Art & Geometry

I recently was fortunate enough to attend the NCTM conference in Boston and while I was there, I checked out a really cool art museum in Providence, RI.  I took these two pictures while I was there because I thought that they could connect to what we are learning about in our polygons unit.

When showing the pictures to my students, I projected each one and had them brainstorm some questions they were curious to know about.  Of course, I encouraged them to be mathematically based questions, but I did still get, "What color is it?"  Check out some of the questions that they came up with:

How many different squares do you see in the picture?

What is the scale factor used if you dilate the center square to the outer square?  What about from the outer square to the center square?

How many lines are there?

I asked my students to explore 2 different questions.  First, I said, "If I asked you to determine the number of 'spokes' in the picture, how might you do it?"  Groups brainstormed ideas for how they might tackle the problem.

The general response was that students would break the picture down into smaller parts, count the number of spokes in that section and then multiply by the total number of those parts.

Once we had shared some ideas of how you might determine the number of spokes, I asked them to actually figure it out.  They came up with 128 spokes.

The second question I had students explore was what would be the degree measure of the angle created by two adjacent spokes?  

This was a little trickier, not because of the calculation they used (360/128), but they had a hard time defending their use of 360.  Was it the Interior Angle Sum Theorem?  The Exterior Angle Sum Theorem?  Or the fact that there are 360 degrees around the center point where the vertex of the angle is?

This was the second picture I shared with them.  I really like this because it's deceiving.  Is it a hexagon?  An octagon?  Neither...more of a hybrid of the two to form a heptagon.  I find the inscribed circle to be really fun, too!

The question that I posed to the students was to find the measure of each of the seven interior angles. Lots of students found the sum of the interior angles (900 degrees) and then divided by 7, but soon realized that that would not work because not all angles are equal in measure.  So, they revised and were able to make a lot of progress.

I find questions/problems like these to be so much more interesting than the way that I've taught this unit in the past.  I feel like polygons have so much potential for interesting explorations, and I was happy to find some of those in art!

What mathematical questions do you have about these pieces?