These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop (http://www.mathdance.org/store.html)

Sorts of Symmetries

Symmetries found when people stand in a line

Time One hour

Groups: duets, trios, quartets

Materials: none

Space: large, free of desks or chairs

Concepts: symmetry, translation or slide, rotation, reflection or mirroring, glide reflection

Related Activities: Threesies, combining symmetries

Using movements from waving a hand to full-body motion, the students learn to distinguish between kinds of symmetries. Most of the activities in this section will be usable for all ages and movement skil levels. Trained dancers can strive to perform the mirroring improvisations with great precision and beauty, but this activity might also look striking when done by untrained movers or by children. Younger children may lack the physical concentration to do the more involved movement work.

7-1. Try this (x minutes)

Facing the class, wave your right hand: start the wave at shoulder height and reach upwards as you wave. Then say, " Do the opposite." You will probably get a variety of responses. For example, one person might wave their left hand, another person might wave their leg, and yet someone else might shake a fist. Make a note of the responses on the board. Notice that asking for the opposite seems, at least on the on the surface, to be a more open-ended question than asking for the same.

Now do something very different from waving your hand: for example, write a note on a pad of paper. Then say, "Do the opposite of this in the same way that you did the opposite of the wave." Again note the responses.

Choose one of the responses and say, "Now do the opposite of that." Continue building a sequence of opposites by choosing one of the class'es responses each time.

7-2. What is Symmetry?

Most definitions of symmetry refer to the ways in which a pattern repeats, is balanced, or is similar in shape to itself.

One mathematical definition of symmetry might be: a distance-preserving transformation of the plane or space which leaves a given set of points unchanged. This is like saying that the entire space is actually moved in such a way that one object ends in the same place and orientation as another object, with no overall change visible. For example, if four square dancers standing in a circle move one quarter turn clockwise, then they will end up making the same shape in space as when they began, each person taking the place of the person to their left.

Symmetry is found throughout dance, music, and the visual arts, but it is also crucial in understanding wide areas of contemporary science from crystal structure to the intricacies of the quantum theory.

Symmetries in dance and music also include symmetries in time: one dancer or instrument repeats a phrase a certain number of beats after another. Or the phrase is repeated in the reverse order.

Other symmetries in dance include those derived from the similarity between the human body's silhouette as seen from the front and the back. Or choreography might make use of the resemblances in shape and motion between the arms and the legs, as in a cartwheel

7-3. Symmetries In a Line

Most of us feel we can tell when symmetry is present, but can we differentiate the kinds of symmetry? Here are four kinds of symmetry that appear when people stand in a row.

(1) Translation or sliding symmetry:

This symmetry involves the same shapes facing in the same direction. The leader begins with simple arm motions. Move slowly. Carefully add movements of the torso. Take one or two steps, being careful to "telegraph" the steps so the others can follow. Add slow and simple twists of the body. Change leaders and try again.

(2) Reflection or mirror symmetry

This is the kind of symmetry you see when you look in the mirror.

The leader faces the class. Again try to progress from arm motions to the torso to a few steps through space to twists outside the plane.

(3) Rotational or turning symmetry

This is the kind of symmetry used in the game Simon Says, or observed when you see yourself in a video monitor in a store: you move your right arm and your TV image moves its right arm.

This is more difficult to follow. The leader faces towards the followers. If the leader raises her right hand, then those following also raise their right hands. This kind of symmetry is often seen in Spanish Dance, especially in duets. This symmetry involves a 180 degree rotation; other rotational symmetries aree explored in the chapter __________.

(4) Glide symmetry or "glide reflection"

This is the symmetry of footsteps: each step is like the last moved forward, and flipped to the other side:

Glide symmetry is sometimes difficult for people to recognize. The leader and the class face the same direction, but move in mirror image forms. That is, if the leader raises her left hand, as in the picture, then the followers raise their right hands.

7-4 Moving Symmetrically

Flocking

Practice the translation symmetry exercise in a group again. Now allow the leader to turn slowly towards the group; as she does the group also turns in the same direction until a new leader is in front. That person now takes over the leadership role. Continue doing this until everyone has had a chance to be leader. Try to develop the sense of ensemble to the point that it is not clear to an outside observer exactly who the leader is.

Many flocks

Divide the class into several flocks of at least three people each. Allow two or more groups to work on the flocking (translation symmetry) exercise at the same time. Let the groups move around the room and pass through each other on occasion.

Switching symmetries.

Have the students work in pairs, first taking turns improvising within each of the four symmetries introduced above, to make sure that they understand the differences between them. Remember that the leader and follower face the same direction in translation and glide symmetry, and face in opposite directions (preferably towards each other) in reflection and rotational symmetry.

Next have the partners practice switching between translation and glide symmetries. The switch ocurs when each person's body position exhibits what is called "bilateral symmetry," with both left and right sides of the body making the same, but mirror image, shapes. The partner who is in front and leading says, "Switch," and they begin moving with the other symmetry. Can you become proficient enough to follow when the leader does not say "swithch?"

Do the same switching between reflection and rotational symmetries. In this case the partners can alternate saying "switch," because they will be facing each other and no one will be "in front."

7-5. Further ideas

Perform the symmetries. Have the students choose a series of movements, mostly exhibiting the symmetries we have been working with, and practice them so that they can show the class. We say "mostly" showing symmetry because sometimes the breaks from symmetry can be refreshing and interesting. Have the students vary the rhythm, size, and dynamics of their movements within their choreography.

