# by Holly Anthony & Amy Hackenberg

## for MATH 7210, Dr. McCrory, spring 2002

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!

# Purpose & Process of the Project:

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, 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.

# Websites on Symmetry

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

Border Patterns

Wallpaper/Field Patterns or Wallpaper Groups

# Wallpaper Patterns

## Asymmetric

Random Butterflies

## *x

if you were examining the pattern as *632, scroll down

Around the World

Blue Pattern

## 0

Sailboats in Hawaiian Sunset

## 2222

Green Pinwheel

Carpenter's Wheel

Double Wedding Ring

Butterfly

Red Pinwheel

Log Cabin

Flying Geese

Red Geese

Winner's Circle

Red Star

Texas Star

# Strip Patterns

## TR (22i)

Red Star

Sailboats in Hawaiian Sunset

Red Star

## THG (i*)

Butterfly

Flying Geese

Random Butterflies

Around the World

Flying Geese

Green Pinwheel

# Investigating Block Designs

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.

Signature

Flywheel

Plainsailing

Signature

Flywheel

## 4*2

Signature

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 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), it’s 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.) 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.

# 3- and 6-fold centers

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

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)

# Our Quilt Resources

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

# Links to Other Quilt Websites

AOL Quilting Community

Quilts: Shape and Space in Geometry

Quilts and Fabric Art

What are Math Quilts?