Cats Tuesday: Cat Symmetry

Cicking on most of the photos in this post will open a new window so that you can see the full-size image.

I teach mathematics to middle school students in public school and also to college students (which is why I am always late on Tuesday visits to everyone's blog: my algebra class is on Tuesday night after working all day!).

Symmetry is a topic in the middle school and I spent one vacation week playing with paint programs and photos of my Possum (of the kitten fight last week) and Scout (one of Amy's beagle dogs) learning how to show facial symmetry.

In each of the following sets of photographs, the original photo is in the middle. The left photo of each set shows each face with two left halves and the right photo shows two right halves.

I like how they all came out. I then continued with photos I grabbed from the web of Denzel Washington and Tyra Banks. I have included instructions for any of you who would enjoy playing with this. Just click the "read more" at the bottom.

One flickerite who saw these said that she has done them but she used quadrants (top/left; top/right; bottom/left; bottom/right) and the results were wild! I haven't had a spare moment to do that yet!

Thank you for reading this rather long post! Since public school is on vacation, I may get to see your posts during the day today! Happy blogging!

Here is Symmetry Amy (my middle daughter):

To do this yourself (I have cut and pasted portions of the paper I wrote):

In the April issue of Mathematics Teacher I found a wonderful article Fostering Mathematical Inquiry with Explorations of Facial Symmetry by Michael Todd Edwards. I immediately grabbed my digital camera and played! What follows here is a synopsis of his article. I offer my commentary at the end.

Collect photos of famous people (models, celebrities). The photos should only be head-on shots of faces. Scan or download the photos into a folder on your computer. You can use Word, AppleWorks, Paint and even PowerPoint for this study.

Cut the left side of the face. Duplicate it. Flip it horizontally and place it on top of the right side of the face. Align both sides.

The more symmetrical the face, the less change you will see in the new photo. The less symmetrical the face, the more change you will see.

Print copies for students to explore reflection symmetry. They will connect corresponding right and left side facial features with line segments. They will measure the line segments. They will then attempt to find patterns in the quadrilaterals that they have made.

Do attractive faces have similar types of quadrilaterals with similar sides? Are ratios involved? Study the angles of the quadrilaterals. Can you write an algorithm for "beauty"? What is "beauty"?

Comments
Students simply love looking at these photos and want to make them of themselves. Of course, parental permission would need to be obtained for that to happen. There are other questions to explore: What qualities in an infant's face make us feel protective towards them? What types of animal faces give us positive feelings?

I am very concerned about the emphasis on beauty in this exploration. It is a valid topic to explore, but be very careful in public school. Too many of our children, especially our girls, have negative feelings about their beauty and worth. Keep the discussion strictly mathematical and discuss symmetry. The students to whom I have shown this work only want to see the symmetry and think nothing of beauty. There is more than enough material in discussions of symmetry to fascinate us.

Tyra Banks

Denzel Washington

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