symmetry group of the uncolored tiling; $H$ be the subgroup of elements of $G$ permuting the colors; $C$ be the set of colors.

Let $O_i$ $(i\in I)$ be the $H$-orbits of colors and $c_i$ a color in $O_i$. Then $O_i=\{h{c}_i:h\in H\}$ and corresponding to this set is the set $\{h{J}_i{X}_i:h\in H\}$, where $J_i$ is the stabilizer in $H$ of the color $c_i$ and $X_i$ consists of representatives of each $H$-orbit of elements of $X$ with the representatives colored $c_i$.

The following are true:

1. The action of $H$ on $O_i$ is equivalent on its action on $\{h{J}_i:i\in I\}$ by left multiplication.
2. In $O_i$, the number of colors is $[H:J_i]$.
3. If $x\in X_i$ then ${\mathrm{Stab}}_H(x)\le J_i$, where ${\mathrm{Stab}}_H(x)=\{h\in H:hx=x\}$.
4. If $x\in X_i$ then $|Hx|=[H:J_i]\cdot [J_i:{\mathrm{Stab}}_H(x)]$.
5. The number of $H$-orbits of colors is less than or equal to the number of $H$-orbits of elements of $X$.

The following steps, based on the general framework, shall be used to obtain the required coloring of the tiling.

1. Determine the finite group $S$ of isometries in $G$ that stabilizes a representative tile $t$ from an orbit.
2. Determine all subgroups $J$ of $G$ such that S < J.
3. If tile $t$ has color $c$, apply $c$ to all tiles in the set $Jt$. This makes $J$ the stabilizer of the color $c$ inside $G$. If $[G:J]=k$, then $Jt$ is $\frac{1}{k}$ of the tiles in the class where $t$ belongs.
4. To complete the coloring, assign a color to every element of the set $\{gJt:g\in G\}$. One element of this set has color $c$, which is $Jt$. There should be $k-1$ other elements or colors.

Hence, the index $k$ of the subgroup $J$ in $G$ is the number of colors that can be used to perfectly color the orbit of tiles containing $t$.

3. Coloring the Hyperbolic Plane

3.1. Tessellating the Hyperbolic Plane

In [1], Aziz created a computer program Coloring the Hyperbolic Plane (CHP) that tessellates the hyperbolic plane with congruent triangles of interior angles $\frac{\pi }{p}$, $\frac{\pi }{q}$, and $\frac{\pi }{r}$, where \frac{\pi}{p} + \frac{\pi}{q} + \frac{\pi}{r} < \pi. Denote by $e$ one of the triangles of the tessellation and call it the fundamental triangle. Let $P$ be the reflection on the side of the triangle opposite the angle $\frac{\pi }{p}$, $Q$ as the reflection on the side of the triangle opposite the angle $\frac{\pi }{q}$, and $R$ the reflection on the side of the triangle opposite $\frac{\pi }{r}$. The symmetry group $G$ of the tessellation is generated by $P$, $Q$, and $R$, denoted by $\ast pqr$.

Given the fundamental triangle $e$, the tessellation may be recovered by getting the images of $e$ under $P,Q,R$, and their products. There is a one-to-one correspondence between the elements of $G$ and the triangles in the tessellation. Each triangle in the tessellation can then be labeled by the corresponding

element of $G$ . The action of $G$ on the triangles of the tessellation, where $g\in G$ acts on a triangle by sending it to its image, is equivalent to the action of $G$ on itself by left multiplication.

Figure: (1) Labeling the triangles in the tessellation

3.2. Coloring Using Right Cosets. If $S$ is a subgroup of $G$ of index $n$ , a coloring using right (or left) cosets of $S$ refers to a bijective map from the set of right (or left) cosets of $S$ to a set of $n$ colors. Triangles labeled by elements of a right (or left) coset are colored using the color assigned to the coset. In Figure 2, we give a coloring of the hyperbolic plane using right cosets of the subgroup $S$ and in Figure 3, we give another coloring using left cosets of the subgroup $S$ , where $S$ represents a subgroup of index 3 of the hyperbolic triangle group *642.

Figure: (2) Right coset coloring using $S$ ; (3) Left coset coloring using $S$

The right coset colorings of a given subgroup $S$ of the symmetry group $G$ of the tessellation plays an important role in studying the subgroup structure of $G$ . $S$ turns out to be the symmetry group of the colored tessellation and $S$ fixes the colors of the tessellation. In this paper, we will use the right coset colorings generated by CHP to determine the subgroups of $G$ that contain the stabilizer of the tiles in the given semi-regular tilings.

4. Perfect Colorings of Semi-Regular $8\cdot 8\cdot 5$ and $4\cdot 10\cdot 8$ Tilings

In this part of the paper, we illustrate how to obtain perfect colorings of semi-regular $8\cdot 8\cdot 5$ and $4\cdot 10\cdot 8$ tilings using the given framework. Both tilings have symmetry group $G=\ast 542$ ; $G$ contains rotations of order 5, 4, 2 with centers of the corresponding rotations lying on mirror lines.

