I finished my Straws Thingy!
(You can see bigger versions of all the photos here.)
It’s made out of 60 straws, 12 of each color. Three straws make a triangle; four triangles of each color are woven into a tetrahedron; there are five interlocking tetrahedrons, each of a different color. The whole thing has the same symmetry as an dodecahedron or an icosahedron (dodecahedra and icosahedra are dual, so they have the same symmetry). It’s easiest to see the similarity to a dodecahedron since the vertices of the tetrahedra correspond to vertices of the dodecahedron.
Here’s a face-on view showing 5-fold symmetry. That is, each “face” is a pentagon:
And here’s a vertex-on view showing 3-fold symmetry. A dodecahedron has three pentagonal faces surrounding each vertex:
And here’s an edge-on view showing 2-fold symmetry:
Of course, all the above talk of symmetry is just referring to the shape: if you take the colors into account, it has no symmetry at all! Any way you turn it will result in a nontrivial permutation of the colors.
Imagine that all the straws in the Straws Thingy were the same color. Now imagine that you put it on a table, tell a friend to close their eyes for a minute, and then you do something to the Straws Thingy. When they open their eyes they can’t tell that anything has changed. For example, you could rotate it by 1/5 turn around one of its pentagonal faces. The individual straws would be in different places, but overall the Straws Thingy would look the same (since the straws are all the same color). However, you couldn’t turn it only 1/10 of a turn, since then it would be in a different orientation and your friend could tell that something had changed. There are also other, more complicated things you could do, like turn it 2/5 of a turn, turn it completely upside-down, and so on.
If you consider the set of all possible things you could do to the Straws Thingy so that your friend could not tell the difference, it turns out that these form a group!1 The combining operation is to do the two things in sequence—this must still leave the Straws Thingy in an indistinguishable position, so there must be a single thing in the set you could have done that has the same effect as doing the two things in sequence. For example, turning 1/5 of a turn around a pentagonal face and then turning 2/5 of a turn around the same face is the same as just turning 3/5 of a turn. The identity element is where you just leave the Straws Thingy alone (which definitely counts). And everything has an inverse, because you can always just do the opposite of whatever you did.
If you have a Straws Thingy, or some other sort of dodecahedron or icosahedron, you might enjoy thinking more about what all these different operations are. How many are there? There is a very deep connection between the symmetry of the Straws Thingy and this group of operations that leave it looking the same. In fact, one way to think about group theory is that it is nothing less than the study of symmetry itself.
This group is often called . It is an alternating group. If you’re curious I’ll let you look up those things to read about them.↩
Nice! I hope it was fun, and I love that you connected it back to your NaBloPoMo series via group theory.
I suggest that you mention chirality. The Straws Thingy is distinguishable from its mirror-image, even when colourless. It was the first thing I thought of when you made the “as symmetric as the icosahedron” claim.
Yes, very good point. You can look at any vertex and see which way the straws weave, AND you can look at any face and see which way the tetrahedra weave. There are two different choices in both cases. I should have been more careful in my phrasing—you’re right that a straws thingy does not have the same symmetry as an icosahedron if you take mirror reflection into account (since I only said “the symmetries of an icosahedron”, mirror reflection should have been included). I was thinking about about manipulating a physical icosahedron, which you can rotate but not turn into a mirror image of itself. This rotational subgroup of an icosahedron is indeed the alternating group , and is the same as the group of symmetries of a straws thingy. The entire group of symmetries of an icosahedron, including mirror reflection, is .
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