Sunday, November 7, 2010

Tessellating Kites

Our study of motions led to one of my favorite topics in all of mathematics:  tessellations.  I've posted some previous work on this blog, have an old webpage with some good tessellation resources, and found a new source of beautiful Alhambra images to share with students.

As a math topic I just love them.  The visual aspect, the geometry, the connections with algebra, the historical context, the art connections with Escher... it's darn near perfect.  Working with 2nd - 12th graders they are amazing for rigid motions because you use the motions to make the tiles, then to repeat the tiles in a pattern, then can see the motions in the finished tessellation.  It's visual and kinesthetic.  There ought to be a song.

Instead, how about some geogebra.  For some reason, this time around, I got interested in kites.  Which tessellate by side to side rotation and what would a mixed glide reflection/rotation tessellation look like.

The question of which kites tessellate by double rotation boils down to what happens at the joint vertices between the two (possibly) different edges.  We can pick the angles so that the kites blossom (tessellate around a point) at the vertices between congruent sides.  In this sketch, you set those numbers, then observe the effect on the other angles.  What condition is necessary for the angles to work out?  Is it sufficient?

Webpage or geogebra file.

This sketch lets you make alterations to that classic 60-120-90-90 kite tiling.  The sketch will adjust and give you a chance to both design and watch the effects.

Webpage or geogebra file.

 This sketch does a tiling that can be done with any kite.  (Pretty good question as to why it works for any kite!)  Two of the congruent sides rotate to themselves around a midpoint, and the other two fit together with a glide reflection.  Escher was fond of this pattern, as it allowed him to create creatures going in opposite directions for his contrasting tilings.

Webpage or geogebra file.

I would love to hear from readers if they prefer these geogebra sketches as links or embedded applets.  Could you take a second to comment?  Also, I love making geogebra tessellations, so if you have any ideas for ones you'd like to see, let me know.


  1. It's pretty cool that you posted this today. It looks to me to be not so far off from linking nature with math. RIP Mandelbrot. he died just a few miles from here (weird).

    I have had a lot of success with a grass growing project that I created. A Wordpress widget allowed me to upload the files onto my blog (super excited!) The project links concepts of ecology to algebra, and engages my students more than any other lesson I have created. I'm very happy to be able to share it with everyone [who reads my blog and teaches algebra] :)

    Students love watching their grass grow and taking care of it, so much so that they forget to complain about the algebra! I hope you check it out!

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    Best regards