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**Title:** The Symmetries of the Square form a Nonabelian Group
with Eight Elements (Migrations)

**Artist and Designer:** Amy F. Szczepanski

**Materials:** Cotton Fabric, cotton thread, and wool-polyester
blend batting

**Technique:** Machine piecing and quilting

**Artist’Äôs Description: **The basic motif uses the Flying Geese
pattern in a nine-patch block that has neither rotation nor reflection symmetry.
The quilt consists of 64 of these blocks (32 of the original block and 32 of
its mirror image) arranged in an 8-by-8 grid. The placement of the blocks was
determined by writing down the 8-by-8 grid of the group table of the dihedral
group (D_4 or D_8, depending on your preferred notation) and applying the group
element in each space to the quilt block. The quilting pattern shows the quotient
of the group by its center; this quotient is isomorphic to **Z**_2
x **Z**_2. The quilting pattern is sewn in two colors of thread
(maroon and light green) and with two designs (loops and zig-zags). The four
possible combinations of color and design are identified with the four elements
of**Z**_2 x **Z**_2, and the quilting pattern of each
block represents the coset it belongs to in the quotient.