Hook

In this automaton, on $512\times 512$ grid, each ($j$-th) cell's state $s_j \in \mathbb{C}$ and the update rule

$$s_j' = M \bigl(1 - \min \{ 1, |S_j'| \} \bigr) S_j'$$

$$S_j' = s_j + \overline{\sum\limits_{k \in \mathcal{N}(j)} s_k^2 } + \zeta$$

where

• $\overline{z}$, as usual, denotes complex conjugation of $z$, $\overline{a + bi} = a - bi$,

• $\mathcal{N}(j)$ is the $5\times 5$ neighbourhood of $j$-th cell with this cell at its centre,

• $\zeta$ is a random isotropic “noise” with modulus bounded by $R$ (or absent, $R = 0$),

• $M > 0$ is the scaling multiplier, here it goes from 0.48 to 0.68, with $M = 0.53 \pm 0.02$ being “optimal”.

Color scheme is the standard representation of $\mathbb{C}$ values: hue denotes argument (e.g. red for $\arg z \approx 0$, cyan for $\arg z \approx \pi$), brightness (luminance) denotes modulus (HSL color model with Saturation $\equiv 1$).

— The setup is quite simple; we consider the gap between this simplicity and the intricacy of emerging behaviour to be of some interest. What is happening, and why? (Spoiler: we do not understand enough.)

Remark 1. For $M \in [0; 4]$, indeed, $|s_j| \leqslant 1$: the state remains inside unit disk $\mathbb{U} \subset \mathbb{C}$. Initial state of each cell is randomly distributed in $\mathbb{U}$.

Remark 2. Actually, the “isotropic” random variable is uniformly distributed in the square $\mathbb{S} = \bigl\{ z \in \mathbb{C} | -1 \leqslant \Re{z} \leqslant 1, -1 \leqslant \Im{z} \leqslant 1 \bigr\}$ (or in $R\mathbb{S}$), but it provides almost the same behaviour as “fair” isotropic one (with uniformly distributed $\arg$).

Remark 3. When $M < 0.6$, randomly initialised field quickly “dies out” into background noise (if there is any) and does not produce self-sustaining filaments (one may also call them hyphae, worms, snakes etc.) you see spreading above. To overcome such degeneration, we start with $M > 0.6$ and then, as soon as filaments appear, lower it to e.g. 0.53.

These remarks should help you reproduce/verify this behaviour if you want to do it from scratch. Alternatively, get our implementation via links below Abstract.

Introduction

There are lots of generalisations of cellular automata (CA), classical Conway's Game of Life [ADA10] being an appropriate “last common ancestor” (in configuration space if not in time because of [NEU66]): see e.g. [ROL24], ???

To change the state space of each cell from $\{ 0, 1 \}$ to $\mathbb{C}$ (or $\mathbb{C}^n$, $M_{n}(\mathbb{C})$, etc.) is certain “low-hanging” generalisation that we focus on here — complex-valued CA, $\mathbb{C}$CA.

Particularities of such generalisation do not hang low though... It seems that the majority of studies in $\mathbb{C}$CA are connected to, and based upon, either

• Quantum Mechanics (QM), beginning with seminal [FEY82]. In particular, there are Quantum CA (QCA), actually 2 kinds of them — 2 ways to interpret “quantum something” phrase — “physical” and “simulated”: in the former, cells themselves are assumed to be physically realised quantum systems, while the latter simulate such systems on classical computers. Naturally, Schrödinger equation that describes evolution of a system in time plays important role in these studies. See [GRO88], [WAT95], [ARR19], [FAR19], [BRO23], [BER24], ???

or

• Artificial Neural Networks (ANN). See [HIR06], [BAS21], [GAO23] (Complex-Valued Neural Networks), [RUO22] (Quaternion-Valued Neural Networks), ???

Although this “Scyllic” QM/ANN restriction is natural, perhaps necessary in certain sense, even from formulaic point of view there are great many other, non-QM/ANN, $\mathbb{C} \rightarrow \mathbb{C}$ mappings, and e.g. Mandelbrot/Julia sets show that even very simple ones such as $z \rightarrow z^2 + c$ can produce quite sophisticated structures. Adding update rules, thus time, thus dynamics, one may find other interesting regions of this vast space... or one may fall into “Charybdis” of complexity for the sake of complexity.

???

Thus our search is guided by formulaic (also computational) reasons perhaps more than it should be. As of this writing, at least, there is no physical/chemical/biological or whatever “prototype” for update rules considered here.

Transition from discrete to continuous domain often introduces “organic” shapes and behaviour (to be more precise, our perception categorises them so), as seen in e.g. SmoothLife [RAF11], Lenia [CHA19], Glaberish [TYR22], ???. Here we observe this phenomenon as well.

