“A mathematician, like a painter or poet, is a maker


of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” (G. H. Hardy)


“...in mathematics, the primary subject-matter is not


the mathematical objects but rather the structures in which they are arranged.” (M. Resnik)


Q


uilts are a familiar set of cultural artifacts for many people. Prized for their unique patterns and labor-intensive production, quilts also happen to be beautifully


mathematical. A primary unit of quilt making is the quilt block,


a square grid filled with patches of fabric according to certain rules. Quilters commonly refer to quilt blocks as 9-square or 16-square because of the underly- ing grid. In this exploration, we will consider the 16-square. Te reader is invited to extend (or should I say reduce?) the exploration to the 9-square.


Introduction: Build a Quilt Block To launch the investigation, I ask participants to imagine that they are planning a 16-square quilt block. Now, suppose that we restrict each quilt block to a 4-by-4 grid of squares. Each square can be filled in one of six ways, as shown in Figure 1. One possible quilt block configuration is shown in Figure 2.


Once participants understand the rules by which


quilt blocks can be constructed, I ask them to color a quilt block using a blank 16-square grid.


Compare Your Blocks, Find Your Partners Once everyone has created a quilt block or two, I ask participants the question, “From a mathematical per- spective, how might your quilt block be similar to or different from the one your neighbor has constructed?” Inevitably, this starts up a conversation about fractions, decimals, and percents. Aſter we agree that a classifica- tion scheme could be imposed using this framework, I try to steer participants to something deeper by asking, “Why do humans find quilts beautiful?” Usually, this is enough to get participants thinking about symmetry. I ask, “What symmetries do your quilts possess? Do all quilts possess the same symmetries?” From here, I ask participants to circulate around the room, compare quilt blocks, and find a “partner” whose quilt block possesses exactly the same symmetries.


Developing a Taxonomy Once participants have become familiar with the idea that all quilt blocks are not necessarily the same in terms of the symmetries they have, I bring the group back together and ask, “What sorts of symmetries can a quilt block possess?” Here, I hope to elicit the four possible line symmetries and the three possible turn symmetries. Tese symmetries, together with the iden- tity symmetry, are displayed in Figure 3 below.


Figure 1. Quilt block restrictions


Figure 2. An example quilt block


Figure 3. Available symmetries of the square American Institute of Mathematics · Summer/Autumn 2016 · MTCircular 05 ⊇


Stock Photo: Natalya Belinskaya, 123rf.com.


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