The story behind this discovery dates back to the 1970’s when I used to volunteer as a math resource specialist at a small school near my office. One day, a teacher introduced me to a 10 year-old girl (we will call her “Ann”) who was (in the teacher’s words) “bad at math.” I found that to be strange announcement since, in my view, Ann had never been exposed to math, but only to arithmetic. So one day I invited my class to do an experiment.
I brought my own bucket of pattern blocks to school, in which I added some regular pentagons I had made. Piles of the same shapes were put on several desks, and students were asked to tile the surface with just one shape – and I made sure Ann was at the table with the regular pentagons. After a few minutes, all the students had succeeded – excepting those at the table with the pentagons. No matter how hard they tried, there was always going to be a gap.
Ann said, “Well, we can do it, but we are going to need a lot of grout.” And, after looking at the other tables, she said, “This is a strange kind of math – 3 works, 4 works, and 6 works, but 5 doesn’t work. Why is that?”
I was delighted to hear that question because this is the kind of question mathematicians ask themselves often. Ann told me she wanted to experiment more with the pentagons, and I gave her a bunch to take home so she could report her findings the next week when we met again. At this point I didn’t know what to expect, but it sure wasn’t what she showed up with.
The next week, she started off with the following pattern:
Ann pointed out that this shape, while not a tiling pattern, looked like a pentagon if you “squished” your eyes a little bit. “So,” she said, “suppose we start with this pattern, shrink it in size and build a new pattern with the same shape.”
First generation pattern
Then, she said, just keep repeating this process. You will always need some grout, but the picture should be very pretty.
The next two generations of patterns are shown below:
As you can see, Ann was quite right. Yes, you still need grout, and, the resulting pattern is quite pretty.
Basically, what Ann had discovered (and accurately described) is a fractal – a shape with a non-integer dimension. I’ve told Ann’s story many times, but never before constructed the fractal patterns to show people. I decided to call this the Trinity fractal, named after the school where I volunteered (Trinity Parish School in Menlo Park, California.)
I lost touch with Ann, but talked with her on her first day of college at UC Berkeley, where she was majoring in mathematics.
I’m glad I may have played a small role in helping her see the beauty in this subject, and have no doubt that she has gone on to do great things.
I’m also happy to finally share her discovery with others in the hope that it encourages other teachers to move beyond arithmetic to see the beauty in real mathematics, as encouraged by the Common Core State Standards for mathematics. Our workshops on CCSS Math can be scheduled by e-mailing me at firstname.lastname@example.org Also, our work with pentagon tiling has continued in a new way. See pentiles.wordpress.com to see what we’re doing in this area!