The Mandelbrot set is probably the world's most famous fractal.
Zooming in one sees filaments, with smaller copies of the whole inside.
The Mandelbrot set is a map of Julia sets, which are different fractals associated with each point in the complex plane.
Points with bounded iterates have connected Julia sets, and are coloured black.
Other points are in bright colours, varying according to how quickly their iterates escape to infinity.
Zooming near a baby Mandelbrot set one sees embedded Julia sets.
They look similar to the Julia sets for points in corresponding places near the whole Mandelbrot set.
These are good places to make the first decision to go off-centre, and find a new point to zoom in on.
Zooming continues, and now history repeats, only doubled up rotationally and twice as fast.
Everything you see has been seen before, including the doubling up, until a crisis point when a new two-fold symmetric shape appears.
Often this is when artists choose to make the next decision to go off-centre.
After going off-centre, history repeats once more, getting longer.
Keep repeating the pattern, zooming in and going off-centre.
At the next crisis the new centre has the pattern from the previous decision wrapped around twice rotationally.
But you have to wait until history has repeated.
This can be tiring.
Advanced fractal exploration software has special features to accelerate navigation.
At the centre of each cascade of history repeating with doubling lies a baby Mandelbrot set.
Each baby Mandelbrot set has an integer period, so-called because its iterates repeat periodically.
Different baby Mandelbrot sets can have the same period, but each baby Mandelbrot set has exactly one period.
There are numerical methods for finding the period of the main baby Mandelbrot set in a view.
Once you have the view and period you can find the coordinates using Newton's root-finding method.
With the period and coordinates you can find how far to zoom in.
Finally you can warp to the place where the next decision should be made to continue the pattern, skipping the repeating history.
This allows the fractal artist to focus on pattern construction instead of manual zooming.
As well as a period, each baby Mandelbrot set has a pair of external rays.
Unlike the period, external rays uniquely identify a particular baby Mandelbrot set.
External rays are a map between the outside of the Mandelbrot set and the outside of the unit disk.
Some deep unsolved conjectures are related to this mathematics.
External rays are best expressed as binary expansions, because squaring doubles angles on the circle.
The biggest circle has period 2 and external rays .(01) and .(10).
The biggest baby Mandelbrot set has period 3 and external rays .(011) and .(100).
The number of bits in external rays matches the period of the component.
External rays can be combined using tuning, which copy/pastes blocks of binary digits.
Tuning the rays of the period 2 circle (.(01), .(10)) by the period 3 baby Mandelbrot set (.(011), .(100)) gives (.(011100), .(100011)).
These are the rays of the period 6 circle that is attached to the period 3 cardioid...
...in the same way that the period 2 circle is attached to the period 1 cardioid of the main body of the Mandelbrot set.
Baby Mandelbrot set cardioids (and their attached disks) each have 2 rays with strictly periodic binary expansions.
Tips and the centres of spirals in the filaments, have rays with strictly pre-periodic expansions.
The tip of the antenna has one external ray .1 = .0(1) = .1(0).
The binary expansion is not unique, so tuning gives two different rays.
And so the hairiness of the Mandelbrot set is revealed: there is more of the Mandelbrot set between tuned tips.
Patterns of zooming and going off-centre are also reflected in binary expansions.
The periods of the central baby Mandelbrot sets go up exponentially in Fibonacci-like sequences.
The binary expansions of rays at each successive level are formed by copy/paste of earlier levels rays.
"Old Wood Dish" by JWM copy/pastes rays like (.(O), .(I)) to (.(OO11), .(OI00)).
Its pattern of periods follows the sequence ..., 34, 70, 142, ..., P, 2 P + 2, ....
Compare with "Evolution of Trees" by DinkydauSet. Its pattern of periods seems to follow a sequence ..., 70910, 141965, ..., Q, P, P + 2 Q + 145, ....
"Mandelbrot Deep Julia Morphing 1" by Microfractal has two patterns, one similar to "Old Wood Dish": ..., 137, 278, ..., P, 2 P + 4, ....
And a later one similar to "Evolution of Trees" in two stages: ..., Q, P, P + 2 Q + 4, ..., interleaved with ..., Q, P, P + 2 Q, ....
In practice ray analysis is not feasible for deep patterns.
The periods go up too quickly, often doubling or more each level: ray cost goes up like period squared.
This image is a recreation of "Polefcra" by Jonathan Leavitt.
Its pattern of periods follows an alternating sequence: ..., P, 2 P + 41, ..., interleaved with ..., P, 2 P + 44, ....
Its rays copy/paste like (.(O), .(I)) to (.(IO (100)^(13 or 14) 10), .(II (011)^(13 or 14) 01))
Once the patterns have been analyzed, synthesis of new variations is possible.
Many thanks to artists for sharing their coordinates, without which this analysis would have been impossible: DinkydauSet, Jonathan Leavitt, JWM, Microfractal, ....
Software for deep Mandebrot set exploration includes: FractalShades, FractalShark, FractalZoomer, Fraktaler-3, Imagina, Kalle's Fraktaler 2+, M-Perturbator-GTK, ....
The final baby Mandelbrot set approaches!
Thus end many Mandelbrot set zoom videos. (And who am I to break with tradition?)