Math and String Games: Part 1

I have been exploring string games again, this time with an eye for the mathematics involved in them.  I have an old kids book on games, and most of them start off with what is called "Opening A" that looks something like the picture here. To make it put a loop of string behind your thumb and pinkie fingers of both hands but so that the string lies across the front of your other three fingers. Then take one index (or in some cases middle) finger and loop it through the "palm string" of the other hand. Then do the same thing with the other hand. You end up with three loops.

Opening A

In this picture I have my fingers pointed outwards, but if you point your fingers upwards it looks a bit like two wedges because the strings around the middle of index finger tend to go up at an angle from the middle outwards. 

If you look at the strings that go over the index fingers as being sort of open ended loops or sideways "U", you can see that one of them is inside the other. If you used your right hand to take the palm string of the left hand first then the loop around the right index finger should cross on top of the loop around the left index finger. Which one is on top makes a difference in some string games.





two loop loom

Some string games involve dropping the thumb loop to end up with this "two finger loom." It struck me as a bit funny that it is simplest to have three loops over six fingers and drop down to two loops over four fingers than to just create the two loops in the first place.  So I started playing with other ways of creating the two finger loom.

One of the ways is to have one loop going over both pinkie fingers and then move one index finger in, point it down towards the wrist of the other hand (in other words, point it down) and through the loop, then twist the finger to point up and put the other index finger in the same loop. Pull hands apart and presto - the two finger loom!

What I'm trying to do is see the string game both as a set of movements - a finger dance - that creates a certain configuration of string but also as a string in a particular configuration that can be created in multiple ways.

Now I've got the two loop loom and I spread my fingers wide and I move them closer together. If my fingers were impossibly thin I say the pattern was two triangles with their flat sides apart and their points pointing in together. Or maybe its four triangles all pointing together. My thick fingers make the traingles' points not really points.  I watch how if I spread my fingers apart the relative size of the triangles change. Since I cannot spread my fingers very far, they don't change a huge amount, and it feels to me like my hands are staying in the same places - the same distance from each other. Could they really be? Am I just changing the angles of the lines or am I also changing the lengths? I apply the Pythagorean theorem to understand what is going on and see that I must be leaning my hands in slightly when I'm spreading my fingers. My son helped me make this diagram. Note that the string length is 2b plus twice the square root of a squared plus b squared. The string does not actually run along line a.

Continue on to String Games part two.