Wish I Had Said That
The Northern Virginia Math Teachers Circle (NoVaMTC) met for the first time last night. Professor Bob Sachs from George Mason University led us through a fantastic activity about Conway Rational Tangles. A Google search will provide lots of information about them; however, you’ll have more fun by investigating them yourself. Let me give you a quick overview…
Take two pieces of rope, each about 10 feet long. (It’s easier to see what happens if the ropes are different colors, but it’s not a requirement.) Give each end of the rope to a different volunteer. Of course, this implies that you’ll need four volunteers. But to explore, you can just take two small pieces of string and perform the moves yourself. Call the four volunteers A, B, C, and D, and assume they are arranged as shown below.
With the ropes arranged in this way, there are two possible moves:
Twist: The two people on the left switch positions; the person in front lifts his rope, and the person in back goes underneath the front person’s rope. It is very important that this move is always done consistently. That is, it must always be the two people on the left who switch positions, and the front person must always lift the rope over the back person.
The results after one and two twists are shown below.
Rotate: All four volunteers move a quarter-turn clockwise; from the original position, A would move to B, B would move to C, C would move to D, and D would move to A.
There is also a third move, Display, which just means that the volunteers closest to the audience lower their ends of the rope, and the volunteers further away raise their ends of the rope. That way, everyone in the audience gets a good look at the ropes. This move is only important if doing the activity with a group.
And basically, that’s it. Now do a few moves to sufficiently tangle the rope. The question is:
Is there a series of twists and rotates
that can be used to untangle the ropes?
You might find it surprising that it is always possible to untangle the rope using only twists and rotates. But even knowing that, figuring out how to do it is far from trivial.
Try it for yourself. Put just one twist in the rope. Now, what series of twists and rotates will untangle the rope?
Try it with two twists. Or maybe twist, rotate, twist. Or maybe twist, rotate, rotate.
Is there a way to codify this system? Given a series of twists and rotates to sufficiently tangle the rope, could you describe the series of twists and rotates that would untangle the rope? That’s the ultimate goal. Play with it for a bit.
Okay, that’s enough of an introduction to Conway Tangles. Take some time to play. If you want to learn more about it, I recommend Tom Davis’s description of how to use this activity in a classroom.
As the NoVaMTC group was exploring tangles, some of the members were hung up on the notion that just untangling the ropes was not sufficient; they also wanted the four volunteers to be returned to their original position. When they asked Professor Sachs about this, he responded that it’s like the Albert Einstein quote:
Make things as simple as possible, but not simpler.
He explained that when solving a problem, he’ll make some assumptions that he might revisit later. But at the beginning, he’ll analyze a simplified model first, hoping that it offers insight for how to solve the original problem. Such is the case with tangles: Let’s explore the problem by first figuring out how to untangle the rope, then later maybe we’ll return to the problem of getting all four volunteers back to their original positions.
This reminded me of the George Polya quote:
If there is a problem you can’t solve, then there is an easier problem you can solve: Find it.
Have fun exploring Conway Tangles!
For a diversion, here are a few other mathy quotes that you might enjoy…
A math student’s best friend is BOB (the Back Of the Book). Just remember that BOB doesn’t come to class on test days.
Tell students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
What is now proved was once only imagined.
Those who say it cannot be done should not interrupt those who are doing it.