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ESCHER #1: What is Symmetry?

Math Circle 1.6.2015(January 6, 2015)  How do you define symmetry?   This question became the crux of our first Math Circle on Escher.  Our 11-13 year-olds debated this question for well over an hour.

It all started when I passed a bunch of images around:

  • >beehives and other natural patterns,
  • >examples of Moorish architecture,
  • >various Escher works, and
  • >a guy staring at his reflection in a puddle…

I asked each student to hold one picture up and say something about it to the group.

“What do all these pictures have in common?” I then asked.*  Initial responses began with tessellations and the like.  “But how would that categorization include the Narcissus image?” I countered.

“Symmetry?” asked J, after some brief discussion.  Everyone agreed that every image involved some sort of symmetry.

“By the way,” I asked, “what do you think Escher is trying to communicate or explore with his work?”  Responses were quite varied and, drat, I didn’t write them down.  I read them a list of Escher’s themes from a helpful paper I found online**  We discussed whether his themes involved art, math, both, something else?  “Which themes do you think we’ll be focusing on most here?”  I asked with a smile.

“Regular divisions of the plane,” quoted G, from the list I had just read.  Some of the kids looked confused, so I asked G to explain what this means.
Then I gave each student a small slip of paper with the ABCs typed in capital letters.  “How might this be connected to what we’re already talking about?”  Silence for a bit.  Perplexed looks.

Then A posited (both bravely and tentatively), “The word symmetry is spelled with letters?”

I was excited by A’s risk-taking with this response.  Risk-taking is such an important skill in mathematics. (Sadly, I don’t think I made this point to the group.  Hopefully I’ll remember in a future session.)  “Yes!  I didn’t even think of that!”  I responded, then put her conjecture on the board.  “What else?”  No one said anything, so I said, “Then let’s just look at the first letter, A.”  I drew a big capital A on the board.***  Now came the aha moment.

“A is symmetrical!” announced multiple voices at once.

“How?” I asked.

“You can draw a line through it and it’s the same on both sides,” responded many of the kids.  As they answered, some of them started calling out some other letters that are symmetrical:

H, A, I , V, Z, E, C, K, X, S, D, N, B…

“Wait a minute!” protested someone.  “How is Z symmetrical?” (Let the games begin!)  Debate began.



So many arguments for and against the symmetry of Z – I can’t even remember them.  Some of the kids were taking turns coming up to the board, taking the marker out of my hand, and explaining their points.  The first person who jumped up to the board stood directly in front of the board with back to the group and addressed me off to the side.  “Speak to the group, not me, and don’t block the board,” I gently directed.  “If your audience can’t get your point, there’s almost no point in making it.  Communication is one of the biggest skills a mathematician needs.”  Presentation styles immediately adapted to this feedback.  The kids presenting were now addressing the group, with their backs to me.  I was no longer in charge.  (Phew!)

Many soon realized that if you split the Z with a straight line, no matter where you do it, you do not get a mirror image.  Someone pointed out that you get a mirror image but flip-flopped.  Not everyone understood.  I passed out some props:  sharpies, clear plastic strips, and pins.  I asked them to draw Zs, trace them onto the plastic Zs, put the pin in the middle, and rotate the plastic Zs 180⁰ about the paper Zs.  Now everyone saw that with that rotation, every point on the original Z was now located on the repositioned Z.

Is this symmetry?  The group was split.

Still not everyone was convinced of the absence of a mirror image without rotation, so I asked them to fold their Zs (drawn on paper) and hold them up to the light.  Now everyone was convinced.

A new idea was emerging from the group discussion:  could there be multiple types of symmetry?



M posited that the letter there is a “perfect symmetry,” which is possessed by the letter O.  G defined O as a “totally symmetrical letter.”  I asked for a definition from the group.  Someone said you “can cut them anywhere and get a mirror image.”  Everyone agreed.

“Then,” said M, “there’s imperfect symmetry.”  He was referring to the symmetry possessed by the letter A.  I asked the kids to define that:

  • >“You can cut it in half with a horizontal or vertical line and get mirror images” (… followed by arguments that diagonal lines too might produce symmetry)
  • >“You can cut it down the middle with any straight line and get 2 equal images, equal in size and shape.”

Everyone agreed that these were sound definitions of symmetry, and that A passed that test.  We still had dissension about Z.  I asked the kids to vote and the group was split.  Half were convinced that Z is not symmetrical.  Period.  The other half thought it is, but for some reason it wasn’t fitting into the definition.  But the definitions of the two types of symmetry felt pretty good to almost everyone.  Hmmm…  was the problem in the definition, or its application?  After some heavy thinking, M posited a revised definition:

  • >“Both sides are the same when cut; you can move it and it retains it’s…. (silence)”

No one had a word for exactly what it retains, but people suspected it might be size and shape.  And despite the fact that the Z was retaining something when rotated, kids still disagreed about its supposed symmetry.



I usually try to lead a math circle by saying as little as possible, and providing pretty much zero information. My goal is for kids to invent and discover math.  But sometimes things get too frustrating.  Or abundant curiosity is coupled with insufficient know-how.  Our group was approaching both of these conditions.  Sometimes when this happens I put the current problem aside and do something new for the rest of the session.  Then I revisit the problem the following week, in the hopes that sleeping on it for a week will shake loose kids’ assumptions and/or generate new insights.

This time, however, I suspected that individual percolation wouldn’t yield progress, so I did something I’m pretty reluctant to do:  I asked a leading question.

“Do you think there are only 2 types of symmetry?”  Eyes widened with pleasure: maybe this was a way out of the conundrum.  We started a new list:  types of symmetry.  The kids dictated 5 types of symmetry they had discovered thus far:

  • >Horizontal
  • >Vertical
  • >Diagonal
  • >Perfect
  • >Rotational (I supplied that word after many kids named it first with some variation on the word “rotate.”)

I used another leading question to further elicit a distinction between the first 4 types (“mirror”/”reflectional”/”line”) and the last.



I showed the kids some friezes, or “strip patterns,” which are patterns of a single image repeated linearly, frequently used in wallpaper edging.  On the board I drew a line of bunny faces.  After laughing, everyone agreed that this pattern had reflectional symmetry:  you can fold a strip in half and get the same number of bunnies on each side, and the bunnies themselves possess reflectional symmetry.  (An interest in infinity arose from this conversation, but I steered the topic back to symmetry.)

Then I drew another strip, this time lined with one-eared bunnies (see photos).  The reflectional symmetry of the individual bunny went out the window with that.  The kids pointed out that the pattern as a whole could still be symmetrical if the bunnies on one side of the strip had ears on the opposite side of their heads from the bunnies on the other side.  So I had to be a pill and tell them that each bunny’s ear is in the same position.  The kids grouped this pattern, therefore, in the “asymmetrical” group, along with the letters F and G.  To be continued next week…


*Email me ( if you’d like for me to send you a word doc with many of these images pasted in.

** Understanding Escher: A look into the mathematical principles behind his tessellations, by Vivien Foo Huimin, Lynn Xu Hanni, and Wilson Zhu Ming-Ren, of the National University of Singapore.  Their instructor, Helmer Aslaksen, posted many student papers from his course Mathematics in Art and Architecture.  According to this paper, 4 of Escher’s goals were

  • >“penetration of the worlds,
  • >the impossible,
  • >the infinite,
  • >regular divisions of the plane.”

*** thanks to James Tanton for the idea of using the ABCs to teach symmetry in his pamphlet The Mathematics of Symmetry:  Smart Phones, Frieze Patterns, Fractals, and More!

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