Disclaimers

Any part of this “paper” and/or simulator may change at any time, to reflect further developments of the investigation; you can already see it in videos here, with slight changes in UI from one video to another. On the other hand, it may suspend and turn into $\mathfrak{Delineamentum}$ $\mathfrak{Aeternum}$ — eternal outline.

And some parts should appear, clearly absent for now. In this “paper”, they are marked by “???”

The automata considered here are defined by update rules structurally simple enough to have been already considered by someone, somewhere, some time ago, at least once over decades. We have not found these exact considerations... yet, perhaps because we looked for wrong keywords, or into wrong places, or with insufficient thoroughness. This search in the past continues though.

FilamentiComplex 2D

Consider the class of $\mathbb{C}$CA represented by the one defined in Hook section, with its characteristic filamentiferous structures that move in “void” ($|z| \approx 0$) while overall preserving their shape — F$\mathbb{C}$CA.

???

Taxonomy

Types of filamentic structures that appear in F$\mathbb{C}$CA.

Even before one goes to differentiate shapes and movements, one immediately notices the distinction in color, which here, we remind, represents the argument of $\mathbb{C}$ value. We see red, green, and blue filaments, corresponding to 3 values of $\sqrt[3]{1} = \cos (\frac{2\pi}{3}k) + i \sin (\frac{2\pi}{3}k)$, $k = 0, 1, 2$, in $\mathbb{C}$: $\cos 0 + i \sin 0 = 1$ ($\arg \approx 0$) for red, $\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2}$ ($\arg \approx 120$°) for green, and $\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} = -\frac{1}{2} - i \frac{\sqrt{3}}{2}$ ($\arg \approx -120$°) for blue.

On the other hand, $\mathbb{C}$ness is necessary: when you look closer, each filament has smaller regions of colors different from its main color, especially on its edges for larger $M$, in particular, 2 colors alternating on left/right sides. They are critical for functioning: if we project state of every cell on $\mathbb{R}$ ($a + bi \rightarrow a$), filaments immediately “die out”. In the presense of noise, they then “reborn” quickly, in different types; without noise, the homogeneous real value $M/4$ fills the field:

The simplest structure that, in a sense, preserves itself in this F$\mathbb{C}$CA when $M$ is small is a propeller:

Position aside, 2 directions of rotation (clockwise/counter-clockwise) and 3 colors exhaust all 6 types of this structure.

As $M$ increases from 0.5 to 0.55, this structure turns into moving filament. And if $M$ decreases, at approx. $0.46$ even propellers die out.

Tadpole, short:

This structure is stable, perpetuating itself indefinitely.

Tadpole, long:

This structure and its longer versions are unstable, they tend to branch into filaments or collapse into short tadpoles, the latter then dying out.

Horseshoe:

This structure itself is stable, but from time to time produces more spreading filaments, and in a finite looped field they sooner or later swallow it.

Junction (one pair is enlarged inside maglass, and there are others):

Not a separate “entity”, junction exists between 2 filaments of different colors, here red and green, where they touch each other. Note its alteration between two single-step states, with darker “slit” appearing on both sides.

Its relative stability is demonstrated by the following episode, where 2 short filaments grow and branch, but the junction is preserved until much later:

???

Certain measure of complexity

How “hard” it is to generate a filamentic structure “from scratch” in the absence of omnipotential, in a sense, noise? To be more precise, how many cells, in what relative positions, and to what values we have to set manually at “empty”, i.e. zeroed, field in stopped time, so that when time is unstopped and noise $\zeta \equiv 0$, this generative set of cells turns into single propeller? or single tadpole? Indeed, we aim at minimal generative set, otherwise we could simply recreate entire structure in question cell-by-cell. Note single, because, as demonstrated below in Conclusion, from 1 seed cell set to $\frac{1}{2}$ a kaleidoscopic system with many filaments develops.

At this writing we have found the following embryo that consists of 13 short stripes:

$$c_k = \frac{M}{8} \bigl(1 + \cos(\pi \frac{k}{7}) + i \sin (\pi \frac{k}{7}) \bigr), \quad k \in \{ -6, -5, ..., 6 \}$$

These are vertices of regular 14-gon inscribed into the circle with centre at $M / 8$ and of radius $M / 8$, except 0. Typically for soft CAs, you can adjust this construction, if the deviation is not too big, it still develops into tadpole (or horseshoe). For example, you can increase the number of stripes, or decrease the length of “horizontal” part a little, or assign values to stripes differently; what matters is “progression” of arguments going through 0 along central stripe and modulus attaining maximum along that stripe.

