SYMMETRY EXPLAINER

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 symmetries made by reflection.

OBJECTIVES:

1. The visitor will experiment with patterns made by slides, flips, turns, and glide reflections.
2. The visitor will understand that symmetry is more than single mirror reflections.

MATERIALS:

red Reflectas (3 per person)
Cuisenaire rods white (bone color), red, lime green, yellow, purple (at least 4 of each)
laminated activity cards:
addition puzzle, Timothy and Rebecca, different styles for a head, reflection puzzles (lettered on the back in order of difficulty)
water soluble markers
spray cleaner and rag
hinged mirror
40 cards of asymmetrical design (half arrows and half mushrooms)
paper strips of patterns from a variety of cultures around the world
reference books:
some of the mirror activity books by Tarquin that are also sold in the Science Store
Washburn, Dorothy K. and Donald W. Crowe; Symmetries of Culture, University of Washington Press, 1988

DEFINITIONS:

center of two-fold rotation- in strip symmetry, a point which is at the axis of a 180 degree turn
Cuisenaire rods - a commercial name, known by elementary
school teachers, for a standard set of rods made by the
Cuisenaire Company. Each length of rod, 1 cm to 10 cm, is
color coded. As a mathematics manipulative learning aid, the Cuisenaire Rods have multiple uses. They are most traditionally used to teach properties of addition, multiplication and fractions. We use them to teach about Fibonacci Numbers, symmetry and minimum spanning trees.
dimension - the number of parameters needed to specify the position of a particular point in space
frieze - a decorative band with lettering or sculpture. In symmetry, it refers to one dimensional patterns.
fundamental region - the smallest design that, when repeated in one or more directions, composes a pattern. In periodic tiling this repetition region is always surrounded by other fundamental regions in identical manner as the first.
"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
glide reflection - in symmetry, a transformation that is a combination of "slide" and "flip," a composite of reflection over vertical and horizontal lines
horizontal reflection - a symmetry transformation
which produces a mirror image across a
horizontal line
n-fold symmetry - a rotation that can bring a
pattern back to its starting point after n
transformations
one dimensional - able to move or grow along
one axis only; having one degree of freedom
periodic - appearing at regular intervals; in reference to tiling, having patterns appearing at regular intervals
Reflecta is the commercial name for the red translucent reflecting device.
reflection - a mirror image, also called a "flip"
rotation - a repeated image that appears to be a 180 degree turn of the original, also known as a "turn"; a composite of two reflections over intersecting lines
tiling - an infinite plane completely covered (with shapes) without gaps or overlaps
Penrose tiling - a kind of quasi-periodic tiling devised by mathematician Roger Penrose.
translates - slides - for very young children, moves over in the same way. For older children and adults, the result of two mirror reflections along one direction. (see the strip symmetry explainer)
translation - a motion that involves sliding without rotation or reflection; a composite of two reflections over parallel lines; a repetition of an image with the same up and down orientation, also known as a "slide,"
turn - in symmetry, it is a transformation that can be
made by reflection over intersecting lines. With
frieze symmetry the intersecting lines are
perpendicular.
two-fold rotation - another word for turn as in the
diagram above
two dimensional - having two degrees of freedom,
able to move or grow both in length and height
vector¹ - a point on a plane or the shortest path from the origin to that point.
vertical reflection - a symmetry transformation which produces a mirror image across a vertical line

BACKGROUND:
The subject of symmetry is not easy! But there is a wide variety of activities in this explainer. Think of this as more than one explainer. Some are pretty rigorous, some come easily and with intuition. You and the visitors are not expected to do them all, but if you have learned all the activities, you will be prepared for questions. The materials are not bulky, so keep them on hand, even if you don't plan to use them.
A quilt is a two dimensional pattern. It has length and width. There are at least two vectors of translation in a two dimensional pattern. A frieze (also known as a strip, or a border) is a one dimensional pattern. There can only be one vector of translation.

In addition to the translations, which are transformations present in every frieze pattern, there are four other transformations of a region - (see rules below). These can be performed singly or in combination. There are 16 combinations, but some duplicate themselves and some are impossible under the rules of translation. After elimination there are seven combinations left.

The attached color pages show most of the possible combinations and how they are eliminated to leave seven.
So what are the rules? In non-mathematical language:

A design can repeat in only one dimension (it may have only one vector of translation).
Each repeated design must be next to designs that have undergone the same transformation (process of change).
The transformations that are allowed are translations, reflections (horizontal or vertical), turns, and glide reflections ( horizontal reflection/ translation combinations).
Designs that differ in the pivot point of rotation or length of translation will appear to be different, even when they are classified in the same symmetry group.

One interesting and direct application of this mathematics is in anthropology. Particular patterns of pottery borders, clothing edges, wall friezes, etc. are often favored by certain cultures. There patterns appear with greater frequency and some seem not to appear at all. Along with other evidence, the symmetry classification of an ancient pottery shard can help determine its origin. Did this piece of pottery come from the region in which it was found? Or does it indicate far-reaching trade?

EXHIBIT REFERENCE:
In addition to the following stand-alone exhibit components, there is an explainer on Quilts that is about 2-dimensional symmetry patterns.

