Link to the main Beyond Numbers Table of Contents
Link to Teacher Manual Table of Contents
Link to Maryland Science Center
Link to Eisenhower Consortium
Cathy Brady's home site
cathysfiddle@yahoo.com

PURPOSE:

This explainer will present information to acquaint visitors with the variety of possible patterns that can be made with an asymmetric design. This explainer related to some of the Finding Patterns components of the Beyond Numbers Exhibit.

OBJECTIVES:

1. To provide the visitor with a basic understanding of tiling patterns.
2. To provide the visitor with examples of quilt patterns.
3. To provide the visitor with examples of reflection and rotation combinations.
4. To raise an awareness of the variety of patterns that can be created with a quilting square.
 

MATERIALS:
 

Photographs of quilt designs made by previous visitors "quilt" pieces - 8 of each and 8 mirror images pennies (20+) Polaroid camera and film magnetic white board on an easel chart of possible patterns
	These are Poleroid snapshots of visitors' efforts.	These are "Log cabin" and "Jack-in-the-box" pieces. Most of the pieces in the snapshots were "Clay's choice". The back of each piece has magnets attached. Mirror images have an x on the back.

reference books on hand:
Symmetry: A Design System for Quilt makers, by Ruth B. McDowell, ISBN 0?914881?78?7
 
 

fundamental region - "A fundamental region for a pattern must satisfy three conditions.
(A) The entire pattern is composed of identical copies of the fundamental region or its mirror image, without gaps or overlapping...
(B) Every copy of the fundamental region must be surrounded in exactly the same way by its neighboring fundamental regions...
(C) A fundamental region must be as small as possible, consistent with conditions (A) and (B)." - John Rigby

center of two-fold rotation- a point which is the axis of a turn

line of mirror symmetry- line on which a mirror could be placed perpendicular to the plane so that the mirror image is the same as the other half of the plane image.
 

BACKGROUND:

The structure of this activity comes primarily from a workshop designed by John F. Rigby, School of Mathematics, University of Wales College of Cardiff, Senghennydd Road, CARDIFF CF2 4AG, WALES, U.K. He generously gave me a copy of his brief notes on CREATING PATTERNS WITH A SQUARE TILE.
- Cathy Brady Math Specialist/Education Maryland Science Center

PROCEDURES:

Simplest Activity

1. Invite visitors to try to create a pattern that does not appear in any of the other photographs. If they create one, take a picture of them with it.

2. Show examples of fundamental regions. Patterns are named by traditional quilters Clay's choice (the one in the photographs), Jack-In-the-Box (the one that most resembles Clay's Choice) and Log Cabin.

 
3. Point out that none of these shapes could be simplified any further into symmetrical regions. There are no lines of symmetry, no reflections, flips, or turns. (which satisfies (C) in the definition.

4. Ask: Given a quilting square, how many distinct patterns can we make using the square as a fundamental region? (there are 36)
 

Learning More about Tiling the Plane

1. Use a translucent reflector to show how some of the sewn quilt pieces are mirror images of each other. The backs of the mirror images are marked with an ÏXÓ.

2. Place 4 identical pieces in a 2x2 grid. Show how a "windmill" motif can be made by turning the pieces at 90°, 180°, and 120°.     3. Starting with the fundamental piece in a different orientation, use the same method to create a second visually distinct image.

4. Show:

	and
The chain shows a mirror line and the penny shows a rotation point.

5. Have the visitor identify the mirror lines and lay a piece of chain to indicate the location of the lines. Have the visitor identify the centers of twofold rotational symmetry ("rotation point") and place pennies on that point(where there are circles)

6. Point out that any pattern with mirror lines in one direction and with twofold centers in between the mirror lines is said to symmetry type pmg. Show how both do that. They are the same symmetry type but not produced by the same method.

7. 16 different methods of arranging tiles produce 36 different patterns and exhibit 11 symmetry types.

8. Have visitors describe the transformations on fundamental regions that occur in the different photographs and in the patterns they are making.

BIBLIOGRAPHY:

• For All Practical Purposes: Introduction to Contemporary Mathematics, COMAP, Inc. 1988
• Grunbaum, Branko and G.C. Shepard; Tilings and Patterns, W.H. Freeman and Company, 1989
• McDowell, Ruth B., Symmetry: A Design System for Quilt makers,
• Washburn, Dorothy K. and Donald W. Crowe; Symmetries of Culture, University of Washington Press, 1988

AUTHOR:

Cathy Brady, Math Specialist
Maryland Science Center