Last Time | Next Time |
You may bring your own knots to the quiz.
Some reading, and try to tricolor some knots.
Some of you evidently didn't realize it, and so have not submitted that. If you would like to, submit it by Thursday.
Then we'll have another art contest. But we have an early "judge's choice" winner: Mae's knot pretzels. Here they are,
Now, if
They were tricky!
On this problem I ended up just looking for good reasoning: Both of these wallpapers actually have \(R_2\) symmetry, and the one on the left has two lines of reflection (whereas the one on the right has none).
Then we look into the types of centers of rotations....
while hexagons (with their 6 sides) suggest that an \(R_6\) is possible, the inside "flower" has 16 petals. In order for both of these patterns to fall back on themselves, the rotation must be a common divisor of 6 and 16. The ONLY common divisor of both is 2: \(6=2*3\) and \(16=2*2*2\).
The wallpaper also has two perpendicular lines of reflection.
In the end, that drives us to conclude that this is \(D_{2_{kgkg}}\).
But our pattern only has symmetry about two centers, whereas \(C_2\) has three. Thus, by deduction, we've discovered the pattern for this wallpaper, and it's \(D_{2_{gggg}}\).
How do we distinguish knots?
This is the most beautiful piece of mathematics I know:
It turns out that there are golden rectangles in the heart of an icosahedron, interlocked as Borromean rings!
We'll make some using 3x5 (or 4x6) cards -- which are not quite golden. Their corners are the 12 vertices of the icosahedron. The tricky part is locking them together!
But we can easily figure out which is which, because the knot is a single continuous piece of material, whereas the link is two separate pieces of material.
It's all about overs and unders!
Type I | Type II | Type III |
The third images on that page stems from this picture from a recent Science issue:
In order to consider the picture on the left a knot, we have to know what its ends are doing. In the figure at right, I assumed that they are just connecting to each other in the simplest way.
The succession of steps then go on to show that the knot is actually an unknot! That's good news for the hypebusy people!
Here's another picture of an unknot, which could trick you -- but knowing the R1 move saves you:
We'll just try a few with a string, to see what we can learn.
Reidemeister Move I is tricolorable. | Reidemeister Move II is tricolorable. | Reidemeister Move III is tricolorable. |
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So, for example: if you've got your picture of a knot down to three crossings, and it's not tricolorable, then it's the unknot.