Coloring Uniform Honeycombs

Year: 2009 Authors: Glenn R. Laigo; Ma. Louise Antonette N. De las Peñas; René P. Felix

Core claim

Subgroup structure of the honeycomb symmetry group enables systematic cell-colorings of uniform honeycombs.

Topics

color symmetry, uniform honeycombs, subgroup structure, three-dimensional tilings

Domains

group theory, combinatorics, geometry, symmetry groups, geometric patterning, visual coloring, 3D pattern design

Methods

subgroup analysis, orbit-stabilizer construction, symmetry-group action, coloring framework

Media

three-dimensional honeycombs, truncated cubic honeycomb, bitruncated cubic honeycomb

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 2009: Mathematics, Music, Art, Architecture, Culture

Coloring Uniform Honeycombs

Glenn R. Laigo, glaigo@ateneo.edu Ma. Louise Antonette N. De las Peñas, mlp@math.admu.edu.ph Mathematics Department, Ateneo de Manila University Loyola Heights, Quezon City, Philippines

René P. Felix, rene@math.upd.edu.ph Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines

Abstract

In this paper, we discuss a method of arriving at colored three-dimensional uniform honeycombs. In particular, we present the construction of perfect and semi-perfect colorings of the truncated and bitruncated cubic honeycombs. If $G$ is the symmetry group of an uncolored honeycomb, a coloring of the honeycomb is perfect if the group $H$ consisting of elements that permute the colors of the given coloring is $G$. If $H$ is such that $[G:H]=2$, we say that the coloring of the honeycomb is semi-perfect.

Background

In [7, 9, 12], a general framework has been presented for coloring planar patterns. Focus was given to the construction of perfect colorings of semi-regular tilings on the hyperbolic plane. In this work, we will extend the method of coloring two dimensional patterns to obtain colorings of three dimensional uniform honeycombs. There is limited literature on colorings of three-dimensional honeycombs. We see studies on colorings of polyhedra; for instance, in [17], a method of coloring shown is by cutting the polyhedra and laying it flat to produce a pattern on a two-dimensional plane. In this case, only the faces of the polyhedra are colored. In [6], enumeration problems on colored patterns on polyhedra are discussed and solutions are obtained by applying Burnside’s counting theorem. The works [14, 19] highlight edge-colorings of the platonic solids. There are studies on colorings of three-dimensional space using an algorithm that makes use of the group structure of the Picard group [1, 2, 21]. Cross sections of the colored three-dimensional patterns were used to produce colored two-dimensional Euclidean patterns.

We find the occurrence of colored honeycombs (space filled with polyhedra) in different places; for instance, as representations of geometric constructions, or as models of chemical structures. Shown in Figures 3(a), 3(b) and 4(b) are illustrations of colored honeycombs. Interestingly, the colorings shown are representations of three different uniform constructions of the bitruncated cubic honeycomb, a honeycomb consisting of truncated octahedra. For example, the coloring with two colors in Figure 3(a) represents two types of truncated octahedra: half are obtained from the original cells of the cubic honeycomb and the other half are centered on vertices of the original honeycomb. In Figure 4(b), this colored honeycomb is referred to as the cantitruncated alternate cubic – there are 3 types of truncated octahedra in 2:1:1 ratios. In Figure 3(b) there are 4 types of octahedra in 1:1:1:1 ratios; each type is represented by a different color. In [18], a bitruncated honeycomb is used to represent a spongy graphite network of carbon atoms in 3-dimensional space.

In this paper, we present the construction of colored honeycombs where an entire cell gets one color.

Laigo, De las Peñas and Felix

In recent works [8, 16], a method for determining subgroups of three-dimensional symmetry groups in spherical, Euclidean or hyperbolic 3-space was discussed. The approach, based on concepts on color symmetry theory, allows for the characterization of each subgroup in terms of the symmetries it contains. This development is helpful in the construction of colorings of honeycombs especially in hyperbolic space, since the subgroup structure of hyperbolic symmetry groups is not widely known. As will be seen in this work, the subgroup structure of the symmetry group of a given honeycomb plays a significant role in arriving at colorings of the honeycomb.

