David A. ReidSchool of Education Acadia University Wolfville NS B4P 2R6 CANADA
We then look for other transformations that leave the pattern unchanged. We find three types: vertical reflections through the centre of each brick, horizontal reflections through the centre of each brick, and 180° rotations. At the beginning diagrams (e.g., fig. 2) are sufficient to describe the locations of mirror lines and centres of rotation . I am always modifying the activities I use, and so when I introduce precise language depends on the students and on the way the activities have developed. When the time comes I distinguish between elements of the bricks, and elements of the pattern. Most importantly, the four segments bordering a brick are its sides, but the sides of a brick might contain several edges of the pattern, an edge being a segment joining two adjacent points (this terminology is used by Grünbaum and Shepard [1987] to describe tilings of the plane). For example, in Running bond each brick is bordered by six edges, as the upper and lower sides of each brick are made up of two edges. The rotations occur at the midpoints of edges and at the centre of each brick (see fig. 2). Those at the midpoints of vertical edges and at the centre of each brick are compositions of the two reflections. Fig. 2. Symmetries of Running bond (only one example of each type of symmetry is shown)
After Running bond, the next monohedral pattern we encounter on our tour is the one called "Stack bond" (see Photograph 2).
The symmetries of Stack bond are shown in fig. 3. They include reflections vertically and horizontally though the centre of every brick, and along every edge. There are also 180° rotations that are compositions of two reflections, at the intersection of the reflection lines.
Bricklayers classify the most common brick patterns into three groups. Patterns made up only of stretchers (e.g., Running bond and Stack bond) are called Stretcher bonds. Patterns in which courses of stretchers alternate with courses of headers are called English bonds. Patterns in which the courses consist of alternating stretchers and headers are called Flemish bonds (see Photograph 4). The arrangements shown in Photographs 1, 3 and 4 are the prototypical patterns of each type. Other patterns in a group are often given names based on the name of the group (e.g., English Cross bond and English Garden Wall bond are types of English bond, see Photographs 9 and 12). Not all brick patterns have standard names (two different patterns are referred to as "Dutch bond"), and verbal descriptions found in texts on bricklaying are sometimes ambiguous. I have developed a notation to describe patterns more exactly. The letters S and H indicate the pattern of stretchers and headers in each course. Numbers indicate how much each course is shifted over or offset from the previous course, as a unit fraction of the total length of the pattern of headers and stretchers. Table 1 shows the notations for the four patterns illustrated above.
Table 1. Notations for four common brick patterns
Note that Raking Stretcher bond has a "handedness", either it shifts to the right as one moves down one course, or it shifts to the left as one moves down.The most common offsets in Raking Stretcher bond are one fourth and one third of the length of the stretcher (S4 and S3). The symmetries are the same whatever the offset (see fig. 5). They are four rotations of 180 degrees, one at the centre of each brick and three at the midpoints of the edges. As any one of the four is the composition of the other three, only three need to be marked to completely define the set of transformations. This pattern provides an interesting basis for a classroom investigation of the composition of rotations. Composing any two rotations produces one of the translations in the pattern and composing three rotations produces a fourth. Fig. 5. Symmetries in a Raking Stretcher bond (S3 in this case) In Raking Stretcher bond the offset is in the same direction in each course. My tour also includes a brick pattern in which the offset alternates left and right (see fig. 6 and Photograph 6). As with Raking Stretcher bond offsets of one third and one fourth are most common. Unlike Raking Stretcher bond these patterns have no handedness.
Fig. 7 shows the symmetries of One-Third-Running bond (and related patterns like S4S). They are two 180 degree rotations about the midpoints of the horizontal edges, and a reflection across a horizontal line through the centre of each brick. Fig. 7. The symmetries of One-Third-Running bond (S3S) and related patterns S3S can also be the starting point of an interesting classroom investigation. All the patterns on the tour are periodic, but S3S is the first one in which two bricks must be translated to produce it. In the other patterns we have encountered only a single brick has to be translated to produce the entire pattern. Grünbaum and Shepard [1987] call the parallelogram defined by the two basic translation vectors of a periodic tiling, the period parallelogram of that tiling. In fig. 8 the period parallelograms are shown in blue. For S0, S2 and S3 the period parallelogram has the same area as one brick (see fig. 8, a, b, c). For S3S the period parallelogram has the area of two bricks. The bricks shaded in light green in fig. 8d are the images of the brick shaded in dark green under the two translations.
In Basketweave there are two perpendicular reflection lines through the centres of every module, as well as a 90 degree rotation centred at the corners of the modules (see fig. 9). The figure shows only one reflection line as the composition of the reflection shown and the 90 degree rotation produces the perpendicular reflection. Fig. 9. Symmetries in the Basketweave pattern In Herringbone the symmetries are harder to identify. My students are usually quick to observe that there are 180° rotations (See fig. 10a) at the centre of some edges. By copying the pattern on tracing paper and flipping it over, they can see that there is a reflectional symmetry of some sort, but investigations with mirrors make it clear there are no reflection lines. Careful work with tracing paper then leads to the discovery of perpendicular glide reflections (see fig. 10b).
