classes to the list that had been “discovered” through construction. But the question remains: Are there more?


Are There More? Part II To investigate this question further, I encourage par- ticipants to “imagine” other possible symmetry groups. For example, “Could a D-D-90 quilt exist?” Usually participants come to the conclusion that if a quilt block has 90-degree rotational symmetry, then it must have 180-degree rotational symmetry—so, no such group exists. Tis naturally leads to the idea that a quilt block’s symmetries can be composed to produce other symme- tries—a 90-degree rotation composed with a 90-degree rotation is a 180-degree rotation. Tis sort of investiga- tion can lead to many interesting results. Past partici- pants have discovered that a quilt possessing any two line symmetries automatically inherits a rotational sym- metry (disallowing types such H-V or H-V-D or D-D). Participants are also quick to note that any quilt with 90- turn symmetry inherits all turn symmetries (disallowing types such as 180-270 or 90-270 or H-V-90). Proceed- ing via elimination, we can conclude that there are only seven quilt block “types” that possess at least one sym- metry. Each of these types is found in the list above.


Conclusion A nice way to conclude the activity is to look at some ac- tual quilts through the lens of symmetry groups. If you have no quilts on hand, pictures of quilts are easily found on the internet. Questions for further study include:


• How many quilts can be constructed according to the quilt block restrictions in Figure 1? (Here you might want to look for an upper bound first.)


• How many quilts that possess all possible symmetries can be constructed according to the quilt block restrictions in Figure 1? (Consider black and white “negative” images for an even more interesting result.)


• How many quilts that possess all the rotational sym- metries (and no others) can be constructed accord- ing to the quilt block restrictions in Figure 1? (You might want to answer the previous question first!)


• Is the number of D-D-180 quilt blocks the same as the number of H-V-180 quilt blocks? (Te answer might surprise you.)


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American Institute of Mathematics · Summer/Autumn 2016 · MTCircular 07 ⊇


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