Mandelbrot: two views of a Quantum Zoom within the tree of a mu

This actually includes a classic zoom followed by a two quantum zooms. The first quantum zoom spins around the center of a tree within a tree, taking a cryptic logarithmic spiral towards the center that repeatedly passes through all of the branches in sequence. It employs 1 pallet rotation per bifurcation for each mu, stressing the similarity that exists between the levels of bifurcation as well as successive mus, creating the appearance of near continuity. One feature of this approach is that the trees in the embedded Julia that surrounds the sequence of mus appear to revolve around the "current mu". Each layer of trees has twice the number of trees of the layer directly above it. However, each of these layers also appear to be revolving at only half the speed. As such, trees pass the spire on the left of the current mu at the same rate independently of the depth of each layer. The second quantum zoom goes back out along the current branch from the the center,, employing the same rule for the assignment of colors. The third quantum zoom reverses the second, descending along the branch, but employs a rule that includes the depth within a tree along the branch in its formula for the assignment of color. Specifically, pallet rotation is iterations of escape divided by iterations per bifurcation times depth along the branch where the beginning mu has a depth of 1. And by depth I mean the count of mu in the sequence of mus. The results of the new color assignment rule are striking. The plateaus appear to be all tuned to all look alike - save for the revolution of the pallet relative to the center of the tree. One could say that the structure of the mu encodes the position in the sequence - as suggested by the iterations per bifurcation of the mus. See below. The veins that give the mus texture are inherited from the branch of the Mandelbrot in which we found in the parent parent mu. Those veins appear prominent even in the deepest part of the zoom. The veins about the borders between neighboring plateaus become dense as one approaches any given border and emphasis them. Finally we take a classic zoom back out to the Mandelbrot. It will give you a clearer idea of how deep you had gone and where you had been. The video was the result of work done in 2023 but got set aside, lost, forgotten and then rediscovered. There will be more to say about it in a later video. A note on calculating the iterations per bifurcation... The number of iterations per bifurcation increases by a constant 260 = 2 (for the double spirals that result from the primary disk of the parent parent mu) times 5 (4 main branches of the disk of the disk of the main disk of the parent parent mu + 1 for the location being the first branch) times 2 (for the 2 branches of the double spiral that result from being one of the double spirals off the main disk of the parent) times 13 (for the 13 branches of disk 12 of that main disk). But the iterations per bifurcation of the first mu in the sequence isn't 260. Its 250. The subtraction of 10 is accounted for in a formula for the iterations per bifurcation that has been tested against a variety of heterogenous trees (including a tree with 5 times 3 times 4 times 7 principal branches) and all their principal mus (181 of the afore mentioned heterogenous tree - I explored the tip of the first principal branch in the video "A Tesseract of Variation"). The formula I've found, partly through trial and error, appears to work for disks of arbitrary descent, but can likely be proven through analytic means by someone with a bit more talent than myself. Most of the work has already been done by mathematicians, but to make use of their work, begin with the point that iterations per bifurcation of a mu is equal to the cycle length of its main cardioid. For those who might be interested in simply puzzling it out the same way I did, I would suggest trying to construct the formula from a hybrid of extended products and extended sums, using Farey fractions for the disks, focusing on disks with numerators of 1 since other numerators can be handled by order of visitation as mod coprime numerator and thus can be derived from the trees of disks with all Farey addresses where the numerator is 1 by means of permutation matrices. And while the Farey fractions of the disks when expressed as decimals arrange the disks counterclockwise, a naming convention best visits branches clockwise, rather than the counterclockwise convention that I've seen used elsewhere, beginning with the branch that immediately follows the trunk. However, given the peculiar nature of the infinite combinatorial complexity of the Mandelbrot, no formula will ever be able to take into account all its structure with all the variations in kind, but a tool bag of formulas should at least scratch the surface.