Symmetry and Balance

COMING SOON - an expanded version of this article  in our  Extras section!

(This post is meant to point out just a few ways a parent or teacher can utilize the concept of symmetry to illustrate all kinds of learning at different age and ability levels. We hope you find it useful and that it generates ideas of your own for its use. More detailed suggestions will be added in our EXTRAS for different age groups, including an Open Letter to High School Students.)

Reflected landscape, photo by J.J.

In its simplest form, symmetry means having parts that exactly match each other. This can mean the reflection of a scene in clear, calm water, the wings of a butterfly, or a square bandanna folded into two triangles. But it can also mean much more. Just look at some of the words used in definitions of it: similarity, balance, harmony, proportion, rhythm, equality, order, agreement, equilibrium. Can you see opportunities to use the idea in art, music, mathematics, writing, science, human relationships and comparisons of almost anything?

Everybody has two knees

It is possible to start teaching about symmetry to children as soon as they understand the number 2. You can get the idea of 2 hands and 2 feet across and proceed from there. This can be done one-on-one, but is faster and more entertaining for the child if it is done with two or more children. Make it a game like "I Spy". I also use a song with a pattern like "'Old MacDonald Had a Farm". [Ital] Hold your hands up, palms together and have the child do the same. Establish that we have 2 hands. Say (or sing) "Everybody has 2 hands, eieio". Get the child to clap hands and say or sing the phrase with you. Point to your feet and stamp them, followed by "Everybody has 2 feet". Every few days introduce a new body part: ears, elbows knees, lips, etc. Repetition builds remembrance.

Motorcycle vs. bicycle

Working with symmetry leads directly to comparing and contrasting things: objects, living things, words, conditions, events, even ideas. This is one of the most useful skills you can teach. Constantly ask:  "How are these alike? How are they different?" This promotes observation, comparison, and consideration about how the likenesses and differences may matter in the long run. Don't be surprised if a child comes up with a difference that you haven't even considered. Praise them for correct observations, and gently correct misconceptions. Be prepared to answer "Why?" questions. If you don't know the answer, say so and discuss how you and they might find out what the correct answer is. (Introduce children to a library as soon as this is possible.) The more adept children become at sorting out similarities and differences, the better they will be at organizing their thinking and solving problems — in school and out.

Butterfly, photo by J.J.

Symmetry in Nature can be an interesting way to begin learning about Science. Studying pictures of butterflies and moths or making a leaf collection can can give children a hands-on experience with wings or leaves that are or aren't symmetrical.  Observing the growth habits of shrubs and trees can also be useful. For more about trees, see our posts for April 18 and May14, 2020. Emphasize that such characteristics are what scientists use to classify living things and establish how they are related to each other.

Pushme - pullyou?  (Photo by J.J.)

Navajo storyteller statuette

Symmetry is very important in the visual arts. I always tried to discuss symmetry when we were talking about cultures,  Both symmetrical and asymmetrical features are important when looking at the art of Native Americans. For example, in understanding why certain techniques developed in the Arctic and different ones in the desert Southwest, even though the tribes who now live in both places are closely related.  Recent studies have traced the people who are commonly called Navajo have strong genetic and artistic connections to tribes in Canada. Their art has made it possible to trace migrations made centuries ago and to reunite people who share an ancient heritage.

Original pastel sketch in center, "mirror" sketches on sides

As babies, our faces are very symmetrical, but as we grow older, the sides of the face become less alike.  If students have one of their class pictures, have them use a small mirror or aluminum foil to double one half of their faces. *We used to do this when school pictures were delivered.) Repeat this by doubling the other half. Students may already begin to see differences in their "two faces".  Have some photographs of mature and older people that they can experiment with. Exercises like this will make your students more discerning. Ask them to draw "self-portraits", either as they really are today or as they think they will look in the future. To illustrate the process let me show you a sketch of a lady who let me sketch her years ago in Panama. I thought her face was very interesting and made the other two sketches, one with the left half of her face doubled and one with the right half.

Symmetry is vitally important in any study of either history or current events. There is a great deal of truth in the idea expressed by the statement that "History is written by the conquerors." Unfortunately, textbooks tend to give only one side of an event or series of events, when there are always at least two sides involved — and often many of them.  I have been distressed to hear and read that the latest generations of our young people are not familiar with the Great Depression, World War II, the Holocaust, the Korean War or the Cold War (which is sill going on) and, consequently, are not in any position to put these events and their subsequent effects in perspective.

Guitar picker and audience, photo by J.J.

Most of those events have a great deal written about them and even old newsreels from the times which students, particularly, should be made aware of.  I lived through most of them. I encourage students to interview and perhaps record conversations with "senior citizens" while we are still around, as well as consulting the resources mentioned above.  The same thing is true for understanding the contentious issues of the present, many of which have very long roots. Debate is a great activity to practice that kind of symmetry. Not only must a team produce a compelling argument to support their proposition, but they must understand and anticipate the arguments that will be put forth by their opponents.