As seen in the video, this structure remains “embryonic” with another neighbourhood, rotated square; a tadpole develops there as well.

(Lack of) organisational hierarchy

Single filaments, or, in the best case, several filaments ephemerally connected via junctions, seem to be the highest level of such hierarchy. Higher levels do not appear when a field is enlarged, either.

In biological terms, it does not reach even the level of a single cell, perhaps only something like lipidic droplets. In this regard, FilamentiComplex is less interesting than e.g. Lenia.

???

Resilience and regeneration

In the presence of noise, when its amplitude is small, all structures we have considered above preserve themselves, occasionally changing direction of movement or producing “offsprings”, usually long filaments. This resilience is limited though: as the amplitude of noise grows, these structures “dissolve”.

You can nullify quite large parts of them (set states of cells to 0), and they will “restore” themselves, perhaps in slightly different type, and continue their movement:

On the other hand, these properties are common for “soft” CA.

???

Game of Life analogues

Tadpoles and horseshoes correspond to gliders and spaceships, while propellers are similar to flip-flops.

There are no direct analogues of still lifes such as $2\times 2$ squares; motion is necessary to persist in FilamentiComplex.

Existence of glider gun analogues, especially “regular” (periodic) ones, is uncertain.

FilamentiComplex 3D

Straightforward generalisation, with $5 \times 5 \times 5$ neighbourhood and “optimal” $M \approx 0.31 \pm 0.02$:

No apparent filaments, only volumetric propellers at lower $M$.

Not-so-straightforward generalisation, with $3 \times 3 \times 3$ neighbourhood, $M \approx 0.5$:

The behaviour of this one is closer to that of 2D version: persistent structures — sheets — are larger. It seems like $N = |\mathcal{N}(j)|$, i.e. the size of neighbourhood, has its own optimal values for such update rule, independent of dimensionality of underlying space.

Other OrganiComplexes 2D

Quadratic-Square

$$\sigma(x) = \bigl( x_1 (1 - x_1) \bigr)^2$$

Quadratic-Square-Root

$$\sigma(x) = \sqrt{x_1 (1 - x_1)}$$

This looks rather like some solid “tissue” and is well-known from discrete-stochastic CA world too. Note characteristic “phase transitions” as the amplifying multiplier $M$ crosses certain thresholds.

Pseudo-2-layer

for e.g. $\alpha = 0.05$, $\beta = 50$ (and $M \approx 0.55$) gives the automaton seemingly consisting of 2 layers — brighter, quicker “foreground” filaments and darker, more fixed “background” mosaic. But there is no explicit 3D, single $512 \times 512$ layer “splits” into these two. They correspond to distinct “stability regimes” w.r.t. update rule, regimes that differ in what exactly they preserve over iterations. At that, mosaic pseudo-layer alternates between 2 states.

???

Critical factors

(Critical for what exactly? — For the behaviour observed in FilamentiComplex.)

• Non-linearity. Square, $s_k^2$, is “special” in that neither cube and higher powers, nor other non-linear functions such as $e^{s_k}$ produce filaments, at least when we do not adjust other elements of an automaton.

• Reflections. Why them, why not translations, rotations, inversions etc.? Well, for now, ad hoc — because they in particular induce this “organicity” — not just in shape, but also in behaviour. (And they are computationally fast.) The conjugation comes to mind first, but other reflections of $\mathbb{C}$ work too, for example $a + bi \rightarrow -a + bi$ or swap $a + bi \rightarrow b + ai$:

• Modulus normalisation. Here we do it by means of the mapping $|z| \rightarrow \sigma_Q (|z|) = (1 - \min \{ 1, |z| \}) |z|$, where Q is for Quadratic (also called logistic). Filaments do not appear under many even slightly different $\sigma(x)$ such as $\frac{1}{2} - |\frac{1}{2} - x_1|$ or $\sqrt{x_1(1-x_1)}$ (see previous section, “Other OrganiComplexes 2D”), while e.g. for $\sigma_S(x) = \sin(\pi x_1)$ and $\sigma_L(x) = \ln \bigl( 1 + x_1(1 - x_1) \bigr)$ they do. (Here $x_1 = \min \{ x, 1 \}$.) Recall that these mappings play key role in dynamic chaos researches [DEV89].

$\sigma(x)$ with unbounded support $\{ x | \sigma(x) \ne 0 \}$, $\mathbb{R}_+$ instead of $(0, 1)$, sometimes work too, i.e. $\sigma(x) = xb^{-x}$ or $\sigma(x) = 1 / (a + x^p)$.