Fig. 10. Symmetries in the Herringbone pattern A further investigation of the Herringbone pattern can explore how the rotations are produced by the perpendicular glide reflections.
The six monohedral patterns my students encounter on our tour are those listed by Coxeter (1969) in his brief discussion of symmetries in brick walls. They are examples of the plane symmetry groups cmm, pmm, p2, pmg, pgg, and p4g. Fig. 11 shows another brick pattern made only with stretchers that has symmetry p1. One question for further exploration is: - Are p1 and the six symmetries Coxeter (1969) lists the only ones possible in brick patterns involving stretchers only? Can you prove this?
Fig. 11. A brick pattern representing plane symmetry group p1 Many brick patterns are not made up of stretchers only, but instead include headers and stretchers. This raises a second question: - Are there other symmetries possible in brick patterns involving both stretchers and headers? If not, why not?
These two questions provide the basis for mathematical activity, including proving, in a context that is more directly accessible to students than traditional Euclidean geometry tasks. A more extensive exploration of plane symmetry is possible if non-rectangular paving bricks and tiles are considered. I usually give my students a taste of the possibilities during the tour, by examining a paving of octagons and squares, but time does not allow a complete treatment.
A possible classroom activity is to compare the most frequently occurring brick patterns with a class in another community. Email has now made such cross cultural comparisons simple and quick and they are a source of many opportunities for mathematical explorations (see http://www.stemnet.nf.ca/~elmurphy/emurphy/math.html for another example).
A brick pattern like English Garden Wall bond (S2S2S4H, see Photograph 12) is 4-isohedral. The headers form one isohedral set. The stretchers in courses adjacent to headers form another isohedral set. The stretchers in courses one course away from the headers form a third isohedral set. Finally, the stretchers along the horizontal mirror line between the headers form a fourth isohedral set. A brick pattern is said to be isogonal if the symmetries of the pattern can map any edge onto any other edge. Because the edges in brick patterns are usually of two or more lengths, most brick patterns are not isogonal. Stack bond, Stretcher bond and Basketweave are 2-isogonal, as all their long edges can be mapped onto any other long edge and all their short edges can be mapped onto any other short edge. In Herringbone all the edges are the same length, but the pattern is also 2-isogonal, as the edges that contain the centres of the rotations (see fig. 10a) cannot be mapped onto the other edges. An interesting exploration is to try to create a brick pattern that is isogonal, or to prove that this cannot be done.
Photograph 13. English Cross bond showing cracking along joints These two modes of failure suggest two ways to quantify the strength of a bond. The resistance to vertical cracks can be quantified as the ratio of the length of the shortest possible crack along joints between bricks to the vertical distance. For example, in Photograph 13 the crack travels down 4 courses and has a length of 7 (using the height of each brick as the unit of measure) so English Cross bond (S4H2) has a ratio of 7:4. Stack bond (S0), on the other hand, has a ratio of 1:1, and because of its weakness it is not used structurally. The resistance to the separation of layers can be quantified by determining what percentage of the pattern is occupied by headers. The more headers, the greater the strength of the bond between the surface layer and the layer behind it. The extreme cases are Header bond (H2) which is 100% headers, and Stretcher bonds (0% headers) which are used only when the bricks form a single layer over some other material (in this case the bricks are connected to the materials behind with metal ties embedded in the mortar between the bricks). These numbers oversimplify the problem of measuring the strength of a brick wall, but they serve as an introduction to two important issues and the use of mathematics in analysing structures.
[2] I encountered this name for this pattern on the
web site of the Acme Brick Company (http://www.brick.com).
I would be interested in hearing from anyone who knows of other
names for it.
Grünbaum, B. and Shepard, G. 1987. Murphy, E. 1995 E-mail math [Online]. http://www.stemnet.nf.ca/~elmurphy/emurphy/math.html
[referenced 8/10/2004]. Originally published in Reid, D. A. 2003a. Brick Patterns [Online]. http://plato.acadiau.ca/courses/educ/reid/Geometry/brick/Index.html [referenced 8/10/2004]. Reid, D. A. 2003b. Frieze Patterns [Online]. http://plato.acadiau.ca/courses/educ/reid/Geometry/Symmetry/frieze.html [referenced 8/10/2004]. Joyce, D. E. 1997. Wallpaper Groups [Online]. http://www.clarku.edu/~djoyce/wallpaper/ [referenced 8/10/2004]. Washburn, D. and Crowe, D. 1988.
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