An Evening of Math II

Tonight I gave the kids the choice of doing more partitions, or something else. They chose something else. In the book Math Tricks, Puzzles & Games by Raymond Blum I found this problem:

TUNNELS

Try to connect each rectangle with the triangle that has the same number.  Lines cannot cross or go outside the diagram.

When I showed the problem to the kids, A was very upset and said that she didn’t know how to do it because she had never learned.  She threw a fit and started drawing a picture instead of working on it.  B immediatly started working on it.  He worked for about 20 minutes, drawing lines, erasing, drawing more lines.  He eventually got frustrated, saying that it was impossible.  In spite of A’s protests I brought her over next to B and had him explain what he had tried, and why it was impossible.  B connected rectangle 1 to triangle 1, then connected rectangle 2 to triangle 2, but at that point rectangle 3 was completely cut off from triangle 3.  Then A took a pencil and a copy of the puzzle and without pausing connected rectangle 1 to triangle 1 and rectangle 3 to triangle 3.  Just as B was saying that she wouldn’t be able to connect the last pair, she did!

Getting A to look at a problem is 95% of the battle.  I reminded her that it was B’s work on the problem that helped her to her solution.

Math with Balloons

In demonstrating what a mathematical knot is, balloons are the best medium that I’ve found.  They have enough stiffness to keep the knot from collapsing so that you can’t see its structure (a string or rope would either be floppy and hard to work with, or you would have to pull the knot tight, which would make it hard to see it’s structure).  They have enough give to allow knots to be tied in them.  They are very visisble and the ends can be twisted together to complete the loop.

Here I am with not a knot, but a link: the Borromean rings.  It has beautiful symetries that have been recognized for centuries and is important to low dimensional topology (in fact, an early incarnation of my dissertation featured the Borromean rings).

Knot Tying Schema

I’ve noticed that whenever my two-year-old wants to tie a knot, she takes the two strands that she’s interested in and repeatedly twists them around each other.  Of course this doesn’t produce any kind of knot at all (or I should say it only produces the unknot).  But this twisting appears to be the only basic move that she knows (and she learned it herself) for producing knots.

It made me wonder what the basic moves are that we use to tie knots.  Every knot theorist knows that there are three basic moves for transforming one picture of a mathematical knot into another (possibly quite different looking) picture of the same knot.  These moves are called the Reidemeister moves.  Any two pictures of the same knot can be made to look like each other using just these moves (and “plane isotopies”).  Similarly it seems to me that mathematically you really only need two moves to tie any knot, or what I mean to say is to go from a picture of a straight string to a picture of any knot.  Here is an illustration of those two moves:

Of course after doing several of these moves, you will want to glue the two ends of the string together if you wnat to get a picture of a mathematical knot.

When people tie knots, it does seem like there is some finite set of moves they use, but the catalogue of moves seems to be bigger than these two moves I illustrated above.  For instance there are moves like “wrap one end around a loop” or “put one end through a hole” or “follow one strand through a whole series of moves.”  Of course each of these can be broken up into a sequence of the two basic moves, but it is not always useful to break things down into the simplest possible moves in practice.  It makes me wonder what the catalogue of moves is that people use.

More Experimental Topology and Experiments in Topology

I visited my friends Peter and Liz (I stayed a few nights) and came back with a laundry list of things to post about:

1) They have some really cool polyhedra and mathematical quilts, so sometime I am going to have to go over there with a camera and click some photos. Peter’s latest quilt project is one tiled with spidrons, which is in the design phase now.

2) Peter was a little upset that I failed to credit him for pointing out to me originally that and expanded base is (see synchronicity) and asked if there were any other examples of this. Anyway, thanks, Peter, for pointing this out and inspiring so much recreational math.

3) I finally carried out that idea I had that I mentioned back in the post more experimental topology about making a five pointed star. It took some trial and error (hey! that’s what experimental topology is all about. Right?) but here is what I came up with:

Start with ten strips of paper, with one end cut into points. The points should come to an angle of approximately .

Draw a line down the center of each and tape them together so that the tips are all touching:

Now start taping opposite ends together, like so:

When you have four of the five opposite pairs taped together, the last one needs one full twist. If you’ve been proceeding in a clockwise direction when taping opposite pairs together, you should make one full twist in the last pair by turning the strip closest to you in a counter-clockwise direction. Anyway, in the end you should get something like this:

Now, cut along all of those center lines, and what do you get?

A mess, but if you untangle the mess, you should get:

A pair of stars that are linked!

Topologically this is the Hopf link.

4) Peter also loaned me a couple of books:

Experiments in Topology

I haven’t read much yet, but it has some cool stuff, like if you have a strip of paper, one inch thick, what is the shortest Moebius strip that you could make?

Also he lent me

Goedel, Escher, Bach: an Eternal Golden Braid

(gosh, that’s a really small image of the book, but, oh well). I’m about a fifth of the way into this book. Excellent so far. Won the Pulitzer Prize! I’m learning lots about the works of Goedel, Escher and (believe it or not) Bach.

That’s all for now, but soon I will have to blog about my visit to my kids’ classes for career day, which was last week….Stay tuned.

More Experimental Topology

I was thinking that there might be a way to get a hexagon along the lines of the methods of the post Math with scissors. I told my kids about the idea and they were excited to do some math experiments.

I started with three strips of paper.

Taped them together like this:

Then I wanted to tape the three ends up like this,

but I knew there had to be some twisting of the strands involved. In the end we are going to cut each of the strands down the middle.

So we experimented. I knew that at least two of the strands had to have some odd number of half twists in them to have any chance of getting a hexagon (can you see why?). It took us several tries, but my son and I both came up with our own solutions for how to get a hexagon:

Try it and see if you can get it. Here are our solutions:
My son’s solution: Put a single half twist in each of the three strands, twisting two in one direction and the other one in the opposite direction.
My solution: Put a half twist in each of two of the strands, twisting them in opposite directions. Leave the third strand untwisted.

Of course, the experts will want to conjecture and prove necessary and sufficient conditions to get a nice flat hexagon. I also have an idea for making a five-(or more)-pointed-star along similar lines. I’ll let you know what I come up with.

Math with scissors

The other day I visited my son’s class and did the following math demonstration.

Take a strip of paper and draw a line down the middle. Our line is red.

Tape one end to the other, but put a “half twist” in it, just like shown here. This is a Moebius strip. It only has one side to it. A pretty cool object in and of itself.

Now for some real fun. Take a pair of scissors and cut the Moebius strip in half along the red line–But before you do that try to guess what you will end up with!!! I’m not going to show you. You have to try it yourself. Just make sure that you only cut along the red line!

There is more that we can do. Let’s make a + out of paper with two red lines drawn as shown here.

Now tape two opposite ends to each other with no twists. Like this.

Tape the other two ends to each other, but put in a half twist. What you have here is actually a Klein bottle (whatever that is) with a hole in it.

Now cut along the two red lines. Make sure not to make any other cuts except along the lines.

Keep cutting…

Keep cutting.

In the end what do you get?

(Make a guess first.)

(!)

One more: Make a star shape just like this. Make sure that all of the strips are the same length and that the angles between the strips are the same. Draw three red lines. One down each of the strips.

Here is what it should look like.

Now tape two opposite ends together with no twist.

Take another two ends…

…and tape them, with no twist, just like this.

Now for the tricky part. You have to do this part right to get nice shapes in the end. But, don’t worry too much. If you do it “wrong” you’ll just get something other than what we got. Flip your paper over. Tape the last two opposite ends with a full twist to the left. Just like shown here.

Here is another view. Now cut along the red lines and see what you get. Bailey’s class loved it.