The Hierarchy of Hexagons

February 11, 2014 7:13 AM

The Hierarchy of Hexagons. School geometry seems to me one of the most lifeless topics in all of mathematics. And the worst of all? The hierarchy of quadrilaterals.
posted by Wolfdog (36 comments total)

Fantastic. This is exactly how math should be taught. I'm going to be happier all day for having read that.

This is definitely the kind of classroom exercise Lockhart was after in A Mathematician's Lament (link).
posted by So You're Saying These Are Pants? at 7:28 AM on February 11, 2014 [4 favorites]

I thought this might be a Flatland thing where some of the hexagons were leaders, some were workers, etc.
posted by GuyZero at 7:39 AM on February 11, 2014 [5 favorites]

Freaking beautiful.
posted by en forme de poire at 7:58 AM on February 11, 2014

That's a beautiful proof they produced.
posted by benito.strauss at 8:11 AM on February 11, 2014 [2 favorites]

Reminds me of naming genitalia on those lazy Sunday mornings in bed with a special partner. And so my penis was named Jeff.
posted by Splunge at 8:16 AM on February 11, 2014

This is fantastic. Quadrilaterals are kind of ... dull. It makes sense to introduce them early on, so many things in human life now are boxes. However, it's not a difficult or particularly interesting set of things to categorize. Hexagons, at least, can be weird and funny enough for kids to be interested in.

Also, I am surprised one of the hexagons was not named Jayden.
posted by adipocere at 8:21 AM on February 11, 2014 [2 favorites]

By a happy coincidence, this is my (6×6×6)th post.
posted by Wolfdog at 8:25 AM on February 11, 2014 [2 favorites]

"Not all Stacys are Concave"
posted by chavenet at 8:34 AM on February 11, 2014 [6 favorites]

A long and careful exploration of the hierarchy of quadrilaterals is actually great preparation for a long life of jockeying cascading style sheets.
posted by kaibutsu at 8:42 AM on February 11, 2014 [1 favorite]

I'll have to look at this tonight.

That said... My roommate once said of me, and how true this is:

"Hexagons are my totem shape"
posted by symbioid at 8:43 AM on February 11, 2014 [1 favorite]

Love the visualizations, love that kids at that age are writing proofs, love that elementary teachers blog about their work, don't love the unnecessary implicit gender essentialist dichotomy Bob and Stacy intersection is impossible stuff.
posted by oceanjesse at 8:44 AM on February 11, 2014 [1 favorite]

the unnecessary implicit gender essentialist dichotomy Bob and Stacy intersection is impossible stuff.

wow
posted by jpdoane at 8:53 AM on February 11, 2014 [5 favorites]

don't love the unnecessary implicit gender essentialist dichotomy Bob and Stacy intersection is impossible stuff.

what?
posted by rebent at 8:54 AM on February 11, 2014 [2 favorites]

This really is excellent.

I have love-hate memories of Geometry class. At the time, I thought it was horribly boring (and even then, I usually loved math); but in hindsight, it was the only math class that most grade schoolers in my district took that really focused on proofs, and so it stood as a unique introduction to the type of content you find in typical college level math-for-math-majors (and math-for-CS-majors, and advanced math-for-engineers-and-scientists) classes. Yet I wouldn't want to tell students that, because if they were like me they might erroneously conclude that advanced math must not be any fun! There's a core of what mathematicians do that high school students only get exposed to in the most unpleasant of contexts, and it's nice to see ways of fixing that.
posted by roystgnr at 8:54 AM on February 11, 2014

I thought this might be a Flatland thing where some of the hexagons were leaders, some were workers, etc.

All vertically opposite angles formed when two lines intersect are equal, but some vertically opposite angles formed when two lines intersect are more equal than others.
posted by Celsius1414 at 8:58 AM on February 11, 2014 [7 favorites]

Getting elementary school students to develop novel proofs is pretty great stuff.
posted by [expletive deleted] at 8:58 AM on February 11, 2014 [1 favorite]

I don't think the shapes needed gendered names! It's a little thing, but it stuck out to me.
posted by oceanjesse at 9:10 AM on February 11, 2014 [1 favorite]

They aren't elementary school students, but students studying to become elementary school teachers. It's still a nice proof.
posted by Elementary Penguin at 9:12 AM on February 11, 2014 [6 favorites]

They aren't elementary school students,

Well now I feel a little embarrassed. Sorry for the derail!
posted by oceanjesse at 9:14 AM on February 11, 2014

Awwwww. It's still a cool proof, and still impressive that education student came up with it, but I'm sorry to lose my image of a kick-ass group of 9th graders named Stacey, Bob, and Mercedes. (I had assumed they named the hexagon classes after their fellow students.)
posted by benito.strauss at 9:20 AM on February 11, 2014 [1 favorite]

I don't think the shapes needed gendered names! It's a little thing, but it stuck out to me.

I happen to know this class. Stacy was named for Stacy Keach; Bob was born Robert, but is now legally Roberta, and is actually a trans woman polygon. It will all be okay.
posted by Pater Aletheias at 9:26 AM on February 11, 2014 [15 favorites]

don't love the unnecessary implicit gender essentialist dichotomy Bob and Stacy intersection is impossible stuff.

