A Brief Word from Holly:
As a young girl, I had the good fortune of being blessed
with two very talented grandmothers who lived nearby. My Granny Gene loved to
piece quilts, especially quilts with lots of small pieces. My family never really
knew where her patience came from! My other grandma, Granny Laura pieced and
handquilted quilts not only for her daughter and two grandchildren, but she
also handquilted for others as a source of additional income. Both spent 'many
a day' piecing and quilting quilts. I now have twenty or so beautiful quilts
made by my grandmothers that I can use and pass on for many generations to come.
The memories that they envoke make them priceless!
Grandma Laura Garrett with Red Star quilt.
For this project, Holly wanted to look at her quilts in a mathematical context and analyze their symmetry patterns. Amy joined enthusiastically in this task and this webpage is a result of our endeavors. We hope you enjoy it!
To investigate the strip, wheel, and wallpaper symmetry patterns
that we find in a collection of approximately twenty handpieced quilts made
by two Tennessee quilters, Holly’s grandmothers. In particular, we have:
1. Identified and analyzed strip, wheel, and wallpaper symmetry patterns in
the collection, looking both at the quilt (a collection of blocks which
in turn are made of pieces) and the quilting (the stitching itself) and/or
pieces, depending on the quilt.
2. Identified the fundamental region of the symmetry pattern and analyzed how
it relates to the block used to make the quilt.
3. Investigated whether the quilt symmetry and quilting symmetry are in harmony
or disharmony (that is, we looked at the symmetries of the block, the collection
of blocks, and the quilting to examine their compatibility as a symmetry pattern.)
In some cases we looked at the pieces or block in comparison to the entire quilt.
4. Identified the strip and wallpaper symmetry patterns that are “missing”
from this collection, and speculated about why. This aspect of the project led
to investigating the generation of quilts using different blocks and other sources
(books and a quilter!) about making quilts.
For background information on symmetry and wallpaper patterns before you look at the quilts, check out the links below. (Some of these links were written with reference to oriental rugs, but are appropriate for our purposes as well.)
Asymmetry and Symmetry Breaking
Wallpaper/Field Patterns or Wallpaper Groups
Asymmetric 
*xFlowerbasket 
*442 
0 
2222 

** 
TR (22i) 
TVRG (2*i) 
From a quilt book (Malone, 1985), we chose some block designs that we thought might allow us to generate symmetries we hadn't seen in Holly's quilts. We generated "quilts" (16 blocks) on The Geometer's Sketchpad using various isometries or combinations of isometries. We were able to generate the symmetry patterns listed below, as well as *442 (we did not include *442 below because we had so many examples of that pattern in the real quilts above.) Below the examples shown we have described a summary of our findings.
2222 
2*22 
4424*2 
Summary of our Findings:
In general, it seems that 2*22 comes from reflection
of the original block when the original block contains only a cyclic rotation
center (i.e. no mirrors; for examples, see chain link, flywheel, and signature.)
Since quilts are not often made through reflecting blocks (you cannot just “flip”
a group of pieces stitched together!), it is not surprising that 2*22 is not
common.
However, if there are 2 mirrors, i.e. a 2fold dihedral center, (for example,
in plainsailing), then translation can produce 2*22. Quilt blocks are often
translated to produce quilts! But since it seems more common for quilt blocks
to have 4 mirrors (rather than 2), it’s still not surprising that 2*22
is not often produced. (Translation with a 4fold dihedral center in the original
block appears to produce *442.)
The symmetry pattern 2222 seems to result from translation of the original block
when the original block contains only a 2fold cyclic rotation center (i.e.
no mirrors; for examples, see chain link and signature. In our real quilts above,
see Green Pinwheel.) If the original block contains a 4fold cyclic rotation
center, the symmetry pattern 442 seems to result (for an example, see flywheel.)
If the original block is completely asymmetric, then º will result (for example,
see Sailboats in Hawaiian Sunset.)
A composition of reflection and rotation was needed to produce 4*2 (see signature)—again,
not a common way to assemble blocks when making a quilt.
Amy met a mathematician and quilter, Jo Hoffhacker (in UGA's
Mathematics Department), who has made quilts that aren't based on squares. However,
in general, 60 degree and 30 degree angles (triangles) are less common because
they can be more difficult to quilt with; certainly they are less common in
tradtional quilting because they are not constructed with traditional blocks,
which are generally squares. Thus we did not produce:
333, 3*3, *333 (except for our example of Rumple quilt below, ignoring color),
or
632, *632 (except for the field of hexagons in Flowerbasket, ignoring color)
*333 Rumple, by Jo Hoffhacker
*632 Flowerbasket (ignoring color!)
Though we experimented a good deal, we did not produce the following
symmetry patterns:
xx, *2222, 22*, 22x
We thought we might at least produce *2222! We were not surprised that we didn't
produce symmetry patterns with an“x” because glide reflections would
not be common when assembling quilt blocks. However, we did find *x, so we should
not rule out the possibility entirely!
We also did not see these border patterns:
T (or ii), TG (or ix), TV (or *ii)
Beyer, J. (1999). Designing tessellations. Lincolnwood, IL: Contemporary Books.
Bonesteel, G. (1982). Lap quilting with Georgia Bonesteel. Birmingham: Oxmoor House Incorporated.
Malone, M. (1985). 500 Fullsize patchwork patterns. New York: Sterling Publishing Co., Inc.
Singer. (1990). Quilting by machine. Minnetonka, MN: Cy DeCosse Incorporated.
Venters, E. & Ellison, E. K. (1999). Mathematical quilts. Emeryville, CA: Key Curriculum Press.
Wagner, D. (1995). All quilt blocks are not square. Radnor, PA: Chilton Book Company.
Our links to the Websites on Symmetry come from
Quilts: Shape and Space in Geometry