We’ve been talking a lot about 72 degree angles recently. Yesterday’s project was about a question our friend Paula Beardell Krieg asked:
Paula Beardell Krieg’s 72 degree question
In that project we learned that a right triangle with angles 72 and 18 (pictured below)
Is nearly the same as a right triangle with sides of 1, 3, and
Today I wanted to show the boys a neat surprise that I stumbled on almost by accident. The continued fraction expansion for the cosine of the two large (~72 degree angles) are remarkable similar and lead to the “discovery” of a 3rd nearly identical triangle.
We got started by reviewing a bit about 72 degree angles:
Now we did a quick review of continued fractions and the “split, flip, and rat” method that my high school teacher, Mr. Waterman, taught me. Then we looked at the continued fraction for 
Now we looked at the reverse process -> given a continued fraction, how do we figure out what number it represents? Solving this problem for the infinite continued fraction we have here is a challenging problem for kids. One nice thing here was that my kids knew that they could do it if the continued fraction had finite length – that made it easier to show them how to deal with the infinitely long part.
Finally, we went to the computer to see the fun surprise:
Here’s that 3rd triangle:
I love the surprise that the continued fractions for the cosine of the (roughly) 72 degree angles that we were looking at are so similar. It is always really fun to be able to share neat math connections like this with kids.
Mike's Math Page 
When I used to go whitewater kayaking we would often stop and play with a wave. I always liked that no matter how much we surfed on, did enders in, or challenged the waves those waves just didn’t wear out, they just kept on giving. They way these triangles have stood up to your playing with them in different ways day after day reminds me the waves. Endlessly responding, endlessly surprising. I guess number are like waves.This continued fractions thing you did is so cool.