## Thursday, February 05, 2015

### Speaking of Polygons

Today's comic had its genesis in a class a couple weeks back on finding the Measure of Interior Angles of Regular Polygons. As I have done in the past, the instructor (facilitator) informed the students that they wouldn't have to memorize the names of the polygons to answer questions, such as "Which polygon has seven sides?", but they would have to recognize the name of the polygon when they saw it and know how many sides it contained.

The majority of problems of this nature tend toward 5, 6, 8, 10 and 12-sided figures. (That's pentagon, hexagon, octagon, decagon and dodecagon for those of you playing along at home.) This could be because those are the easiest numbers to work with when it comes to the number 360 (yes, and 9, too, but...), and after five-sided pentagon, it's easier to draw regular shapes with even-numbered sides. (At least, that's my experience.)

Heptagon and nonagon were given, not just for the sake of completeness, but because they do turn up, even if not as often. After asking about how many they had to learn, one student asked "How many names are there?" A reasonable question. There are classifications for larger polygons, but they can default to n-gon, where n is the number of sides. And n can be 10 or 11 or 12, if we wanted it to be, but would more likely be 15 or 18 or 20, a larger number but one that would be easier to work with.

This had me wondering about the prefixes themselves. We had names for them up to 12, which corresponds to English words for numbers. We have a base 10, but we have names going up to 12 (which makes sense given imperial units). At thirteen, we start with "three and ten", then "four and ten", etc. However, in high school Spanish I, we had to learn names up to 15 -- once, doce, trece, catorce and quince -- (Yes, I remembered them, but I did double-check the spellings.) -- before we get to "ten and six", "ten and seven". So it's an arbitrary designation.

Years ago, at the request of a different student (obviously, as it was years ago), I looked up more names and found that an eleven-sided polygon was an undecagon. This made sense to me, as it was "one and ten", the way that dodecagon was "two and ten". And the pattern continued after that.

So when a student asked this class's instructor, "What about 11 sides?", I was a little surprised when she said, "A girl in my other class just looked that up. It's a hendecagon."

What?

Yes, I not-so-immediately, slightly nonchalantly, started typing on a computer in the corner. You learn new things. What did I learn?

Eleven in Greek is hendeka but in Latin it's undecim. The form undecagon is, therefore, a hybrid construction while the "cleaner" hendecagon is not.

This does lead to a problem with thirteen. The hybrid 13-sided name is tridecagon while the all-Greek variation would be triskaidecagon, which, while familiar to anyone who has heard of triskaidekaphobia, is nowhere nearly as easy to spell.