games,  geekiness,  homeschooling,  mathematics

String Games and Mathematics, part 2.

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The strings create loops and spaces, but the divisions are in many ways arbitrary.In my last post on string games I talked about Opening A and dropping the thumb strings to create a two loop loom. Today I want to write about what happens when you take the palm string of Opening A. When you just take the index or middle fingers, put them under the palm string of the opposite hands and pull straight out you get three loops.

Now here I run into a terminology problem. When I said you get three loops, I meant that you get three spaces bordered by string. My first picture represents those spaces with three different shades of blue. However “loops” could represent something different, so maybe I’ll refer to those spaces as spaces. Loops could represent segments of string that wraps around one finger. In the same picture I represented the middle finger loops with a red < and a yellow >.  Of course the bottom part of the red < could be seen as the top half of another > around the corner of the bottom space. Even with this second definition of loop, the segment of string belongs to more than one loop.

Drawing out charts of the string positions helps me see what is happening and why moving a loop here or there creates what it does.If you put the finger between the palm and palmstring from the top, instead of the bottom, and then twist to point the finger up as you pull away you can put a twist in the string.  Put a single twist in both middle finger loops and you’ll get a handful of strings that looks something like this: two parallel lines on the outside and four lines that come together in the center and then branch out to touch your thumb, both sides of your middle finger and your pinky finger. The middle strings all bend.

A beautiful X is created by twisting opening A and then moving the middle finger loop to extend behind the whole hand. Don't you want to play Cat's Cradle yet?I made this little diagram to show how the strings bend, and I’m numbering the places where the strings could touch. Each finger has two numbers, because it has two sides from which the string could touch. Then I can play with the numbers and formulas. If a string segment starts from one hand from a place with a number under 6, then it will touch the other hand at that number + 4. If the starting number is more than 6, then it will touch the other hand at that number – 4. The exceptions are the outer segments of string that start at 1 or 10, in which case they end at the same number.

 If you take the loops that go around your middle finger and gently place them so they loop around your thumb and pinky finger (and across the whole back of your hand) then the bent line segments straiten and you get this neat little figure. Now if you use the same number scheme as explained above, the four inner strings segments all connect to numbers that add to 10.

If you let the strings on the back of your hand slide down your wrist the parallel strings (shown here in black) end up being caught by the crossing strings and pulled down. It creates a cool illusion because it looks like those strings should be the ones wrapped around the wrist. It looks like you should have three separate loops. I’ve done two pictures, one to show what it appears, and the other showing a part of a trail the string takes. Can you picture flipping the black line to make the rest of the path? Could you rotate it?

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