Uniform honeycomb

We start the discussion by defining uniform honeycombs. A polyhedron is called uniform if its faces are regular polygons and it satisfies the property that its group of symmetries acts transitively on its vertices. A uniform honeycomb is a three-dimensional honeycomb with uniform polyhedra as its cells and where the symmetry group of the honeycomb acts transitively on its vertices. Uniform honeycombs are also called Archimedean honeycombs.

In three-dimensional Euclidean space, twenty-eight such honeycombs exist: the cubic honeycomb and seven truncations thereof; the alternated cubic honeycomb and four truncations thereof; ten prismatic forms based on the uniform plane tilings (eleven if including the cubic honeycomb); and five modifications of some of the mentioned by elongation or gyration [11, 13]. In this paper, we illustrate the concept of arriving at colored honeycombs using two examples of uniform honeycombs, the truncated and the bitruncated cubic honeycombs, shown in Figures 1(a) and (b) respectively. Both of these honeycombs are directly constructed from the only regular honeycomb in three-dimensional Euclidean space – the regular space filling of cubes [4]. Interestingly, the centers of the cells of the bitruncated cubic honeycomb coincide with the body centered cubic (BCC) lattice.

The symmetry group of both the truncated and bitruncated cubic honeycombs is the group [4, 3, 4] generated by four reflections $P$, $Q$, $R$ and $S$ satisfying the following relations

The planes of reflections $P$ and $Q$, $R$ and $S$ intersect at an angle of $\pi /4$; the planes of reflections $Q$ and $R$ intersect at an angle of $\pi /3$ and the planes of reflections $P$ and $R$, $P$ and $S$, $Q$ and $S$ intersect at an angle of $\pi /2$. The planes of the reflections $P$, $Q$, $R$ and $S$ are shown in Figures 1(a) and (b), respectively, for the truncated and bitruncated honeycombs.

A method for coloring symmetrical patterns

The following method given in [12], will be applied to arrive at colorings of uniform honeycombs.

Let $G$ denote the symmetry group of the uncolored honeycomb and $X$ the set of cells in the honeycomb. If $C=\{c_1,c_2,\dots ,c_n\}$ is a set of $n$ colors, an onto function $f:X\to C$ is called an $n$-coloring of $X$. To each $x\in X$ is assigned a color in $C$. The coloring determines a partition $\mathcal{P}=\{f^{-1}(c_i):c_i\in C\}$ where $f^{-1}(c_i)$ is the set of elements of $X$ assigned color $c_i$.

Let $H$ be the subgroup of $G$ which consists of symmetries in $G$ that effect a permutation of the colors in $C$. Then $h\in H$ if for every $c\in C$, there is a $d\in C$ such that $h(f^{-1}(c))=f^{-1}(d)$. This defines an action of $H$ on $C$ where we write $hc:=d$ if and only if $h(f^{-1}(c))=f^{-1}(d)$.

132

Coloring Uniform Honeycombs

(a)

(b) Figure 1: The (a) truncated and (b) bitruncated cubic honeycombs with the planes of the reflections $P$ , $Q$ , $R$ and $S$ .

Since $H$ acts on the set $C$ of colors of $X$ there exists a homomorphism $\sigma$ from $H$ to $\mathrm{Perm}(C)$ where $\mathrm{Perm}(C)$ is the group of permutations of $C$ .

Let $c_i\in C$ and denote by $O_i$ the $H$ -orbit of $c_i$ , that is $O_i=H{c}_i$ . Suppose $J_i=\{h\in H:h{c}_i=c_i\}$ is the stabilizer of $c_i$ in $H$ . From each $H$ -orbit of $X$ with an element colored $c_i$ , pick one such element. Put these elements together in a set $X_i$ . Then the set of all elements of $X$ that are colored $c_i$ is $J_i{X}_i=\{jx:j\in J_i,x\in X_i\}$ , that is, $f^{-1}(c_i)=J_i{X}_i$ . A one-to-one correspondence results between the sets $O_i=H{c}_i$ and $\{h{J}_i{X}_i:h\in G\}$ where $h{J}_i{X}_i$ denotes the image of $J_i{X}_i$ under $h$ .

