# Three Algorithms
## Space Concept
The Mandelbrot set is a fractal formed by repeated iterations of the complex
number formula $z \to z^2 + c$. A first naive algorithm is to simply iterate
repeatedly to check if the sequence diverges. An improvement is to check for
convergence, using Newton's method for root finding to check that a periodic
limit cycle has derivative less than $1$ in magnitude. A third optimisation is
to optionally postpone those more expensive checks, depending on the outcome
of the previous pixel - if a pixel diverges, its neighbour is likely to
diverge also, while convergent pixels cluster together too.
These three algorithms are sonified by stereographic projection of the complex
z coordinate (after each $z \to z^2 + c$ iteration) to the unit Riemann sphere
in $3$D space, whereafter Ambisonics takes over. The waveform so placed in space
is a hash noise function of the current iteration number of the pixel together
with the candidate period of the cycle under investigation for interior checks.
## Technical Desciption
The piece is a fixed multimedia video composition. The soundtrack is encoded
in $16$ channel ACN/SN3D 3rd order ambisonics, with a separate binaural render
provided for preview purposes only. The sound is synchronized to the image,
both are generated by the same fractal iterations. One audio sample frame is
calculated per iteration at $48000$ Hz, a pixel may take many iterations,
thus the image appears in scanline fashion.
The video file is rendered at $480\times 240$ pixels, upscaled $4\times$, which
gives a total audio length of approximately $4$ minutes $30$ seconds.
- audio codec: f32_le 16ch 48000Hz PCM
- video codec: yuv420p h264 profile high level 4.1 crf 20
- media container: Matroska (MKV)
## Requirements
- video playback and projection ($1920\times1080p60$), synchronized to
- sound reproduction (Ambisonics decode to Cube speakers)
- setup time: a simple 5-10mins playthrough (a/v line check)
## Download Links
- 800MB
- 100MB
## About mathr
Claude Heiland-Allen () is an artist from London
interested in the complex emergent behaviour of simple systems, and
mathematical aesthetics.
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