In coloring the semi-regular tilings, we will make use of the subgroups of $G$ . GAP [8] is used to generate a listing of the subgroups of $G$ shown in Table 1. For the purposes of this paper, and due to coloring constraints, we will only consider subgroups up to index 5.

List of Subgroups of *542 of Index <= 5: Number of Subgroups = 7
1 Group( [ Q, R, P ])
2 Group( [ Q, R, PRP ])
2 Group( [ RQ, P ])
2 Group( [ RQ, PQ ])
4 Group( [ RQ, PRPQ ])
5 Group( [ Q, P, RPR, RQRPRQR ])
5 Group( [ Q, P, RPQR ])

Table: (1) Subgroups of *542 of index less than or equal to 5

The generators $Q,R,P$ appearing in Table 1 are mirror reflections with axes shown in Figures 4 and 5 for the respective tilings $8\cdot 8\cdot 5$ and $4\cdot 10\cdot 8$ .

Figures: (4-5) Generators $Q,R,$ and $P$

4.1. Semi-Regular $8\cdot 8\cdot 5$ Tiling. The semi-regular $8\cdot 8\cdot 5$ tiling has two orbits of tiles: the orbit of 8-gons and the orbit of 5-gons. To construct the perfect colorings, we color each orbit of tiles separately. We first color the orbit of 8-gons.

First, note that the finite group that stabilizes an 8-gon is of type $D_4$ , the dihedral group of order 8. We need to select the subgroups $J_i$ that contains $D_4$ . The condition that $J_i$ contains the stabilizer is always satisfied by $G$ . To find other subgroups containing $D_4$ , we will use the right coset colorings of the subgroups of $G$ . To obtain the right coset colorings, we use the CHP program.

From the program, Figure 6 shows the right coset coloring using the subgroup $J_1=⟨Q,P,RPR,RQRPRQR⟩$ . Note that the subgroup $D$ generated by the $90^{\circ }$ rotation about the indicated point $x$ and mirror about the horizontal line through $x$ fixes the color of the given right coset coloring. Thus, the subgroup $J_1=⟨Q,P,RPR,RQRPRQR⟩$ contains the group of type $D_4$ and can now be used to color the orbit of 8-gons for the $8\cdot 8\cdot 5$ tiling.

Figure 4 shows the right coset coloring using the subgroup $J_2=⟨Q,P,RPQR⟩$ . Similarly, the subgroup $D$ generated by the $90^{\circ }$ rotation about $x^{\prime }$ and mirror about the horizontal line through $x^{\prime }$ fixes the colors of the coloring. Thus, the subgroup $J_2$ also contains $D_4$ .

We are now ready to color the orbit of 8-gons. We will use the subgroups $J_i$ , namely $J_1,J_2$ and $J_3=G$ . Using $J_3=G$ , we color all 8-gons using one color to obtain the coloring in Figure 8.

Next, we color the orbit of 8-gons using $J_1=⟨Q,P,RPR,RQRPRQR⟩$ . To obtain a perfect coloring using $J_1$ , we first choose a representative tile $t$ from the 8-gons. We then color $J_{1}t$ with black, as seen in Figure 9. To color the rest of the orbit, we apply the 5-fold rotation with center $A$ lying on mirrors $R$ and $Q$ on $J_{1}t$ to obtain a coloring of five colors given in Figure 10.

Lastly, we color the orbit of 8-gons using $J_2=⟨Q,P,RPRQR⟩$ . Coloring all tiles in $J_{2}t$ black, we obtain Figure 11. Then we assign 4 different shades and textures of gray to the tiles in the other orbits by applying the 5-fold rotation about $A$ to obtain Figure 12.

Next, we color the orbit of 5-gons. The finite group that stabilizes a 5-gon is of type $D_5$ , the dihedral group of order 10. We now select the subgroup $J_{i^{\prime }}$ that contains the stabilizer. Aside from $G$ , our choice for $J_{i^{\prime }}$ is the subgroup $⟨Q,R,PRP⟩$ . Figure 13 shows the right coset coloring using $⟨Q,R,PRP⟩$ . The subgroup generated by $72^{\circ }$ about $x^{\prime \prime }$ and mirror reflection about the horizontal line through $x^{\prime \prime }$ fixes the colors of the coloring and is of type $D_5$ .

To color the orbit of 5-gons, we let $J_{1^{\prime }}=G$ to obtain Figure 14 and let $J_{2^{\prime }}=⟨Q,R,PRP⟩$ to obtain Figure 15.

To color the entire semi-regular tiling, we combine all the colorings of each orbit of tiles above. Thus, the resulting perfect colorings of the $8\cdot 8\cdot 5$ tiling are shown in Figure 16.

