Many patchwork quilts have designs which are highly geometric. Some of these
patterns have been analyzed mathematically, ofen in terms of repeating
two-dimensional (wallpaper) patterns. Quilts can also illustrate other
mathematical structures. This talk will display a quilt which has the group
table of the dihedral group $D_4$ in its pieced design and which shows the
quotient of $D_4$ by its center in its quilting pattern. Other group-theoretic
quilt designs will also be presented.
The main design features of a patchwork quilt are the pieced top and the
quilting pattern. The top consists of pieces of fabrics sewn together to form a
design. Often the quilter will make several identical square blocks and sew
these together in a grid, forming a repeating pattern. The quilting consists of
lines of stitching across the top which form a secondary design.
The fiber artist Jinny Beyer has explored how two-dimensional symmetries are
most often used in quilting and how these symmetries can be used to design
novel patterns. Other quilters have used the Penrose Tiles and other
non-periodic tilings in designing quilts. Less traditional mathematical quilts
have unusual shapes, such as Mobius Bands or Klein Bottles. My work explores
how group tables can be used in the layout of a quilt top. The main observation
is that both group tables and quilt tops are often displayed as a grid of
squares.
The group table for $D_4$ (the symmetries of the square) can be thought of as an
eight-by-eight array; each of the 64 spaces in the array is labeled with one of
the group elements. I designed a square image which was not fixed by any of the
non-identity elements of $D_4$, and I sewed 32 copies of the design and 32
copies of the mirror image of the design (a total of 64 blocks). I then sewed
these blocks together in an eight-by-eight grid; the orientation of each block
was determined by applying the element of the dihedral group which appeared in
that location in the array. An illustration of this idea can be seen on a web
page belonging to Jim Conant:
http://www.math.cornell.edu/%7Ejconant/dihedral/dihedral.html
The quotient of the dihedral group by its center is isomorphic to the product
of two copies of $\mathbb{Z}_2$. Thus, each element of $D_4$ is in an
equivalence class labeled by one of the pairs (0,0), (0,1), (1,0), or (1,1).
One color of thread was used to indicate the first element of the pair, and
another color was used for the second element. Zeros were represented by loopy
stitching, and ones by zig-zags. These lines of stitching form a pattern over
the quilt top which gives the group table of the quotient group.
These ideas can be expanded to any group which can act on a square. For example,
the symmetric group $S_4$ can re-arrange the corners of a square.