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 hand-quilted quilts not only for her daughter and two grandchildren, but she also hand-quilted 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 hand-pieced quilts made
by two Tennessee quilters, Hollys 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.)
Symmetry and Pattern
Asymmetry and Symmetry Breaking
The Four Basic Symmetries
Wallpaper/Field Patterns or Wallpaper Groups
From a quilt book (Malone, 1985), we chose four 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 (but 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--basically our conjectures about patterns in the creation of symmetry patterns!
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 2-fold cyclic
rotation center (i.e. no mirrors; for examples, see Chain Link and Signature.)
4*2 results from reflection when the original block contains a 4-fold cyclic
rotation center (see Flywheel.) Furthermore, a composition of reflection and
rotation needed to produce another example 4*2 (see 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 and 4*2
are not common.
However, if there are 2 mirrors, i.e. a 2-fold 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), its still not surprising that 2*22 is not often produced. (Translation with a 4-fold 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 2-fold 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 4-fold cyclic rotation center, the symmetry pattern 442 seems to result (for an example, see Flywheel. Note also that in the case of Chain Link, rotation of a block with a cyclic 2-fold center produced 442.) If the original block is completely asymmetric, then the symmetry pattern º will result from translating the block (for example, see Sailboats in Hawaiian Sunset.) We saw all of these symmetry patterns in the real quilts, because translation and rotation are both relatively "easy" (and possible) ways to arrange blocks.
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
xx, *2222, 22*, 22x
We thought we might at least produce *2222! We were not surprised that we didn't produce symmetry patterns with anx 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 Full-size 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
Math Forum @ Drexel
David Joyce, Clark University
AOL Quilting Community
Quilts: Shape and Space in Geometry
Quilts and Fabric Art
What are Math Quilts?