As a consequence of the orbit-stabilizer theorem given the assumptions above, we have the following:

Theorem:

1. The action of $H$ on $O_i$ is equivalent to its action on $\{h{J}_i:h\in H\}$ by left multiplication.
2. The number of colors in $O_i$ is equal to $[H:J_i]$ .
3. The number of $H$ -orbits of colors is at most the number of $H$ -orbits of elements of $X$ .
4. If $x\in X_i$ and ${\mathrm{Stab}}_H(x)=\{h\in H:hx=x\}$ is the stabilizer of $x$ under the action of $H$ on $X$ then

(a) ${\mathrm{Stab}}_H(x)\le J_i$ . (b) $|Hx|=[H:J_i]\cdot [J_i:{\mathrm{Stab}}_H(x)]$

Thus, using the above framework, we outline the steps to obtain a colored uniform honeycomb, where $H$ permutes the colors of the resulting coloring.

1. Pick a cell $t$ from an $H$ -orbit of the elements of $X$ .
2. Determine the finite group $S^{\ast }$ of isometries in $H$ which stabilizes $t$ , that is, $S^{\ast }={\mathrm{Stab}}_H(t)$ .
3. Choose a subgroup $J$ of $H$ such that $S^{\ast }\le J$ .
4. Apply color $c$ to cell $t$ and to all the cells in the set $Jt$ . If $[H:J]=k$ , then $Jt$ is $1/k$ of the cells in the $H$ -orbit where $t$ belongs.
5. Assign a color to every element of the set $\{hJt:h\in H\}$ . The set $Jt$ is given color $c$ and each of the remaining $k-1$ elements of the set gets a different color. In this coloring of the given $H$ -orbit of cells, $J$ will be the stabilizer in $H$ of color $c$ .

Laigo, De las Peñas and Felix

To obtain a coloring of a given uniform honeycomb, we consider each $H$ -orbit of cells separately, coloring each orbit with a given set of colors such that $H$ permutes the colors. If two $H$ -orbits of cells are to have a color in common, the subgroup $J$ used should contain the stabilizers of representative tiles from the two $H$ -orbits. Combining the colored orbits of cells will give a colored honeycomb where all elements of $H$ effect a permutation on the set of colors.

Constructing perfect colorings of the truncated and bitruncated cubic honeycombs

In this part of the paper, we discuss the construction of perfect colorings of the truncated and bitruncated cubic honeycombs. Given either a truncated or bitruncated cubic honeycomb, we apply the framework to arrive at colorings where the symmetry group $G=[4,3,4]$ of the honeycomb effects a permutation of the colors in the coloring.

In coloring the honeycombs, we will make use of the subgroups of $G$ . (Table 1 gives a list of low index subgroups of $G$ up to conjugacy in $G$ obtained from [8, 16]). In this work, we obtain all perfect colorings where the number of colors used for each $G$ -orbit is at most 4. Any other perfect coloring satisfying the given restriction on the number of colors may be obtained by a symmetry of the honeycomb, a one-to-one change of colors or a combination of both. We consider those colorings where a $G$ -orbit of tiles gets at most 4 colors.

Table 1: The index 2, 3, 4 subgroups of $G=\left[4,3,4\right]$ up to conjugacy in $G$ .

Perfect colorings of the truncated cubic honeycomb. The truncated cubic honeycomb has two $G$ -orbit of cells: the orbit $X_1$ of octahedra and the orbit $X_2$ of truncated cubes. The perfect colorings that we will discuss first will involve those colorings where the $G$ -orbits of cells do not share a color; that is, a color that is used in $X_1$ will not be used in $X_2$ . We will color $X_1$ first, then $X_2$ .

