Math Circle 2.13.2015

(January 13, 2015)  For the most part, I’m going to let the pictures do the talking this week.  They tell the mathematical story of our session via the students’ work and question.

We continue to wrangle over the definition/types of symmetry.  I posed 2 new questions this week:

1)  Can you get a mirror image of an object using rotation only?*

2)  What is a tessellation?

All other questions listed here were student-produced.  We used tracings and manipulations of pentominoes to explore these and other questions.


  • What is a tessellation? (The kids played “Is it or Isn’t It?” to come up with a tessellation definition.  I and other students put patterns on the board for the group to make tessellation conjectures about.  Their definition: “repeating shapes, in a line, touching, equidistant, gaps OK if symmetrical”)
  • What’s the difference, if any, between a tessellation and a tiling?
  • What are some methods to determine whether shapes are identical?
  • Why are pentominoes called pentominoes?
  • How does the point of rotation affect the result?



(“Open” is the word mathematicians use to describe unanswered questions.  The Google dictionary describes an open question as “a matter on which differences of opinion are possible; a matter not yet decided.”)

  • Can any shape be tessellated?
  • Can stars be tessellated?
  • In Escher’s work with the left-oriented smiling light fish tessellated with the right-oriented dark birds, are the fish and birds the exact same size and shape?**
  • Does symmetry require mirror images?
  • Does a row of identical asymmetrical objects posess symmetry?
  • Does the answer to the above question depend on either the shape of the objects, or the shape of infinity?
  • Is linear infinity symmetrical in 2 directions?
  • Does infinity have to be straight?
  • What you call a 2-sided shape, if one exists?
  • Do the patterns in symmetry of numbers/values act the same as the symmetry in patterns of shapes?
  • Can a tiling with a gap in one side be part of a symmetrical pattern?

If you haven’t yet, click here or scroll down to see the photos.



*I have a mathematical concept in mind with this question – the same concept I had in mind when discussing wallpaper edging last week.  If you are a student in this class, DO NOT READ THE LAST FOOTNOTE  BELOW*** – IT IS A SPOILER.  If you read it, you’ll lose the opportunity to enjoy a very satisfactory mathematical discovery, if I set it up well.  We’ve only been doing this for 2 weeks, so be patientJ

**Please, please, please, could someone tell me the name of this work?  I can’t find it anywhere!  I don’t want to post an image of it here since I’ve read that Escher’s descendants have had a really hard time limited unauthorized reproduction of his work, and I want to be sure about fair use.  I could email you the image, though.   Or click on the photos from class – a small image of it is in the photo with the yellow notebook. (

***SPOILER SPOILER SPOILER (DO NOT READ IF YOU ARE A STUDENT IN THIS CLASS):  If you rotate a certain pentomino (the T-shaped one, for instance) about a  a vertex, you end up with a “glide reflection,” which is a rotation plus a translation.  So far, the consensus in this group is that if you move an “imperfectly symmetrical” object without reflecting it, there’s no symmetry in the resulting position/image.  I have some other things up my sleeve for the next few weeks to set the stage for a discovery of translation, and a possible acceptance of translation as a type of symmetry.  But these kids are inventing their own math, so I’ll be in a tough spot if we do weeks of activities involving translation and they still reject a connection to symmetry.  What will I do?  Hmmm….

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