Figures: (6-7) Right coset colorings of $J_1$ and $J_2$ respectively; (8) Perfect coloring of the orbit of 8-gons using $J_3=G$ ; (9) $J_{1}t$ ; (10) Perfect coloring of the orbit of 8-gons using $J_1$ ; (11) $J_{2}t$ ; (12) Perfect coloring of the orbit of 8-gons using $J_2$ ; (13) Right coset colorings of $J_2$ ; (14-15) Perfect coloring of the orbit of 5-gons

Figure: (16) Perfect colorings of the $8\cdot 8\cdot 5$ tiling

4.2. Semi-Regular $4\cdot 10\cdot 8$ Tiling. The semi-regular $4\cdot 10\cdot 8$ tiling has three orbits of tiles: the orbit of 4-gons, 10-gons, and 8-gons. We follow the steps given in 4.1 and color each orbit of tiles separately.

To color the 4-gons, we use $J_1=G$ , $J_2=⟨Q,P,RPR,RQRPRQR⟩$ , and $J_3=⟨Q,P,RPQR⟩$ where D_{4} < J_{i} . We have the three colorings in Figures 17, 18, and 19.

Next, we use $J_{1^{\prime }}=G$ and $J_{2^{\prime }}=⟨Q,R,PRP⟩$ , where D_5 < J_{i'} , to color the 10-gons shown in Figures 20 and 21.

Lastly, we use $J_{1^{\prime }}=G$ , $J_{2^{\prime }}=⟨Q,P,RPR,RQRPRQR⟩$ , and $J_{3^{\prime }}=⟨Q,P,RPQR⟩$ , where D_2 < J_{i'} , to color the 8-gons in Figures 22, 23, and 24.

Next, we combine all these colorings to obtain the perfect colorings of the $4\cdot 10\cdot 8$ semi-regular tiling as seen in Figure 25.

Figures: (17-19) Perfect coloring of the orbit of 4-gons; (20-21) Perfect coloring of the orbit of 10-gons; (22-24) Perfect coloring of the orbit of 8-gons

Figure: (25) Perfect colorings of the $4\cdot 10\cdot 8$ tiling where the orbits do not share colors

Observe that if $J_i$ is used to color one orbit of tiles, it can also be used to color a second orbit of tiles as long as $J_i$ contains the stabilizer of a tile in the second orbit of tiles. Moreover, if a color used to color tile $t$ in the first orbit of tiles is to be used to color tiles in the second orbit, then the tile $t^{\prime }$ that will be assigned the same color as tile $t$ should have a stabilizer contained in $J_i$ .

In coloring the $4\cdot 10\cdot 8$ tiling, the orbit of 4-gons and the orbit of 8-gons can share the same color. These colorings appear in Figure 26. The colorings A and B are obtained using $J_1=G$ to color both orbits of 4-gons and 8-gons. The colorings in C and D are obtained using $J_2=⟨Q,P,RPR,RQRPRQR⟩$ while the colorings in E and F are obtained using $J_3=⟨Q,P,RPQR⟩$ .

Figure: (26) Perfect colorings of the $4\cdot 10\cdot 8$ tiling where the orbits share colors

5. Conclusion

In this note, we give an approach to color semi-regular tilings on the hyperbolic plane. We use the general framework for coloring planar patterns where an orbit of tiles in the given tiling is colored using a subgroup of the symmetry group $G$ of the tiling containing the stabilizer of the tile. We use the GAP program to generate the subgroups of $G$ while a helpful tool in studying more closely the subgroup structure of $G$ is the CHP program.

We intend that the approach provided here in obtaining perfect colorings of semi-regular tilings will provide a springboard in the construction of colorings (both perfect and non-perfect) of tilings in general on the hyperbolic plane.

References

[1] Aziz, Shahid. A Computer Algorithm for Coloring A Hyperbolic Tessellation. A Masteral Thesis, The University of the Philippines – Diliman, 1996.

[2] Coxeter, H.S.M., and W.O. Moser. Generators and Relations for Discrete Groups. 2nd ed. New York: Springer-Verlag, 1965.

[3] De Las Peñas, Ma. Louise Antonette N., René P. Felix, and M. V. P. Quilinguin. A Framework for Coloring Symmetrical Patterns in Algebras and Combinatorics: An International Congress, ICAC ‘97 Hong Kong. Singapore: Springer-Verlag. 1999.

[4] Felix, René P. A General Framework for Coloring Planar Patterns, a paper presented in National Research Council of the Philippines (NRCP) Conference. Philippines. February, 2004.

[5] Grünbaum, B., and G.C. Shephard. Tilings and Patterns. New York: W.H. Freeman and Company, 1987.

[6] Hernandez, Nestine Hope Sevilla. On Colorings Induced by Low Index Subgroups of Some Hyperbolic Triangle Groups. A Masteral Thesis, The University of the Philippines – Diliman. 2003.

[7] Mitchell, Kevin J. Constructing Semi-Regular Tilings, a paper presented in The Spring 1995 Meeting of the Seaway Section of the Mathematical Association of America. 1995. http://people.hws.edu/mitchell/tilings/Part1.html.

[8] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.3, 2002, http://www.gap-system.org.