A problematic proof and more fun with tiles!

How a week flies when you're thinking hard!  My week at the Art of the Islamic Pattern class is coming to an end.  I thought a week would be enough, but MAN do I wish I were staying for week 2!  I feel like we are barely scratching the surface of this.  Thankfully I will have some more time to look at this in Spain.

One of the many interesting things we did this week had to do with this image.

Essentially, our teacher said that to evenly divide the perimeter of a circle into 10 parts, you needed to pack 2 circles into your original circle, and then take another (third circle) tangent to the 2 you just packed in.  The radius of the third circle is what gives you 10 congruent parts. Here's a better picture of what's going on:

It seemed a little too good to be true.  THAT's where math comes to save the day.  I wrote up a proof that it works.  I was really proud of it.  Just a little trig, Pythagorean theorem and BOOM.  Proof done.

Then after class I went to do more art with some peeps where Riswan (another student) also expressed surprise about the same division problem.  Luckily I had my proof in my (pocket protector-ed) proverbial pocket and pulled it out.  While I was pretty proud of it, I knew it wasn't THAT satisfying.  Yeah... it was an accurate algebraic proof, but the geometric proof would get to the WHY behind the concept.  It is too weird that you magically use this new tangent circle (the third one) and it just works out to be one tenth of the circumference... Riswan wasn't satisfied with it either.  So, we went back to the drawing board.  Hopefully I will have an elegant proof of it at some point.  OR maybe someone in the MTBoS wants to take a stab at it.

Talking to one of the art teachers there, I realized that really when we skip constructions OR just do them on the computer, students are missing exploring these in space.  I would LOVE to take 2-3 days at the beginning of the semester and try making some of these with students.  We could really develop some good vocabulary around geometry (perpendicular, bisect, vertical angles, interior/exterior angles, congruency... UNDERSTANDING circles, angles, arcs, chords...) MAN the possibilities are endless!  I still think moving on to geogebra is the right thing to do from there - geogebra allows for a lot more exploring AND I think with the background of having done constructions by hand, students will not be quite as frustrated by geogebra.  Also, maybe students will be more willing to draw out diagrams if they have a lot of experience drawing them in class as "art" rather than math diagrams.

ALSO in class we did THIS massive drawing.

The tessellation of this was a lot of fun too with a lot of manipulatives.

Ahem... check out those wooden tiles!  Those are totally what I would love to do with my Ss.  Pick a tesselation and we will cut one out on the laser cutter.  Maybe this can be a CS thing later in the class...

ALSO, in my inbox today, I got this article about using art in CS.  It made me realize that I can use what I learned in my CS class to talk about recursion.  These patterns require a lot of precision... and computers are good at precision - especially doing precise things over and over again.