In coloring $X_1$ , we start with the cell labeled $t$ in Figure 2(a). The stabilizer of $t$ in $G$ , ${\mathrm{Stab}}_G(t)$ , is the group generated by $Q,R,S$ , a group of type ${\mathbf{O}}_{\mathbf{h}}$ , also known as the octahedral group. We need to select a subgroup $J_1$ that satisfies the condition that ${\mathrm{Stab}}_G(t)\le J_1$ . Using Table 1 we find that the groups A = < Q, R, S, PQP> and B = < Q, R, S, PQRQP> are suitable choices for $J_1$ .

To obtain a perfect coloring of $X_1$ using $A$ , we assign $At$ the color light grey. To color the rest of the orbit, we apply the 2-fold rotation $(PQ{)}^2$ about $x$ on $At$ to obtain a coloring of two colors shown in Figure 2(a). A perfect coloring of $X_1$ using B = < Q, R, S, PQRQP> is given in Figure 2(b). The coloring is obtained by assigning all cells in $Bt$ light grey. Then we assign the colors grey, black and white to the other cells by applying the 4-fold rotation $PQ$ about $x$ .

Coloring Uniform Honeycombs

Next, we color the orbit $X_2$ of truncated cubes. We start with the cell labeled $t^{\prime }$ in Figure 2(c). The stabilizer of $t^{\prime }$ in $G$, ${\mathrm{Stab}}_G(t^{\prime })$ is the group $<p,q,=""r="">$ of type ${\mathbf{O}}_{\mathbf{h}}$. From Table 1, the groups $C=<p,q,=""r,=""srs="">$ and $D=<p,q,=""r,=""srsqrs="">$ contain ${\mathrm{Stab}}_G(t^{\prime })$, thus either $C$ or $D$ may be used to color $X_2$. Using $C$ and $D$, we obtain the colorings shown in Figures 2(c) and (d) respectively.

Note that the group $G$ can also be used to color $X_1$ or $X_2$ since it contains the stabilizer of every cell. Consequently, all the octahedra or the truncated cubes, respectively, will get one color.

The perfect colorings of the truncated cubic honeycomb where the two $G$-orbits of cells do not share colors will be obtained by considering the perfect colorings of the octahedra in orbit $X_1$ and the perfect colorings of the truncated cubes in orbit $X_2$. Using $G$, $A$ and $B$, there are 3 colorings of orbit $X_1$ and using $G$, $C$ and $D$, there are 3 colorings of orbit $X_2$ that will give rise to 9 perfect colorings of the truncated cubic honeycomb where the orbits $X_1,X_2$ do not share colors and both $X_1$ and $X_2$ get at most 4 colors.

(a)

(b)

(c) Figure 2: Perfect colorings of $X_1$ using: (a) $A$ and (b) $B$; perfect colorings of $X_2$ using: (c) $C$ and (d) $D$.

(d)

Perfectly colored honeycombs may also be arrived at by constructing colorings where the $G$-orbits of cells share colors. If a subgroup $J_i$ is used to color one orbit of cells $X_i$, it can be used to color another orbit $X_j$ as long as $J_i$ contains the stabilizer of a tile in $X_j$. Moreover, if a color used to color cell $t\in X_i$ is used to color cells in $X_j$ then the tile $t^{\prime }\in X_j$ that will be assigned the same color as tile $t$ should have a stabilizer contained in $J_i$.

We wish to remark that in constructing non-trivial perfect colorings of the truncated cubic honeycomb, the $G$-orbits of cells $X_1$ and $X_2$ cannot share colors. The subgroups $A$ and $B$, for example cannot be used to color cells in $X_2$ since these groups do not contain a stabilizer of a cell in $X_2$. Similarly, the subgroups $C$ and $D$ cannot be used to color cells in $X_1$ since these groups do not contain a stabilizer of a cell in $X_1$.