"Balance" and "rhythm" are terms associated with symmetry. To be successful in any sport, there are positions and movements that must be practiced over and over to train the body to react and respond correctly without having to stop and think about it. This is true whether you are making a free throw, swimming the backstroke, heading a soccer ball, or sticking a landing.  Gymnastics offers a good example in which not only is body position vital in performing different exercises, but it must look good while doing so.  Almost any kind of dancing requires that two or more people must coordinate their movements with great accuracy.

Square dancers (public domain photo)

When was in college, I belonged to a square dance group. In addition to following all the calls, each eight-person "square" was supposed to figure out a special finale for their performance. I was the smallest girl in our square and weighed the least. My partner was the tallest fellow in the group and very strong. For our finale the other three pairs danced backwards to give us plenty of room while we answered the call for "Swing your partner". We started with a standard "swing", but my partner kept turning and lifting me off the dance floor. We turned until my body was horizontal and parallel to the floor, level with his shoulder at the end of his extended arm. A couple more turns put my feet back on the floor and we danced backward to complete our "square".

Haiku by Basho

In a writing class you can introduce older students to the symmetrical "shaped poem" (also called diamante).  This kind of poem does not rhyme; nevertheless, it has very strict rules for its construction. Prepare a number of examples beforehand so that the students can figure out those rules. Be sure students know what participles are and how to break a word into syllables before they undertake the assignment. For the rules and a couple of examples of shaped poems, see our post from September 25, 2020. Another good activity is creating a haiku. There also is more information about haiku in this post.

When teaching or learning a second language, it is imperative that students realize that just because a word is paired with an English word in the vocabulary list, that doesn't mean it necessarily has that meaning in a conversation.  All languages contain many colloquialisms, in which a word may not convey anything like its dictionary meaning.  Pronunciation is also important. When we were stationed in Germany, my soldier husband went to a store selling car parts to buy an exhaust pipe for our old car. He had read the word for exhaust pipe in a parts catalog and asked for one at the counter.  He was angrily thrown out of the store. It seems his pronunciation was a little off and instead of requesting an exhaust pipe, he had requested, shall we say, a "woman of ill repute".

Now I'm going to turn this discussion over to J.D. to tell you about some other areas of his expertise...

J.D.: In music, it is very easy to find examples of symmetry, from the simplest exact repetition of phrases in "Frere Jaques" to the call-and-response of early Christian religious music to the inverted melodies used as base lines in things like Pachelbel's "Canon". It is easy to see these symmetries in the pitches of musical notes (melodies and harmonies) in written music notation, but nearly all elements of music — rhythm, volume (loudness), timbre, and many more — can demonstrate various symmetries. But one thing can never be symmetrical: because music, like dance or films, always changes from one moment to the next it must have a beginning and and end. Even if you make a piece of music where the second half is an exact mirror of the first, there is a point in time when no-sound becomes music and a point where music becomes silence.

 

 

Of course, particle and quantum physics tells us that it doesn't have to be that way, that the 'arrow of time' can go the other direction, but they have yet to find a way to demonstrate this in any practical way. And that is why the 'time travel' in our previous post was just an intellectual exercise.

Particle physics interactions

Physics, especially the exotic realms of particle and quantum physics, and those disciplines that try to reconcile Einstein's General and Special Theories of Relativity with them, thrive on symmetries. Particles have antiparticles with opposite charges; quarks (a staple of quantum physics) have properties like color and charm — which are not at all what they sound like they should be — which can be mirrored; and ... that is why Annake asks me to write about subjects like these which make her eyes glaze over when she hears me going on about them.

Physicists depend heavily on some pretty esoteric mathematics for their work, but even the simplest math — things we use every day — are based on symmetry. Not really a surprise: math is basically measuring things, and the word "symmetry" is derived from ancient Greek for "same measure"! And the first thing any one learns in mathematics (beyond simple counting) is the equation:

1 + 2 = 3 ;  "The sum of the squares of the sides equals the square of the hypotenuse"; x + y =  7

And even though the expressions on either side of "equals" don't look the same, they must be for the equation to be true.

The term "symmetry" takes on additional meanings in mathematics and the sciences as students get into more technical work. Be sure that they are aware of these. This occurs even in literature and the arts.   But once your students are accustomed to looking for symmetries, they can use this principle to increase their understanding of art. mathematics, science, music, current events — even culture.  But they need to have developed a skill for determining a sensitivity to how things are the same and how they are different. You can help them do that.

Good luck. Stay safe and well.

 This post by Annake's Garden is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.