Loop Growth Lab
A string is looped at one crossing point. The first loop sits in the middle, and each new set of loops is added on the two loose ends, one end turning one way and the other end the opposite way. All of the loops start stacked at the crossing point and then grow on the surface of a sphere. On a sphere a loop that grows far enough becomes a straight line, a great circle, and that is the moment it unfolds. Choose the rule and see whether everything unfolds at once.
The loops · stacked at one crossing point, growing on a sphere
When each set unfolds
One connected string. Here the whole stack is a single string of fixed length, and only one loop is driven. As the driven loop is opened wider it gets longer, and every other loop gets shorter by the same amount, so one loop moving sets all the others moving. With the proportional pull, the driven loop grows by a factor and every other loop shrinks by that same factor: when loop 1 has doubled to 2, loop 2 is down to 1, loop 3 to 1.5 and loop 5 to 2.5. The driven loop is then the same length as loop k when both are √k long, and no loop is ever used up. As the driven loop opens, the + coils are pushed round to one side and the − coils to the other, at right angles to the driven loop at the fold, and they come back together on the far side as the mirror image of the start. With the full pull, each other loop gives up the whole length the driven loop gains: when loop 1 has grown to where 2 was, 2 is down to 1, 3 to 2 and 5 to 4. With the shared pull, the gain is divided among the other loops, which keeps the string's total length fixed but shrinks each loop much less. A loop pulled down to nothing is gone, with its + and − coils vanishing together, and its share of the pull passes to the loops that are left. On a sphere a loop is longest when it is straight (90°), so the pull is greatest at that moment: that is the fold. Past it the driven loop curls onto the opposite side, gets shorter, and hands the string back, and the system runs out again as its own mirror image. Lengths are counted in units of the middle loop's starting length, so loop k starts at length k. The two "how each loop grows" and "when the sets start" choices do not apply to this layout.
One stack. This is the layout with a single string: a middle loop for the number 1, then for each of your numbers a coil wound one way (+, warm colours, drawn just above the sphere) and a coil wound the opposite way (−, cool colours, drawn just below it). The two coils for a number are the same size and sit on top of each other, so with the lift set to 0 the stack looks like a single set of loops. The moving beads show which way each coil is wound. Every loop grows, and a loop unfolds when it reaches 90° and becomes straight. When every loop grows by the same proportion, bigger numbers unfold first, and the wait between two loops unfolding is the logarithm of the ratio of their numbers. That is how multiplying shows up here: the wait from 6 to 1 is the wait from 6 to 3 plus the wait from 3 to 1.
How it is built. Every loop is a circle on the sphere that passes through the crossing point (the white ring). The middle loop is set 1. Each later set is added on both loose ends: the loops on one end lean to one side of the crossing point and the loops on the other end are their mirror images, turning the opposite way. With the fan set to 0 the loops lie exactly on top of each other; a few degrees of fan spreads the stack so each loop can be seen. A loop's size is the angle from its centre to its edge. When that reaches 90° the loop is a great circle, which is a straight line on a sphere: the loop has unfolded. Past 90° it curls up again on the opposite side, as the mirror image of what it was.
The rules. "k loops, all the same size" gives set k that many equal loops on each end. "One loop of size 1 ÷ k" gives set k a single loop that is k times smaller than the first. Loops can gain the same amount (every loop's size rises by the same number of degrees) or grow by the same proportion (every loop's size is multiplied by the same factor). They can start together, or the growth can begin at the middle loop and reach the outer sets later. ζ's own rule is there for comparison: the slider is the shell size σ, loop k has size 1 ÷ kσ on one side and its mirror has size 1 ÷ k1 − σ on the other, drawn here at one fixed moment and without the extra factor the mirror rule adds. In that rule nothing has to reach 90°: the fold is the shell where every loop is the same size as its mirror.
What is not in here yet. One set unfolding does not change its neighbours. Each loop grows by its own rule only, so there are no "negative impressions" left in later sets. That part of the idea needs a rule for what an unfolded set does to the next one before it can be built.