Constructing Modern Mathematics or is it History? English?

Kate shares an example of Islamic tiling uring the project brainstorming

Chicago middle school English teacher extraordinaire, Kate Tabor, returned to Constructing Modern Knowledge for the second consecutive year. In 2010, Kate spent her time at CMK using MicroWorlds to program randomly generated Shakespearean insults. Computer programming and mathematical thinking were used to explore linguistics and literature in a whimsical fashion. Read her recollections from CMK 2010 here and here.

Prior to CMK 2011, Kate traveled to Spain where she became fascinated by Islamic tiling. When project ideas were brainstormed, Kate already knew that she wanted to use Logo programming to recreate Islamic tiles. She even brought along photos of tiling she wanted to explore. This would be done by communicating formal representations of  the visual images to the computer in the Logo programming language. She decided to use Turtle Art, a special Logo microworld for exploring visual art via geometry. I spent a bit of time showing how procedural inputs and variable sliders could be used in MicroWorlds EX could be used to repeat geometric figures of variable size. Turtle Art creators, Artemis Papert and Brian Silverman convinced Kate that all of her objectives could be achieved in Turtle Art, despite its surface simplicity.

Project ideas are posted on the wall during the brainstorming process

I will use Kate’s own words, from her blog posts to tell the rest of her learning story.

Day 2: CMK 2011 – Inspiration and Renewed Enthusiasm 12Jul11

We worked on our projects today. I had a change of plan. I was planning on switching to MicroWorlds and the logo programming language to create the tessellated patterns that mimic

"See, more complicated and much less trial and error. I’m learning."

the tiles that I saw in Islamic Spain. But Artemis Papert and Brian Silverman arrived and I was reassured that it was possible to do what I wanted to do with Turtle Art. So with renewed enthusiasm I worked on my project to relative success. I managed to create a shape, repeat the shape, and understand the relationship of the shapes to each other on the xy plane. I’d say: success.

Kate works on her tile programming project (Christa Gordon also pictured)

Day 3 at CMK11: Ways of Knowing 14Jul11

Which of course leads me back to my few hours of working with Turtle Art yesterday. I think I cracked the puzzle on tessellated figures. I almost have the shape right,

and I’ve figured out the arc needed to turn and interlock the figures.

I need just a little more time to tinker, and I think it will all come together. I’ve been watching new friends work with Turtle Art, PICO Crickets, and Prezi. Another new friend is trying to figure out a way to charge his cell phone with his bike (and to ride a recumbent bicycle). It’s all been good, hard, fun.

Programming in Turtle Art

Logo co-inventor & CMK faculty member, Cynthia Solomon, checks-in

Starting With a Blank Page 14Jul11


Cynthia provides assistance

When I left the project yesterday I was pretty sure that I had a clear picture of what I needed to do. All I had to figure out was how to populate the figure across the page. I was working with the program that I had created, and I was looking for a simpler way to figure out the positions of the figures on the xy axis of the field.

Cynthia Soloman stopped to see how it was all going. I showed off my progress and told her what I was up to. “Wouldn’t it be easier if you started your figure at a point of symmetry?”


So, back to the blank page. Fortunately I knew how to make the shape now, I just needed to start in a different place of the figure. She also showed me how I could fill in the figures. So I set about reflecting and repeating the shape across the field.

After spending three days looking at the geometry of this problem from the outside in (circles and inscribed squares first), from repetition of form, from the basic shape, and now from a point of symmetry, it was surprisingly easy to fill the page with the figure. it’s not perfect. The program is still more brute force than elegant computation, but it works (pretty much). So with a little code assist from my table mate, I even changed the background color.

Photo of Islamic tile found in Spain (left) - Computer graphic recreation (right)

You may read Kate’s takeaway lessons from Constructing Modern Knowledge 2011 in her blog post, Tinkering.

Photo of Islamic tile found in Spain (iPad) - Computer graphic recreation (Macbook)

Screenshots of tiling created by Kate in Turtle Art

CMK 2011 Reflection on Islamic Tiling in Turtle Art from Gary Stager on Vimeo.

By Gary S. Stager, Ph.D.
Founder & Director
Constructing Modern Knowledge