Artifacts for Stamping Symmetric Designs

It is well known that there are 17 crystallographic groups that determine the possible tessellations of the Euclidean plane. We approach them from an unusual point of view. Corresponding to each crystallographic group there is an orbifold. We show how to think of the orbifolds as artifacts that serve to create tessellations. 1. TESSELLATIONS. The history of human civilization gives an enormous number of examples of tessellations or, equivalently, tilings or mosaics. A tessellation is a design made up of one or more types of tiles (tessellas) that are repeated several times and are placed in such a way that they cover completely a surface without gaps or overlaps. Since humans started to build walls, floors, and ceilings, the placement of stones or bricks (as tiles) has naturally given rise to tessellations. The search for beautiful designs, the selection of shapes and colors of the tiles, and the systematic repetition of motifs produced symmetric patterns as examples of tesellations. Translation Rotation Reflection mirror Figure 1. Some rigid moves on a plane. Even though we have infinitely many ways to make symmetric designs, basically there are 17 “methods” which suffice to construct all symmetric tessellations on a plane. In fact, the crystallographer E. S. Fedorov proved in 1891 that, up to a natural equivalence, there are only 17 plane crystallographic groups. Having a tile on a plane, it is possible to translate it, to rotate it about a point, and to reflect it about an axis. These moves are rigid, i.e., they neither deform nor change the size of the tile but change the place of the tile on the plane; see Figure 1. Any other rigid move can be obtained using a combination of translations, reflections, or rotations. Taking these rigid moves as generators, we generate 17 different algebraic groups. They are called the two-dimensional or plane crystallographic groups. Each one of the 17 plane crystallographic groups gives one possible way of making a symmetric tessellation. See [8]. To study symmetric tessellations there are two points of view. On the one hand, we can use algebraic language, and in this sense we speak in terms of crystallographic doi:10.4169/amer.math.monthly.118.04.327 April 2011] ARTIFACTS FOR STAMPING SYMMETRIC DESIGNS 327 This content downloaded from 195.187.72.155 on Wed, 19 Feb 2014 05:43:17 AM All use subject to JSTOR Terms and Conditions groups; on the other hand, we are able to use geometric or topological language, and in this sense we speak in terms of orbifolds. Historically, symmetry was first understood from the point of view of algebra. Following the ideas of W. Thurston (see [11]) it is possible to understand the symmetry of designs by means of the concept of orbifold. This concept belongs to the world of geometry and topology. Both languages are equivalent. The advantage of the geometric language is that orbifolds can be visualized as artifacts that stamp the corresponding symmetric design. For each Euclidean crystallographic group there is basically one Euclidean orbifold. In the interest of rigor, at this point we would like to define some terms with more precision. A plane crystallographic group is a subgroup of the symmetry group of a tessellation of the Euclidean plane such that its translation subgroup is generated by translations in two independent directions. Two such groups are said to be equivalent if they are conjugate in the larger group of affine transformations of the Euclidean plane. In two dimensions these groups turn out to be equivalent if and only if they are isomorphic as groups. The idea of this equivalence is that mere changes of scale and shears should not give rise to inequivalent groups. The reader is referred to the website http://en.wikipedia.org/wiki/Wallpaper_group where the classification into 17 groups is explained in great detail. Given such a group G, a fundamental domain for G is a subset D of the plane (in this paper always a convex polygon) such that for any point x in the plane there is an element g of G with g(x) in D and such that for any two distinct points x and y in the interior of D there is no element in G with g(x) = y. The tiles in this paper are fundamental domains. As the group G acts on the Euclidean plane there is an equivalence relation on the points of the plane. Point x is equivalent to point y if there is an element in G with g(x) = y. The quotient space of this action, the set of equivalence classes of points in the plane, turns out to be a two-dimensional Euclidean orbifold and can be constructed from a fundamental domain by pasting edges. A two-dimensional Euclidean orbifold is a compact surface with or without boundary. Points in the interior that are not singular have neighborhoods that are isometric to Euclidean discs. Singular interior points are called cone points and have neighborhoods isometric to Euclidean cones with cone angle an integral fraction of 360, that is to say, spaces constructed by starting with a radial disc segment of angle 360/n and gluing the two linear sides. Boundary points that are not singular have neighborhoods isometric to Euclidean half discs. Singular boundary points have neighborhoods isometric to radial disc segments of angle 360/n. Alternatively, two-dimensional Euclidean orbifolds can be thought of as surfaces that are locally isometric to the quotient spaces arising from the actions of rotation groups or dihedral groups on a Euclidean disk. We can illustrate these concepts with a couple of examples. Consider the tessellation of the plane by squares with integer coordinate vertices. Let G1 be the group generated by reflections in the lines x = an integer and y = an integer. It can be shown that the translation subgroup T of G1 is generated by translations by 2 in the x and y directions and that a fundamental domain for G1 is the unit square {(x, y) : 0 ≤ x, y ≤ 1}. No two points in the square are equivalent, so the quotient orbifold is the square itself. The vertices of the square are singular points. Later, in Section 3.1, you will see that this orbifold is D2222; see Figure 6. Let G2 be the orientation preserving subgroup of G1. It can be shown that G1 is generated by 180-degree rotations about points with integer coordinates and that a fundamental domain for G2 is the rectangle {(x, y) : −1 ≤ x ≤ 1, 0 ≤ y ≤ 1}. There 328 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118 This content downloaded from 195.187.72.155 on Wed, 19 Feb 2014 05:43:17 AM All use subject to JSTOR Terms and Conditions are equivalences on the boundary of this rectangle. The point (−1, y) is equivalent to (1, y), (−x, 0) to (x, 0), and (−x, 1) to (x, 1). Gluing points to their equivalent points we obtain a topological sphere. Thus the quotient orbifold is topologically the 2-sphere. There are four singular points, all with cone angle 180. Later, in Section 4.3, you will see that this orbifold is S2222. We invite the reader to see [5] and [6]. Table 1 contains the list of the 17 plane crystallographic groups. On it we find both the notation used in the literature of algebra and the notation used in the literature of the geometry of orbifolds. See [2] and [9]. Table 1. The 17 crystallographic groups. S236 D 2 3 6 S244 D 3 3 3 D 3 3 D22 D 2 2 2 2 D 2 4 Crystallograpic Groups Orbifolds