Dihedral Group

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Reuse a gag? I'll even put them in Order for you.

I was going to wait until next week to post this, but I couldn't wait. (Plus, it was quick for posting.) There will be something on Twitter in a few days.

A dihedral group is the group of rotational and reflectional symmetries of a regular polygon. If the polygon has 12 sides, as the dodecagon in the comic, then it has 12 rotational symmetries, because you can map it onto itself with every 15 degree rotation, and 12 reflectional symmetries, one for each of the rotations, plus the original. This gives it a total of 24.

In general, for any n-gon, the order number will be 2n.

And, of course, since January, the store now has a screen, and everyone is wearing masks.

Come back often for more funny math and geeky comics.

Polygon Gone!

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(C)Copyright 2016, C. Burke.

Trapezoid! I choose you!

Okay, let's get the math out of the way first.

Polygons: A closed shape, consisting of straight lines that don't overlap each other. They can be concave or convex. Regular polygons have all sides congruent, and all angles congruent. Thus, regular polygons must be convex, with interior angles less than 180 degrees.

You might want to avoid this Definition: Polygon. Or you might want to read the comments for your own amusement.

As mentioned in the comic above, if you draw all the diagonals from one point, you divide the polygon into triangles, each with 180 degrees, and the sum of those is the sum of the interior angles. There will always be two fewer triangles than there are sides, which is the source of the n less two line in the song. Note that a triangle has no diagonals and is composed of a single triangle of 180 degrees, the trivial case for this rule.

About the comic:
In case anyone reading this is new to this blog or my webcomic (did you come from a #PokemonGO or #TMC16 hashtag?), I did want to explain a little about the characters above. They are NOT supposed to be the characters from the show. They are semi-occurring characters made up to imitate the characters in the show ... a little ... sort of. Except for Melissa (aka Missy), she's new. I won't comment on Mr. Wayne's anime-styled hairdo.

The two in the bottom panel, as regular readers know, are math teachers who regularly appear in the strip, but it's summer break, so they're in the yard, having just been in the pool.

For the long-time readers who think they missed something: Judy is the other English teacher who is sometimes seen with Michele. Her boyfriend is Chuck, who appeared a few times in the early years. He works in an office with Frank, aka "Uncle Frank", who has actually appeared more often -- generally, any time I need someone old and tired to express something. Frank, a leap baby, was introduced as being 40, but he's apparently a bit older.

One of these days, I have to do a character map, or at least update a wiki page somewhere.

As for the Polygons in the picture: only two are supposed to be something. The beaked one on the bottom came from comic #957.

I wanted to include a Pentagon with an army theme, but I've done that before. (In actuality, it just wasn't going to fit. You understand, right?)

Finally, I was going to have one of the kids catch a Polygon in a Polyball, and show the figure (the above-mentioned Pentagon) inscribed inside the circle. But I couldn't fit "inscribe" or "circumscribe" in the song! There aren't a lot of good rhymes.

While I'm at it, there aren't a whole lot of useful rhymes for shape either!

Forgive the long missive. I'm not usually this wordy. If you're new here, click on the "Comic" tag to see more. Or just explore. Thanks for stopping by.




Come back often for more funny math and geeky comics.

(x, why?) Mini: Septagon

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(C)Copyright 2015, C. Burke.

I think the only reason Septagon, as in September, isn't used is because then we'd have to use Sexagon for six-sided.

This leads to misunderstandings with Sexy Primes, of course.
Sadly, there are no Septy Primes

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.

Hendecagon comic.