now i begin to flesh out the possibilities when exploring a simple abstract 3 loop system of variations. in this particular example i use variations of spinning the index loops with a canonical system of operations:

• thumb loops up through index loops and return

• little finger loops down through the index loops and return

then the strings on the hand rotate the three loops so the figure has been rotated + 180 degrees if looked at from the side. i call this my iterative move since i use it in many of my figures.

Iteration Move

• Pass each thumb away from you under all the strings, pick up the far little finger string, and return; drop the little finger loop (this transfers the little finger loop to the thumb and introduces a +1/2 spin).

• Transfer the middle finger loop, inserting the thumb from below, then re-transfer this loop to the little finger, inserting the little finger from above (this move shifts the middle finger loop to the little finger while simultaneously introducing a +1/2 spin).

• Transfer the index loop to the thumb, inserting the thumb from below, then re-transfer this loop to the index, inserting the index from above (this introduces a +1/2 spin).  The entire figure has now been rotated a half turn away from you.

this investigation shows how to form figures with the following iterative set of operations

• a rotation  of the index loops +1/2

• a rotation  of the index loops +2/2

• a rotation of the index loops -1/2

• and mixtures of these simple rotations

the first figure uses a +1/2 rotation of the index loops, thumb loops up through index, little fingers down through the index, followed by the iterative move, and the other figures are variants of the 4 options of rotation noted above.

i also introduce the use of two colored strings to look at the intricacies of the particular ways the circle of string flexes on itself and how the division of the circle into six linear strings allows interesting topological inter-weavings using the simple operations listed above

then i show how murphy’s tennis net illustrates how the circle can divide itself into two color areas when you start with six bands of alternating colors making up the string.