Alhambra Mosaics 2 Manual Interactives

Alhambra Nook Escher

Alhambra Nook Geometric Grids Manual Interactive

Seven Strip Interactives

Hall of Mirrors

Periodic Tiling Manual interactive

Penrose Tiling Manual interactive

4 Color Mapping 2 Manual Interactives

PROCEDURES:

THOUGHT PROVOKERS:
REFLECTIONS ON REFLECTIONS

Show visitors the card with the "incorrect" addition problem on it. Show how a mirror makes the problem read "correctly".

Show visitors the card with the names Timothy and Rebecca. Have them look at the names in a mirror.
Why is TIMOTHY not reversed?
Which of the letters of the alphabet look the same when seen in the mirror?
Which will not look the same no matter how you orient them?

Show some of the "ambigram" examples by Douglass Hofstadter. (We have permission to use them as photo copies.)

N-fold symmetries: let the visitor place cuisenaire rods in between the hinged mirrors and experiment with how many images she can make.

INVESTIGATIONS WITH MULTIPLE REFLECTIONS :
LEARNING TRANSFORMATIONS (MOTIONS)

Show pattern card 1a and have green, red and white cuisenaire rods available.
Show how the pieces fit over the design on the left.
Challenge the visitor to use a reflector to find where it is placed to produce the design on the right.
Point out that this is a reflection (also known as a flip) and that it is placed on a line of reflection.

Have the visitor do the same for 1b and explore why the reflection is further away. (The mirror line of reflection is further from the fundamental region.)

Suggest that two reflections can be used to created a translation (also known as a slide).
Present card 2a and the yellow, green and white Cuisenaire rods and two Reflectas.
If the visitor needs help, propose that the slide is a reflection of a reflection.
Ask the visitor to locate the lines of reflection. there are many possibilities but all reflection line choices are parallel.

Have the visitor identify lines of reflection for card 2b.

Suggest that two reflections can be used to create a turn (Known as a "turn")
Present card 3 and the red, yellow, and purple cuisenaire rods and 2 reflectors.
If the visitor needs help, propose that the turn is a reflection of a reflection (the two lines of reflection intersect ? in this case at 90 degree angles)
Ask the visitor to locate the lines of reflection.

Suggest that three reflections can be used to create a 'glide reflection", which is a combination of a slide and a flip.
Present card 4 and the purple, green and red Cuisenaire rods and 3 reflectors.
Ask the visitor to locate the lines of reflection.

Present the 40 laminated cards of asymmetrical design. Challenge the visitor to make as many combinations as they can of flips (one reflection), slides (two parallel reflections), turns (two perpendicular reflections) and glide reflections (two parallel reflections and one perpendicular).

translation

vertical reflection

horizontal reflection

turn

turn (at a different point of two-fold rotation)

horizontal + vertical reflection and rotation

translation + horizontal reflection
called a glide reflection

translation + vertical reflection + rotation + glide reflection

DRAWING ACTIVITIES

There are four cartoon heads, two cats and two bears, one of each is for right-handed drawers and one of each is for left-handed artists. Let the visitor use the Reflectas to help them draw the undrawn side of the cartoon face.
Let the visitor draw a half head freestyle and use the Reflecta to complete the head.

FRIEZES FROM OTHER CULTURES

If you have Washburn and Crowe's Symmetries of Culture, show it to the visitor. Let the visitor copy and color some of the patterns in the front part of the book and discuss the symmetry types.

BIBLIOGRAPHY:

Coxford, Arthur et al; Geometry, University of Chicago School Mathematics Project, Scott, Foresman and Company

Coxford, Arthur. F. Jr.; Geometry from Multiple Perspectives, NCTM, Addenda series, Grades 9-12

For All Practical Purposes: Introduction to Contemporary Mathematics, COMAP, Inc. 1988

Grünbaum, Branko and G.C. Shephard; Tilings and Patterns, W.H. Freeman and Company, 1989

Hofstadter, Douglas R.; Metamagical Themas: Questing for the Essence of Mind and Pattern, Basic Books, Inc., 1985

Kappraff, Jay; Connections, McGraw?Hill, Inc. 1990

Kay, Cynthia S; of University of South Carolina at Spartanburg, "Slides, Flips and Turns with Kaleidoscopes, Escher Tessellations, and Navajo Fabrics" , a workshop presented at the National Council of Teachers of Mathematics Southern Regional conference in Richmond, Virginia, February 24-26, 1994

Theissen, Richard; "Reflection and Symmetry", a series of articles appearing throughout the 1989 issues of the AIMS newsletter

Washburn, Dorothy K. and Donald W. Crowe; Symmetries of Culture, University of Washington Press, 1988

Wiltshire, Alan. The Mathematical Patterns File, Tarquin Publications 1988

AUTHOR:

Cathy Brady, Math Specialist
Maryland Science Center

¹ taken from the glossary of Fractal Vision, by Dick Oliver, Sams Publishing, 1992

MARYLAND SCIENCE CENTER - Beyond Numbers

REFLECTIONS SYMMETRY 12/16/00