D Combinatorics
2 days ago
Sunday, January 28, 2007
Identify this
There are two objects to identify. The first three pictures are of the first; the next three pictures are of the second. If you can identify it other than "a knot made out of straws and dental floss for your thesis" in a mathematically intelligent way, you get a prize, like a pipe cleaner or something.
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7 comments:
They're like, uh, twisted prisms. Kind of. The first one looks like taking a pentagonal prism, giving the top a 2/5 twist with respect to the bottom, and then superimposing it with its mirror image. The second one is a hexagonal prism with a 5/12 rotation superimposed with its mirror image. I don't know if that counts as being mathematically intelligent, but I do like pipe cleaners.
Hint: The white straws are "virtual sticks," meaning that they are only there to hold the clear red striped sticks together in an appropriate way. The clear red striped sticks are the important part of the object.
Ah! The topic of your thesis is beginning to come back to me. Respectively, they are a "regular" decagon and dodecagon -- "regular" in the loose sense that they are composed of identical line segments with identical angles between each, except they are embedded in 3-space. Intuitively, by staring at the pictures it seems like they would also be sort of "flexy" in the sense that you can continuously deform them in such a way as to change the angles while still keeping them "regular". Unfortunately, I don't remember what the significance is.
Right. We call it "alpha-regular," where the angle at each vertex is some alpha. So this is an alpha-regular knot for my thesis. Well done.
Now the next thing you might notice is that all the sticks that are on the outside go "counterclockwise and up" while the ones on the inside go "counterclockwise and down."
Also you might notice is that if you start with a vertex at the bottom, then follow it up, then back down, you end up at an adjacent vertex, having gone 4/5 or 5/6 of the way around depending on if you are looking at the pentagon or the hexagon.
These observations would help to identify what sort of knot it is.
Brent, this is not going to help your bridge game.
So the answer is a (4,5) torus knot and a (5,6) torus knot. They are called this because of the 4/5 and 5/6 I mentioned previously.
Ah yes. I knew what kind of knots they were, I just forgot to check back here (there seems to be no way in blogger to subscribe to the comments on a particular post, like there is with wordpress etc.). Anyway, I'm going to go read your thesis now. =)
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