Here are three ~~dodecahedra~~ dodecagons (that is, 12-gons) a group ofÂ students I work with put together out of pattern blocks.

I have some thoughts about them, but I’d like to throw out a challenge first. What questions can you concoct about this picture?

List them in the comments I’m curious to see what we’ll get. More later.

1. Why on earth would you break the symmetry by using a parallelogram instead of two triangles in the biggest one? I’m convinced you must have run out of triangles.

2. How many ways can you make a dodecaheron out of regular polygons?

3. What are the angles on the rhombuses?

4. What’s the interior angle of a dodecahedron?

5. If you try to make a tiling out of dodecahedrons, what shapes get left in the gaps?

6. Are the two smaller dodecahedrons the same size?

7. How much bigger is the bigger one?

8. Did you know that you can find the bottom left one in Lockhart’s book, Measurement?

In the biggest one, we’re depending on the edges of adjacent diamonds forming a straight line. Do we know that they do?

(I think these are dodecagons, not dodecahedra. I think the -hedra means 3-dimensional.)

Are the two smaller ones the same size? Is the bigger one an even multiple of them? How can you figure out the area of these shapes? If the two smaller ones are the same size, what (if anything) can you infer about the relative area of the square and the skinny diamond?

I had a couple of different questions.

For the first one, I can see why by symmetry it is a regular dodecagon.

For the second (bottom left), I can see why it is equilateral, but it takes a moment or two of thought to see why it is equiangular, so how about “Is this second one a regular dodecagon?”

For the third one, it’s not even completely obvious that it’s a dodecagon and not whatever you’d call a 24-gon. I mean, there are 24 line segments on the edges, so there’s some claim here about certain angles being straight angles. So “Is this third one a regular dodecagon?” Hm, now I see that Sue already asked this question.

As an extension of that question: Are there any holes inside these dodecagons? I mean, I can see holes … but with ideal geometric shapes would they fit together perfectly?

The tiling of the larger one, after pairing the green triangles, is entirely done with rhombuses. Can the smaller one be made entirely of rhombuses?

This last question reminds me of question 2 from the 2002 Bay Area Mathematical Olympiad at http://mathcircle.berkeley.edu/newsitedocs/bamodocs/bamo2002examsol.pdf

which in turn makes me wonder about the question of how many ways there are to make the smaller one entirely of unit-length rhombuses.