Use music.Try playing recordings of various types of music while the students perform their movements. Beautiful patterns are easily created and can give students the understanding of what it takes to produce a dance phrase. Again, it is important to stress that non-dance vocabulary is wonderful - perhaps even preferable - everyday movement can take on beautiful qualities when performed in unison and with clarity and commitment. Examine the symmetries found in the music as well as using it to accompany movement.

Look at dances. View videotapes of contemporary dances and discuss the uses of symmetry in the choreography. There are many more kinds of symmetry than we have explored here, for example, the symmetries between arms and legs, and symmetries in time.

A puzzle with the hands. Grasp the fingers of your left hand with those of your right, with the left thumb pointing up and the right thumb pointing down. Does this shape have one of the types of symmetry we have been discussing?

Notice body symmetries. Notice the symmetries around you everyday. Find each of the symmetries we have described, and make notes about what you or the people you were watching were doing when you observed these symmetries. In addition to looking for symmetry as you walk down the street, you might look for them in sports activities, public areas, or the classroom.

Visual Arts. Explore the symmetries in the work of artists, like M.C. Escher, who studied the symmetries found in Islamic art and applied them to his own work. Examine the symmetires found in public art, or in the craft work from various cultures. Create artistic designs utilizing symmetry.

Literature. Symmetry ocurrs in writing as well as visual arts. Poetry often repeats rhythmic structure. Palindromes such as "A man, a plan, a canal, Panama!" read the same backward and forward. How else can you find or create symmetries with wods?

7-6. Assessment

The key idea is that symmetry is not monolithic, but appears in various forms, and that we can learn to distinguish the forms. Students may be questioned individually to see whether they can correctly name the symmetries present in a movement or visual pattern. Simple visual designs allow for written tests. For example, the letters p,q,d, and b, as the usually appear in many typefaces all exhibit symmetries of one another. Which symmetry is present here?

p b

Ask students, working in groups, to make a short movement phrase which exhibits each of the four kinds of linear symmetries studied in this section. Did they actually use all four, and can they correctly say when they switched? Can they correctly name the symmetries they see in a video clip, and critique its use? Ask them to write a short paragraph detailing the differences and similarities between reflection and glide symmetries. Can they see the symmetries present in other art forms?

Students should be assessed in all four areas:

• Understanding and naming the symmetries

• Creating and performing symmetric patterns

• Observing, identifying, and critiqueing the use of symmetry in movement

• Making connections by finding and naming symmetry in other art forms and disciplines.

7-7. Resources

Godel, Escher, Bach: An Eternal Golden Braid, Metamagical Themas: Questing for the Essence of Mind and Pattern, and Fluid Concepts and Creative Analogies: Cmputer Models of the Fundamental Mechanisms of Thought, by Douglas Hofstadter. Hofstadter is obsessed with the symmetries in art, literature, and science, and has created wonderful and playful studies. See his chapter on self-reference in Questing, and his game Tabletop in Fluid Concepts in which two people try to imitate each other's actions.

Transformation Geometry, An Introduction to Symmetry, by George E. Martin. Springer-Verlag, 1982. An excellent mathematical reference.

Symmetry Discovered: Concepts and Applications in Nature and Science, by Joe Rosen, Dover Press, 1975. An accessible introduction to the ideas of symmetry.

The New Ambidextrous Universe, by Martin Gardner, W. H. Freeman and Company, 1990. A beautifully written exploration of symmetry.

Symmetry, by Hermann Weyl, Princeton Univ. Press, 1952. A classic illustrated introduction to symmetry.

Geometry: An Investigative Approach, and Laboratory Investigations in Geometry, by Phares G. O'daffer and Stanley R. Clemens, Addison-Wesley.1976. A text for prospective teachers which includes many activities dealing with symmetry.

Move this to rotational symmetry chapter:

Rotational symmetry

This one is like doing the "Hokey Pokey." The group arranges themselves around a central point and each person uses the same limb or movement, for instance "Put your left foot in and shake it all about," as in the song. The two person version we practiced above is sometimes called180 degree or "two-fold" rotational symmetry, and is a shape seen often in the Tango. With three people we get what is sometimes called "three-fold rotational symmetry." This is explored more in the section we call "Threesies." Four people create a pattern often seen in square dances. For example, here is a pattern with four people that exhibits four-fold rotational symmetry:

Divide the class up into groups of size 3, 4, 5, or more. Have each play with improvisations that exhibit rotational symetry. Ask them to distinguish between those shapes which have rotational symmetry only and those that also have reflection planes (when each person exhibits bilateral symmetry the overall shape will also have reflection planes).

For example, for a four person group, you might have them create shapes which show only these symmetries, using all four people:

(a) Four-fold rotational symmetry

(b) Four-fold rotational symmetry and four reflection planes

(c) Two fold rotational symmetry and no reflection planes.

(d) Two fold rotational symmetry and two reflection planes.

When the groups have learned how to differentiate between these symmetries, have them work more free form, inventing a series of shapes which they are able to cycle through smoothly without giving verbal cues. It is important to say, expecially for non-dancers, that movement need not be dance class vocabulary. Ordinary movements work well here when performed with energy and clarity.

 These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop (http://www.mathdance.org/store.html)