Perfect colorings of the bitruncated cubic honeycomb. To color the bitruncated cubic honeycomb, we first note that this honeycomb has only one type of cell – the truncated octahedron. The symmetry group $G$ of the bitruncated cubic honeycomb is cell-transitive. This means that we only have one $G$-orbit of cells to color. Since the symmetry group of the uncolored bitruncated cubic honeycomb is also $G=[4,3,4]$, we will use the list provided in Table 1 to choose the subgroups that we can use to color.

First, consider the truncated octahedron labeled $t$ in Figure 3(a). The stabilizer of $t$ in $G$ is $<p,q,=""r="">$ a group of type ${\mathbf{O}}_{\mathbf{h}}$. Aside from $G$, the groups $C=<p,q,=""r,=""srs="">,D=<p,q,=""r,=""srsqrs="">$ contain $<p,q,=""r="">$ and may be used to arrive at perfect colorings of the bitruncated cubic honeycomb. The colorings of the entire honeycomb using $C$ and $D$ are given in Figures 3(a) and (b), respectively.</p,></p,></p,></p,></p,></p,></p,>

Laigo, De las Peñas and Felix

(a) Figure 3: Perfect colorings of the bitruncated cubic honeycomb using (a) $C$ and (b) $D$ .

(b)

Semi-perfect colorings of the bitruncated cubic honeycomb

In this part of our work, we illustrate the construction of semi-perfect colorings of the bitruncated cubic honeycomb. In this case, the group $H$ consisting of elements that permute the colors in a given coloring is an index 2 subgroup of $G$ . Following the framework presented earlier, the first step is to choose an index 2 subgroup $H$ and determine the $H$ -orbits of cells. Then we proceed by coloring each $H$ -orbit of cells separately.

For our first example, let us construct a semi-perfect coloring where in particular the subgroup $C=<p,g,=""r,=""srs="">$ , permutes the colors in the given coloring. There are two $C$ -orbits of cells. As shown in Figure 4(a), the set of “patched” cells is the $C$ -orbit $X_1$ , while the set of white cells is the $C$ -orbit $X_2$ .

To color $X_1$ , we first choose our starting cell $t$ labeled in Figure 4(a). Note that ${\mathrm{Stab}}_C(t)=<p,g,=""r="">$ of type ${\mathbf{O}}_{\mathbf{h}}$ . The subgroup $D=<p,g,=""r,=""srsgrs="">$ of $C$ contains $<p,g,=""r="">$ , so that we let $J_1=D$ . We assign the color light grey to the set $Dt$ and black to the set $aDt$ , to obtain the coloring shown in Figure 4(b) ( $a$ is the two-fold rotation with axis $x$ labeled in Figure 4(b)).

To color $X_2$ , note that if $t^{\prime }\in X_2$ then ${\mathrm{Stab}}_C(t^{\prime })\le C$ , so we can use $C$ to color $X_2$ and $X_2$ gets a single color. Assuming the color white is used to color $X_2$ , the semi-perfect coloring we obtain is the coloring given in Figure 4(b).

As a second example, let us construct a semi-perfect coloring where $E=<p,rq,=""s="">$ is the group consisting of elements that will permute the colors of the coloring. In this case, all the truncated octahedra in the honeycomb will form one orbit of cells under $E$ , so we only have one $E$ -orbit to color. Consider the cell labeled $t$ in Figure 4(c) where ${\mathrm{Stab}}_E(t)=<p,rq="">$ . The subgroup $F=<p,rq,=""srsq="">$ of $E$ has $<p,rq="">$ as a subgroup, so that we may use $F$ to color the honeycomb semi-perfectly. The resulting semi-perfect coloring is shown in Figure 4(c).</p,></p,></p,></p,></p,></p,></p,></p,>

Coloring Uniform Honeycombs

(a)

(b)

(c) Figure 4: (a) Two $C$ -orbit of tiles of the bitruncated cubic honeycomb ( C = &lt; P, Q, R, SRS&gt; ); (b)-(c) Semi-perfect colorings of the bitruncated cubic honeycomb.

