## Posts tagged ‘rotate’

### Never Ask a Mathematician for Directions

I spent my formative years in a very rural county. Roads didn’t have names, or at least they didn’t have road name signs.

In college, my urban friends used to claim that if you asked someone in my hometown for directions, they’d say:

Go about a half mile, and turn left at Old Man Johnson’s farm. Then take a right at the huckleberry tree.

That very well may be true, but it’s no worse than the directions you might be given by a mathematician:

Well, you’re facing the wrong way, so do a reflection about this cross street. Go about 0.628397154 miles, then rotate π/2 radians and travel orthogonally to your previous vector for 600 light-nanoseconds. But they’ve got radar on that road, so keep your speed between 7

^{2}and 4^{3}miles per hour. Turn left, and you should reach your destination in 2.4 minutes ± 0.3%, if you maintain an average speed of 46.8 mph.

Now, that’s bad. But at least I could understand all of it. Which is more than I can say for the directions that Google Maps provided yesterday. Traveling through the suburbs near Washington, DC, I had just crossed MacArthur Boulevard, and I was heading southwest on Arizona Avenue. As you can see from the screenshot below, Google Maps was suggesting that I turn right on Carolina Place, right on Galena Place, right on Dorsett Place, and then left on Arizona. In essence, it suggested that I reverse direction to take a 10-mile, 45-minute route.

I ignored that suggestion. Instead, I stayed on Arizona Avenue with the intention of turning right onto Canal Road in a quarter-mile. Just as I passed Carolina Place, Google Maps said that it was “Rerouting…,” and within 15 seconds, it confirmed that I had made the correct choice:

By ignoring Google Maps, I shaved 3.8 miles and 23 minutes from my commute.

WTF?

My speculation is that Google Maps attempts to avoid my chosen route because it follows Canal Road, which parallels the C&O Canal National Historic Park; it requires me to cross Chain Bridge, which offers a beautiful view of the Potomac River; and it then winds through an affluent neighborhood, where I can feel safe on tree-lined streets with elegant homes. Honestly, who would choose that when Google Maps is offering double the travel time and an opportunity to drive on the beltway?

I once asked Google Maps which highway I should take to California. It replied…

Oh, yeah. Root 66.

The logic employed by Google Maps reminds me of a college friend…

He would always accelerate when coming to an intersection, race through it, and then brake on the other side. I asked him why he went so fast through intersections. He replied, “Well, statistics show that more accidents happen at intersections, so I try to spend less time there.”

### Swish, Swash, I Was Doin’ Some Math

I’m not sure *what* I want to be when I grow up, but I know *who* I want to be. That would be Bill Ritchie, owner and founder of Thinkfun^{®}, a company that makes and markets games and puzzles, including the wildly popular Rush Hour^{®} game.

Our house has recently become addicted to Swish^{TM}, one of Thinkfun’s newest games. *Swish* is a spatial card game played with transparent cards that can be rotated and flipped to make swishes. Swishes — named after the sound made by a perfect basketball shot going through the net — are made by layering as few as two or as many as 12 cards so that every ball swishes into a hoop of the same color.

For instance, the two cards below can be combined to make a swish. The orange dot on the left card aligns with the orange hoop on the right, and the blue hoop on the left aligns with the blue dot on the right, which forms a double swish when the cards are placed one on top of the other.

Here’s a more advanced example. These four cards can be layered to make a quadruple swish:

The examples above show the cards in perfect alignment. They just have to be placed one on top of another for the dots and hoops to align. What makes the game fun and challenging is that cards typically aren’t in perfect alignment like this. One or more of the cards will have to be flipped or rotated to make a swish. In addition, the set-up arranges 16 cards in a 4 × 4 grid, so the two complementary cards are rarely next to one another. Consequently, the game uses all three geometric transformations — reflections, rotations, and translations.

Each card contains one hoop and one dot. These two objects are placed in one of 12 regions — each card is divided into an imaginary 4 × 3 grid. Theoretically, it’s possible to make a 12-tuple swish… though I’ve yet to pull off such a feat during a game.

There are a number of interesting math questions that can be asked about the game:

- Will there always be a swish in any 4 × 4 array of cards?
- The deck contains 60 cards. Does that account for every possible combination of hoops and dots? [The answer is obviously no, but that leads to some interesting follow-up questions.] How many different cards are possible? Finding the total isn’t as trivial as it sounds, since cards can be rotated or flipped; duplicates need to be removed. Which cards were included in the deck, which ones were excluded, and why?
- How many different double swishes can be formed by the cards in the deck? How many triple swishes? … How many 12-tuple swishes?

I don’t have answers to all of those questions yet. But I look forward to discussing them (and others) when I introduce *Swish* to some colleagues at an upcoming math conference!

### 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.