Stellated Icosahedron

I’ve been motivated by George Hart and Zachary Abel to make my own mathematical sculpture with found objects :-). A few former students dropped by to visit me this afternoon and I put them to work making this (they had no where to be, right!?):

A cardboard stellated icosahedron

It’s a stellated icosahedron, made out of these little triangular pyramids. I did not make the pyramids, they came to me this way. Can you guess what their original purpose was?

Pop quiz: What do you think this is??

My wife and I redid our kitchen a few years ago, and I saved twenty of these from (did you guess it yet?) the packaging our cabinets came in. For each cabinet, there are 8 of these keeping the corners safe. The construction process was pretty straight forward, but here are some photos documenting the event.

Construction begins

Every vertex looks like this on the inside.

Almost done!

The last piece goes on.

Here are some more views of the icosahedron. The icosahedron has a symmetry group of size 60.

There are 15 pairs of opposite edges, each with 2-fold symmetry (for a total of 15 orientations, not counting the identity)

There are 10 pairs of opposite faces, each with 3-fold symmetry (for a total of 20 orientations, not counting the identity)

There are 6 pairs of opposite vertices, each with 5-fold symmetry (for a total of 24 orientations, not counting the identity)

So (1 identity) + (15 edge symmetries) + (20 face symmetries) + (24 vertex symmetries) = 60 total orientations.

Now I just need to find a large enough Christmas tree upon which to put this giant star!

Binder clips!

While daughter Vi Hart is off making crazy videos, including this one she posted today, father George Hart  points us to these incredible scupltures with binder clips, by Zachary Abel. (George Hart is also a mathematical scupltor.) Check out this incredible binder clip sculpture by Zachary, a piece called “Impenetraball”:

He has three sculptures with binder clips, and I thought I’d try my hand at making his simplest one, called “Stressful.”

After doing that, I have a HUGE appreciation for his larger binder clip sculptures. This was not easy to make!

Trapezoid Problem (take 2)

Am I blundering fool? You decide!

It turns out the trapezoid construction I posted earlier today is trivial. Thanks to Alexander Bogomolny for pointing out my error. The construction is quite easy (and it does not require the height), and I quote Alexander:

No, you do not need the height.

Imagine a trapezoid. Draw a line parallel to a side (not a base) from a vertex not on that side. In principle, there are two such lines. One of these is inside the trapezoid. This line, the other side (the one adjacent to the line) and the difference of the bases form a triangle that could be constructed with straightedge and compass by SSS. Next, extend its base and draw through its apex another base. That’s it.

So I redid my Geogebra Applet and posted it here. It’s not really worth checking out, though, since it’s indistinguishable from my previous applet. (In truth, you can reveal the construction lines and see the slight differences.) But I did it for my own satisfaction, just to get the job done correctly :-). Anyway, three cheers for mathematical elegance, and for Alexander Bogomolny*.

*check out Alexander’s awesome blog & site, a true institution in the online math community!

Constructing a trapezoid using the side lengths

Lloyd left a comment on a post of mine yesterday, asking:

how do you draw a Irregular quadrilateral trapezoid with fixed dimensions for the two parallel bases and the two legs with no angles given using geometry tools?
top base= 328
bottom base= 223
left leg =220
right leg= 215

How would you answer this question? It’s not trivial. You’ll quickly find that if you do a straight-edge and compass construction, you’ll need the height of the trapezoid.

If we let a, b, c, and d be the side lengths of a trapezoid with a and c as the bases, can we express the height h as

h=\sqrt{b^2-\left(\frac{d^2-b^2-(c-a)^2}{2(c-a)}\right)^2}

This number is constructable, but would take some work to actually construct it on paper. Perhaps we can return to that particular question later. For now, we can let GeoGebra show us the general idea. I’ve made this applet in which you can change the side lengths and the trapezoid will be constructed. I used the height formula above to calculate the height, and the applet shows this value.

Footnote:

Just to hit home my usual point one more time, the figure above is ALWAYS a trapezoid, even when sides b and d just happen to be parallel. Just remember that a parallelogram is a special case of a trapezoid!!

