12-gon Puzzle Challenge II

January 24, 2013

Here are the questions folks contributed:

From Paul:

  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 12-gon 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 12-gons, what shapes get left in the gaps?
  6. Are the two smaller 12-gon 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?

    From Sue VanHattum:

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

    From Elizabeth:

  10. Are the two smaller ones the same size?
  11. Is the bigger one an even multiple of them?
  12. How can you figure out the area of these shapes?
  13. 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?

Thanks for these great questions! And also, for the demonstration that a well-developed curiosity can lead us from a rich environment in the pursuit of some pretty serious explorations.

Let’s consider question 11: Is the bigger one an even multiple of the two smaller 12-gons? (I’ll assume that the two small ones are the same size… though that’s worth thinking about too–see questions 10 & 6. Also see question 7.)

On the one hand, it seems like it would be pretty incredible if it were. That would mean you could take some number of the small 12-gons, break apart their pieces, and arrange them into the larger 12-gon. Is it possible?

Some bookkeeping should help us. Counting the pieces in the big 12-gon, and taking advantage of its 12-fold rotational symmetry, we see that it consists of:

24 tan rhombuses
22 triangles
12 squares
13 blue rhombuses

We can see, and know from earlier play with pattern blocks, that 2 triangles = 1 blue rhombus, so we could rewrite this list as

24 tan rhombuses
22 triangles
12 squares
13 blue rhombuses = 26 green triangles


24 tan rhombuses
22 +26 = 48 triangles
12 squares

What about the small 12-gons?

Type 1 has 12 tan rhombuses and 12 triangles.

Type 2 has 1 hexagon, 6 triangles, and 6 squares, which is the same as 12 triangles and 6 squares.

Wait a minute! 6 squares plus 12 triangles is the same size as 12 tan rhombuses and 12 triangles! That must mean that 6 squares = 12 tan rhombuses… which must mean that 1 square = 2 tan rhombuses! Not obvious at all (and answers question 13, incidentally).

In any case, can we use 12-gons of type 1 and type 2 to actually build the big 12-gon?

The answer, amazingly, is yes! Two 12-gons of type 2 give you 24 triangles and 12 squares; Two 12-gons of type 1 give you 24 tan rhombuses and 24 triangles. Sum those pieces and you have precisely what you need to make the bigger 12-gon!

So, the big 12-gon is precisely four times the area of the small 12-gon. And you can literally break apart four of the small 12-gons to build the big one. Which raises another question for me:

14. How many of the little 12-gons do you need to build an even bigger 12-gon?

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