References

[1] Chung, K.W., H.S.Y. Chan, and B.N. Wang. Tessellations in three-dimensional hyperbolic space from dynamics and the quaternions, in Chaos, Solitons & Fractals, Vol 12, Issue 7, 1181 - 1197 (2001). [2] . Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics, in Computers & Graphics, Vol 25, Issue 2, 333 - 341 (2001). [3] Conway, J.H., H. Burgiel, and C. Goodman-Strauss. The Symmetry of Things. USA: A. K. Peters. (2008). [4] Coxeter, H.S.M. Regular Polytopes. New York: Dover Publication, Inc. (1973). [5] Coxeter, H.S.M. Wythoff’s Construction for Uniform Polytopes, in Beauty of Geometry: Twelve Geometric Essays. Dover Publications, Inc.(1999). [6] Cromwell, P.R. Polyhedra. Great Britain: Cambridge University Press (1997). [7] De Las Peñas, M.L.A.N., R. Felix, and G. Laigo. An Approach in Coloring Semi-Regular Tilings on the Hyperbolic Plane, in Renaissance Banff, Bridges: Mathematical Connections in Art, Music and Science Proceedings 2005, R. Sarhangi, ed. USA: Central Plain Book Manufacturing (2005). [8] De Las Peñas, M.L.A.N., R. Felix, and G. Laigo. On Subgroups of Tetrahedron Groups. In print. [9] De Las Peñas, M.L.A.N., R. Felix, and G. Laigo. Colorings of Hyperbolic Plane Crystallographic Patterns, in Z. Kristallogr 221, 665 - 672 (2006). [10] De Las Peñas, M.L.A.N., R. 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). [11] Deza, M and M. Shtogrin. Uniform partitions of 3-space, their relatives and embedding, in European Journal of Combinatorics Vol 21, Iss 6, 807 - 814 (2000). [12] Felix, R.P. A General Framework for Coloring Planar Patterns, a paper presented in National Research Council of the Philippines Symposium, February 7, 2004. [13] Grunbaum, B. Uniform Tilings of 3-space, Geombinatorics 4, 49-56, 1994. [14] Jablan, S. Edge Bi-Colorings of Regular Polyhedra, in Z. Kristallogr 210, 173-176, 1995.

Laigo, De las Peñas and Felix

[15] Laigo, G. On the Construction of Perfectly Colored Semi-Regular Tilings on the Hyperbolic Plane, a masters thesis, Ateneo de Manila University, Philippines. (2005).

[16] Laigo, G. On Tetrahedron Groups and their Subgroups, a PhD dissertation, Ateneo de Manila University, Philippines (2008).

[17] Meenan, E.B. and B.G. Thomas. Pull-up Patterned Polyhedra: Platonic Solids for the Classroom, in Sarhangi, R. et al (Editors), Bridges Leeuwarden: Mathematical Connections in Art, Music and Science, Tarquin Publications, Hertfordshire, 109 – 116 (2008).

[18] Umeda, T. and M. Fujita. Polymorphism of Carbon Forms: Polyhedral Morphology and Electronic Structures, in Phys. Rev. B Vol 51 No 19, (1995).

[19] Walo, L. On $(k,l)$ - chromatic colorings of the edges of platonic solids, a masters thesis, University of the Philippines – Diliman, Philippines (2003).

[20] Walo, L. Images of orbits and orbits of images in symmetry groups, a PhD dissertation, University of the Philippines – Diliman, Philippines (2008).

[21] Ye, R.S., J. Ma, and H.L. Li. Symmetric Tiling Patterns with the Extended Picard Group in Three-Dimensional Space, in Advances in Computer Graphics, Lecture Notes in Computer Science Vol 4035/2006. Berlin: Springer Berlin / Heidelberg. 486 – 493 (2006).