Origami Hyperbolic Paraboloid

As  I told my classes today, I went to part of a math conference this weekend (this EPADEL MAA meeting, to be specific).

The closing talk was on the Mathematics of Origami by Amanda Serenevy of the Riverbend Community Math Center. Afterward, Amanda taught some of us to fold Hyperbolic Paraboloids with a square of paper. (They are of course approximations to a hyperbolic paraboloid and the paper actually bends in non-rigid ways, which is a bit devious.) Here’s a link to the MAA website where they have instructions on how to make them. And here’s an instructional youtube video, too:

I’ve actually made one before, but I thought I’d highlight it again, since I advertised it in class today and thought my students would appreciate the instructions.

I might report more from the math conference another time soon. I had a good time!

 

Population Mean & Median

Have you seen this map, which shows the geographic center of the USA and also plots the current median and mean of the population? Very interesting! I got it from another math teacher, but I think the original source is the US Census Bureau (that’s what the bottom of the file says at least!). It inspired me to do some more poking around, and in the wikipedia article I found this map of how the population mean has moved over the last two centuries. Cool.

 

The movement of the population mean 1790-2010

 

And of course, here’s the median’s movement over time, too:

Movement of the Population Median

 

Why I hate the definition of trapezoids (again)

Sorry, I thought I got it all out of my system in my first post about trapezoids last week :-). Allow me to rant a bit more about trapezoids. First let me remind you of the problem. Many Geometry books, our school district’s book included, state the definition of a trapezoid this way:

“A quadrilateral with one and only one pair of parallel sides.”

In case you didn’t catch the point of my first post: I think this is a poor definition and should be abolished from all Geometry curriculum everywhere. Here are some pictures I recently came across on the internet depicting the hierarchy of quadrilaterals. These picture agree with the above definition. Let me just say once more, I completely and totally disagree with these pictures, and I think you should too. That is to say, all of the following pictures are WRONG.

BAD:

 

And I could go on and on. Now here are two good ones.

GOOD:

To be fair, the first set of pictures are only partially wrong. They have good intentions. Typically, the first breakdown of quadrilaterals in those pictures is by “number of parallel sides.” The first lines that come off of the word ‘quadrilateral’ divide quadrilaterals into three categories usually:

  • No parallel sides (i.e. the kite)
  • Exactly one set of parallel sides (i.e. the trapezoid)
  • Two sets of parallel sides (i.e. the parallelogram)

So the pictures aren’t wrong, per say. They just depict different information. The problem comes when teachers ask, “Look at this diagram and tell me: Is every rectangle a trapezoid? Is every rhombus a kite?” The answer to both questions is ‘yes.’ But students instinctively answer ‘no’ when using the first set pictures, and you can see why.

The problem is a historic one. If you go back to Euclid’s Elements, Definition 22 in Book 1, you can see the origin of this problem right away (a translation from the Greek):

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

In the above definition from Euclid, here are the (not perfect) translations of each figure:

  • Euclid’s square –> Our square
  • Euclid’s oblong –> Our rectangle
  • Euclid’s rhombus –> Our rhombus
  • Euclid’s rhomboid –> Our parallelogram
  • Euclid’s trapezia –> Our…trapezia/trapezium?

The last definition is a bit confusing, since we don’t have a very well-agreed upon name for this figure. But notice that ALL of Euclid’s definitions are exclusive. A square is never an oblong. A square is never a rhombus. Each of the above quadrilaterals, according to Euclid, is mutually exclusive. These exclusive definitions persisted into the 19th century.

But sorry Euclid, no one likes your definitions anymore. I hate to say it, because everyone loves Euclid.

In his defense, he wasn’t using these names for the same purpose we do. Nothing about his language is very technical and he doesn’t say ANYTHING else substantial about these definitions. He doesn’t use them to make categorical statements about quadrilaterals or to give properties that might be inherited. The names he uses are of little consequence to the rest of his work.

Can we lay this issue to rest yet? A parallelogram is always a trapezoid. Say it with me,

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

Anything you can say about a trapezoid will be true about a parallelogram (area formulas, cyclic properties, properties about the diagonals). A parallelogram is a trapezoid.

For more posts on this topic, visit here and here.