Wednesday, March 24, 2010

Tessellations and Geogebra

In my Geometry K-8 class we've been study transformations.  Which always leads one place for me ... my love, my joy... tessellations.  Arty, playful, deep underlying structure, corner cases that require thought even still; they're perfect.  To me.  I understand how others have dabbled and grown tired, but for me they are ever fresh.

There's a probably a few too many sketches here, but let's have a look.

First - Look at a tessellation, identify the motions, and consider what properties allow it to tile that way:
Quadrilaterals:  webpage and geogebra file

Hexagons1:  webpage and geogebra file

Hexagons 2:  webpage and geogebra file

Second - Look at a tessellation, identify the motions, and then alter the tile Escher-style!

Isosceles Triangles:  webpage and geogebra file

Quadrilaterals:  webpage and geogebra file

Third - Control the properties of the tile so that it will tessellate with the given motions:

Pentagons:  webpage and geogebra file

 (A midpoint rotation and 2 side to side rotations.)

Hexagons:  webpage and geogebra file

 (Quite challenging!  3 side to side rotations.)

Bonus - Kaleidoscopes!  What's the connection between reflectional and rotational symmetry?

Control the number of Sectors:  webpage and geogebra file

Control the Angle:  webpage and geogebra file

The kaleidoscopes were to investigate an open conjecture we have.  My one disappointment with Geogebra in comparison with sketchpad is that animation isn't as easy.  It's nice to have an animate button on your kaleidoscopes. 
The Leah-Jill Conjecture:  If a shape or design has n lines of symmetry, then it will have n-fold rotational symmetry, for n > 1.  Having rotational symmetry does not imply reflectional symmetry for any n.

I don't think the sketches helped.  I can't decide if we should tackle it another way or if we should just move on.

If you have any ideas for a dynamic tessellation sketch, please let me know.  It doesn't take much of an excuse to dive right in.


  1. I'm no good with geogebra and geometer's sketchpad, but when I taught the math for et class, some of my students loved doing (for e.c.) the thing where you start with a tesselating shape, draw a wiggly line from one corner to the next, cut that edge off along your wiggly line, and attach it to the opposite side, for a very fun tesselating shape. (Hmm, how would you do something similar to a triangle?)

    Also, have you seen the posts on rep-tiles? (Here.) Not exactly tesselation, but similar...

  2. There's a sketch for one triangle tessellation here! Try the Escher style triangles. He actually used hexagons and pentagons a lot because of the intricate relationships that are possible.

    I love reptiles - they're an interesting related case to quasiperiodic tilings that have the same kind of inflation and deflation. Check out my colleague's Penrose tiling applet! David Austin is a Java master, and quite an effective writer. Check out the related article at the AMS.

  3. John, gorgeous work! I admire your obvious immersion and fascination with tesellations. Thanks for sharing.