Yesterday's post on designing four hearts got me wondering about folding and cutting paper. It's easy to fold square into eighths to get four equal hearts. I've been working on a six-petal flower in my captain idea log and, before drawing all those lines, only to find I messed up, I cut a paper model. The trick is folding the square into sixths. While I found origami techniques to fold it into rectangular sixths, I had to turn to making perfect snowflakes to figure it out. Even then, there was a little guess work in folding the thirds. I wondered if I could come up with a reliable way to fold sixths without measuring in units. And, I did as you can see in the picture below!

You might enjoy the challenge of folding perfect sixths to get twelfths so that you can cut a flower yourself. The first step is cutting out a square without measuring. Hopefully, you can. If not, nobody will ever know if you hit a search engine!

There are many ways to solve a problem. I decided to think in degrees. I pretended the square was a circle and there are 360 degrees in a circle. How do I remember that number? I remember it because a compass also has 360 degrees because it is based upon a circle. Why did mathematicians decide on 360? Here are some possible reasons:

There are many ways to solve a problem. I decided to think in degrees. I pretended the square was a circle and there are 360 degrees in a circle. How do I remember that number? I remember it because a compass also has 360 degrees because it is based upon a circle. Why did mathematicians decide on 360? Here are some possible reasons:

- 360 can be divided by 2 or 3 or 4 or 5 or 6 or 8 or 9 or 10 or 12 without getting a remainder. That makes it very useful. Nobody wants to divide by 7 or 11 anyway.
- Ancient calendars were based upon 360 days. They followed a lunar month: twelve 30-day cycles of the moon. (In reality, the lunar month is 29.5 days.)
- The Babylonian number system was based on 60, instead of ten. You can fit six equilateral triangles with a side length of the radius into a circle. 6 x 60 = 360.

I cut out a square and then began to think about the relationships between the side of the square and a hexagon formed by these triangles. That helped me figure out how to fold the square into sixths perfectly without measuring. It might be fun to challenge yourself with this in a low-stakes brain teaser.