Give it credit for acknowledging that assigned sex does not always match gender self-identification: "not all Stacys are concave"
posted by justsomebodythatyouusedtoknow at 9:49 AM on February 11, 2014 [11 favorites]

I too once considered geometry listless. But then I took a geometry class in college that required all homework assignments to be generated using GeoGebra. It's so fun to play around in GeoGebra, it should be illegal.
posted by oceanjesse at 9:56 AM on February 11, 2014 [3 favorites]

I actually enjoyed all the classification stuff, and about a year ago I learned something really interesting: it breaks down if you try to model it using object oriented programming.
posted by A dead Quaker at 10:45 AM on February 11, 2014

Geometry was the only math class I've ever taken (other than Logic 101) that I ever actually liked :( (I'm a visual learner)
posted by Mooseli at 10:45 AM on February 11, 2014 [1 favorite]

I don't think the shapes needed gendered names! It's a little thing, but it stuck out to me.

I'd be willing to bet that when the teacher said that the polygon needed a name, he/she was thinking something along the lines of 'equilateral triangle' or something, and then some kid jokingly said 'Bob,' which was funny and so it stuck. From there it's natural to start naming them after people, and the second one being a traditionally female name was maybe just being fair after the first one got a boy name.
posted by nushustu at 10:47 AM on February 11, 2014 [4 favorites]

love that kids at that age are writing proofs

The students in the blogger's class seem to be elementary educators in training. These "kids" are adults.
posted by RogerB at 11:13 AM on February 11, 2014

I actually enjoyed all the classification stuff, and about a year ago I learned something really interesting: it breaks down if you try to model it using object oriented programming.

Keep in mind that an immutable circle is an immutable ellipse. But a mutable circle has no subtype relationship with a mutable ellipse. In short, immutable data 4 lyfe!
posted by a snickering nuthatch at 11:56 AM on February 11, 2014 [1 favorite]

Oh, man, I think RogerB is right. If this was something done by adults, all of a sudden this is way, way less impressive. It's still cool, but having adults figure out math is a different kettle of fish.
posted by nushustu at 12:08 PM on February 11, 2014 [3 favorites]

Holy shit this is awesome.

I wonder if this is a useful way to think about machine learning.
posted by PMdixon at 3:31 PM on February 11, 2014

Also, in my experience I would expect 10-12 year olds to have an easier time with something like this than adults.
posted by PMdixon at 3:33 PM on February 11, 2014 [1 favorite]

Oh FFS. Five weeks just to prepare for classifying quadrilaterals? And why am I not surprised that the accompanying video has the hierarchy wrong, since parallelograms are underneath trapezoids in the hierarchy?

If these teachers have trouble teaching these subjects, it is because they are bad teachers and do not know math. Dull teachers make every subject seem dull. The easiest way to make a subject dull is to drag out a week's worth of lessons into 6 weeks. Dull teachers make dull students.
posted by charlie don't surf at 7:05 PM on February 11, 2014 [1 favorite]

And why am I not surprised that the accompanying video has the hierarchy wrong, since parallelograms are underneath trapezoids in the hierarchy?

From Wikipedia:

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.[1] Others[2] define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition[3]), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid.

And from one of the external links [3],

Believe it or not, there is no general agreement on the definition of a trapezoid. In B&B and the handout from Jacobs you got the Exclusive Definition.

Exclusive Definition of Trapezoid

A quadrilateral having two and only two sides parallel is called a trapezoid.

However, most mathematicians would probably define the concept with the Inclusive Definition.

Inclusive Definition of Trapezoid

A quadrilateral having at least two sides parallel is called a trapezoid.

The difference is that under the second definition parallelograms are trapezoids and under the first, they are not.

The advantage of the first definition is that it allows a verbal distinction between parallelograms and other quadrilaterals with some parallel sides. This seems to have been most important in earlier times. The advantage of the inclusive definition is that any theorem proved for trapezoids is automatically a theorem about parallelograms. This fits best with the nature of twentieth-century mathematics.

Don't be too quick to get snooty about definitions.
posted by Wolfdog at 3:56 AM on February 12, 2014 [3 favorites]

This is exactly where to "get snooty." If a parallelogram is a type of trapezoid, the hierarchy is easier to learn, and it is consistent with other math where you might actually use this concept.
posted by charlie don't surf at 5:25 AM on February 12, 2014

BTW I am at work and pulled up this article as reference material for the task I am doing today, Five Easy Pieces: Quadrilateral Congruence Theorems. I recommend looking at the first page, which has a Venn diagram that makes this so damn easy, but it all depends on a parallelogram being a trapezoid. And then it goes on to use quadrilateral congruence theorems in non=Euclidean space, where this version of the hierarchy is essential.
posted by charlie don't surf at 7:47 AM on February 12, 2014

Dammit I should have left well enough alone. I checked with our math PhD ABD and he says he likes parallelograms not being trapezoids better. Oh well.
posted by charlie don't surf at 9:56 AM on February 12, 2014 [2 favorites]