Classifying the fractal parameter space of the Lenia Orbium

Ivan Yevenko

iyevenko@uwaterloo.ca

@ivan_yevenko

Kojima and Ikegami 2023

"We observed that the pattern disappeared both when the size of the time step was large (dt = 0.5) and when it was small (dt = 0.002)"

Orbium unicaudatus survival time

Orbium unicaudatus survival time

Orbium unicaudatus survival time

Orbium unicaudatus survival time

Jascha Sohl-Dickstein 2024

"The boundary between hyperparameters for which neural network training succeeds or fails has (gorgeous, organic) fractal structure”

paulborke.net

Intuition #1:

Fractals appear when iterating a function is

1. Computationally irreducible
2. Infinitely sensitive to initial conditions

Maybe the same?

Can we do better than random search?

Can we do better than random search?

I) Orbium Island�

II) Trivial Mass�

III) Chaotic boundary

I

II

III

Heuristics:

– Total mass (basic, works OK)

– Lagrangian (characteristic of behavior)

– Approx. Entropy (e.g. PNG, gzip, 3x3 cell counts)

– ???

Heuristic:�PNG Compression Ratio

Heuristic:�PNG Compression Ratio�(0.09 < ratio < 0.125)

Intuition #2:

Most of the fractal space is not infinitely complex, it’s highly compressible

Fractal spaces can be practically exploitable for search!

Applications of Intuition #1 + #2:

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1. Finding complex discretization-dependent creatures on the fractal boundary

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2. Identifying continuous families of creatures rather than distinct parameter settings

​

3. Improving on evolutionary search (e.g. visualizing space of heuristics)

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4. Better distinguish between areas with complex life forms and trivial patterns