The first sixty iterations (z₀ to z₆₀) of the Anti-Buddhabrot fractal with intermediate orbit path interpolation to visualize the motion of orbit point surfaces in the complex plane. The color scheme is the same as in Six Faces of Buddha so as to better visualize how the different regions of the initial surface move about as the iteration sequence progresses. If you want to see what the interpolated orbit paths themselves look like you can check them out in Plato's Daydream.

The first thing you probably notice is the period-2 "head" of the Mandelbrot/Buddhabrot set oscillating between its two positions: up and down. Unfortunately, the smaller period-2ⁿ bulbs sitting on top of the head are too small to keep track and they get lost in the noise of other orbital periods pretty quickly. But if you have eyes for it, you can see that they keep faithfully oscillating through their set of stable positions.

As we transition from z₂ to z₃ we see the head rise to its original position while the period-3 bulbs come in from the sides to complete their own oscillation. As the 3-bulbs rotate back outwards to start a new oscillation the head descends back downwards to meet all the period-4 oscillations. This rinses and repeats in a similar fashion for oscillations of all other periods, or wavelengths (λ), until the shown sequence arrives at sixty.

All stable orbits in the Mandelbrot set are periodical. This means that each set of z-points will eventually at some iteration depth return to zero, which results in the next value being equal to c. From there onwards the sequence keeps oscillating through all the values between z₁=c and zₜ=0 with mathematical precision. Here t is the wavelength of the orbit in question.

The summation of all these orbits of different wavelengths is the evolving Buddhabrot surface shown in the video. Although the iterative Mandelbrot function only produces surfaces at discrete intervals (there is nothing between z₀ and z₁) with the aid of linear interpolation we can get a rough idea of what those might look like. Although mathematically dubious, the linear interpolation does have the benefit of not cutting the surface as it is being interpolated which makes for a more pleasant viewing experience.

As the sequence progresses certain patterns seem to become discernible. The original complex-plane-surface of z₁ seems to be growing out of main cardioid cusp at c=¼ and flowing symmetrically towards c=-¾ which is where the main cardioid starts its bifurcation process along the negative Cr-axis (as illustrated here). All this new surface flowing out of c=¼ must go somewhere. We know all of it must stay close to zero – otherwise the orbit would not be stable – and we know that the λ=2 oscillation is centered around the the sequence z={0, -1}. Since the surface is never cut or otherwise broken all it can do is start twisting in on itself.

As this "spiral-surface" winds up ever tighter another pattern emerges. New point-like light sources start to appear in ever increasing numbers. How these over-densities form is a subject for later video, but in any case, the most visible point-lights encircle the main cardioid. Funnily enough they also function as a sort of built-in counter — each iteration of the Mandelbrot function adds one. When the λ=2 bulb passes from top to bottom a new point-light is formed on the vertical Zr-axis (its main axis of motion). When the bulb returns upward on the next iteration it looks as if it splits the most recent over-density poitn into two — both halfs ending up slightly to the side of the Zr-axis. As the sequence nears its end the main cardioid has transformed into a disk with an illuminated perimeter. Apparently the perimeter acts as a curved axis along which the Buddhabrot surface continually coils in on itself.

The final point to mention is how the different shapes at different iteration depths map to the properties of just plain old numbers. After the sequence finishes you first see all the shapes which map to so-called anti-prime or highly composite numbers — numbers which have more divisors than any number smaller than it. And all the displayed anti-prime shapes do indeed have something in common: they have a lot of stuff concentrated in the middle around zero. At the final iteration depth of n=60 we know that stable orbits with wavelengths of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 must land exactly at zero because, as explained earlier, that is how they get their respective periodicities. So any number that has lots of divisors will have the period-bulbs matching those divisors converging at zero.

Followed by the anti-primes are the primes — numbers which are divisible only by themselves and one. As the name implies they are effectively the opposite of anti-primes which also translates to a defining difference in their iteration shapes. For prime shapes the middle is relatively empty since there are only two orbital wavelengths which cross over at zero during that iteration (you can probably guess what they are). The odd one out is the number two since it is both a prime and an anti-prime, and furthermore the only even prime number. Visually this translates to two being the only prime-shape where the λ=2 bulb is down around z=0 instead of up around z=-1.

Beyond the entire prime/anti-prime classification is λ=1, or the orbit originating from c=0. Lacking the perturbing effect of an iterative addition it will remain perfectly balanced and forever unchanging regardless of how many iterations we perform.

(On a tangent: the highly divisible nature of the number sixty is exactly why the ancient Sumerians based their numeral system on it and why there's sixty minutes in every hour.)

The accompanying music is "Equinimity" by Nuisance.