Math Circle 2.3.2015

ESCHER #4:  Can Any Regular Polygon be Tesselated?

(February 3, 2015)  The story of today’s session can be told by a list of questions the participants asked in response to one question I had written on the board at the beginning of class:

Can any regular polygon be tessellated?

Resulting questions:

  • >What is a polygon?
  • >Can a line be a polygon?
  • >What kind of tessellation are we talking about here?
  • >What does it mean for something to be two dimensional?
  • >Does a 2-D shape have to have thickness in order to exist? Do 2-D shapes really exist in real life, or are they just thought experiments?
  • >If a 2-D shape does has thickness, is that the same as depth?
  • >If a 2-D shape has depth, then isn’t it really a 3-D shape?
  • >Are circles polygons?

I don’t know if your brain is spinning from these questions, but our room was full of spinning brains at this point.  We did what mathematicians often do when this happens – we made some assumptions:

  • >By tessellation, we specifically mean a tiling pattern with every cell the same
  • >A polygon is a shape with three or more sides
  • >A polygon is 2-dimensional
  • >Circles are excluded

All the while, I had a manipulative set of toys on the table:  Polydrons (see photos).  Most of the students used the Polydrons to explore the questions, although a few kids preferred simply drawing and verbalizing their conjectures.  Deep discussions arose from every question above.

A challenge for me was to not answer the questions.  My job is to become invisible.   But of course, those didactic instincts arose within me – thoughts such as “I better inform them that XXXXXX, otherwise are they really learning anything?”  I do know, of course, that the kids are learning more from thinking about and developing their own assumptions, conjectures, and questions, and from thinking about them for a very long time (hours, days, weeks, months, etc), than they are learning from me telling them anything.  But still, that insecurity is there.  Drat.

Anyhow, the kids came up with their own conjectures, which I neither confirmed nor denied. I’m not prepping them for a test here.  I’m hoping to facilitate mathematical thinking.  Two totally different things.

We periodically voted, to see if we had reached consensus on the answer to the question.  We never did.  We attacked the question systematically, with the following conjectures:

# of sides can it be tesselated? proof
3 yes “I’ve seen it;”  “someone else has proved it;” done here now with polydrons
4 yes common knowledge
5 ? so far, can’t do it with polydrons
6 yes J drew it on the board last week


Of course, the conjectures raised more questions:

  • >Could a 3-D polygon be tessellated?
  • >Should we just look it up online? (this from a student who had already explored this question in another class – everyone else was against this idea)
  • >Do you like math? (not sure how this came up, but it was related to the rest of the discussion.  Many students agreed that they like some types of math but not others.  M said he liked what we are doing here – “political debate style math, where we’re sitting around debating things, versus two plus two.”)
  • >Can non-regular polygons be tessellated, particularly 4-sided shapes with equal sides but unequal angles?
  • >What shapes make up a soccer ball?
  • >Are apples the sacred fruit of mathematics? (I posited, in return, that perhaps pineapples are.)

BTW, I had other things on the board besides that polygon question at the beginning of class – specifically instructions for and an example of a cut-paper tessellation.  The instructions were kind of vague, so I was wondering if we might occupy our 75 minutes with kids wanting to create their own tessellation while engaging in the math skill of asking questions to clarify the instructions.  I was wrong.  I expected an interest in product over process, but witnessed an interest in process over product.  Good to know, shoulda known, needed a reminder of this!

As the kids explored, I read to them from Miranda Fellows’ The Life and Works of Escher.  Her book tells Escher’s story through his works.  We got through three of the works/stories only because of interesting discussion about each one.  To be continued.


PS  We will make up our missed day because of snow on 2/24.  There is no class next week on 2/17 because of a conflict I have with the Carver Science Fair.

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