• Neighbourhood. Watch how things change (or not) when we tinker with neighbourhood:

This (in)sensitivity to substantial modifications of the neighbourhood's shape, in our opinion, hints at dependence on other properties, perhaps those of the space (2D grid) where automaton runs, that are preserved even by transitions between “topologically” different (rotated) square, square/circle border, (whirled) star etc. neighbourhoods.

• Synchronicity (all cells are updated simultaneously), an implicit assumption we have taken for granted, is actually important here, because if $\mathfrak{S}$ falls below approx. 0.93 (i.e. at every step, each cell is updated, independently from others, with probability $\mathfrak{S}$), neither filaments nor tadpoles appear and persist, not even propellers. On the other hand, they (and junctions) sustain larger $\mathfrak{S} < 1$:

Optimality

The $M$ that gives the most “interesting” behaviour seems to depend mostly on $N = |\mathcal{N}(j)|$, i.e. number of neighbours of $j$-th cell, although there is also a weaker dependence on the topology/connectivity and shape of neighbourhood. Such $M$ is 5 to 10 percents larger — super-critical — than $M_0$ that is critical in the sense of the following process: consider the nullified field (all states are 0) with a single cell's (“seed”) state set to $\frac{1}{2}$. When we run such setup without noise ($\zeta = 0$), there are 2 outcomes for nonzero “island”, depending on $M$:

1) for $M < M_0$, it dies out,

2) for $M > M_0$ (but not too much), it spreads to entire field, filling it with real value $M/4$.

Experiments provide empirical dependence of $M_0$ on $N$:

$$\begin{array}{r||c|c|c|c|c|c|} N & 9 & 13 & 25 & 41 & 49 & 81 & 121\\ \hline M_0 & 0.703 & 0.632 & 0.5 & 0.414 & 0.385 & 0.312 & 0.262 \end{array}$$

To obtain rough analytical expression for $M_0$, we consider another, even simpler, setup: certain global fixed point of update transformation without noise, where the states of all cells are real and equal to maximal possible value under given $M$, i.e. $s \equiv M/4$. For this to be a fixed point, update rule should not change it (in other words, should perpetuate it): $s' = s$. In turn, since $M|z|(1 - \min\{|z|, 1\})$ attains its maximum $M/4$ only at $|z| = \frac{1}{2}$, we have $S' = \frac{1}{2}$. Thus

$$s_j + \overline{\sum\limits_{k \in \mathcal{N}(j)} s_k^2} = \frac{1}{2}$$

$$\frac{M}{4} + N \frac{M^2}{16} = \frac{1}{2}$$

$$NM^2 + 4M - 8 = 0$$

$$M_{\pm} = \frac{-4 \pm \sqrt{16 + 32N}}{2N} = \frac{2}{N} \bigl( \pm \sqrt{1 + 2N} - 1 \bigr)$$

$$M_0 = \frac{2}{N} \bigl( \sqrt{1 + 2N} - 1 \bigr)$$

The roughness of this theoretical estimate $M_0^{(T)}$ is clearly demonstrated by its deviation from empirical value $M_0^{(E)}$:

$$\begin{array}{r||c|c|c|c|c|c|} N & 9 & 13 & 25 & 41 & 49 & 81 & 121\\ \hline M_0^{(T)} & 0.746 & 0.646 & 0.491 & 0.396 & 0.365 & 0.291 & 0.241\\ \hline M_0^{(E)} & 0.703 & 0.632 & 0.5 & 0.414 & 0.385 & 0.312 & 0.262\\ \hline M_0^{(E)} / M_0^{(T)} & 0.94 & 0.98 & 1.02 & 1.05 & 1.05 & 1.07 & 1.09 \end{array}$$

However, the ratio is more stable than $M_0^{(E)}$ itself, so we rewrite $M$ as $AP$, where perpetuating multiplier

$$P = P(N) = M_0^{(T)} = \begin{cases} \frac{2}{N} \bigl( \sqrt{1 + 2N} - 1 \bigr), & N > 0,\\ 2, & N = 0 \end{cases}$$

and amplifying multiplier $A$ is adjustable. Thus we pass the buck of optimality to $A$ and call it a parameter instead of $M$. Taking into account what has been said before about (additional) 5–10 %, we suggest optimal amplifier

$$A_0 = 1.08 \pm 0.02$$

Cf. [MIT93].

???

Conjectures

Almostness+ conjecture: F$\mathbb{C}$CA, in particular its update rule or maybe state space of its cell, lacks (only) few tweaks that, if introduced, will make its behaviour more interesting, its emerging structures more sophisticated.

Almostness– conjecture: F$\mathbb{C}$CA nearly attains maximum of self-organisation potential in its class, so whatever tweaks you introduce, nothing significantly novel emerges.

???

Conclusion

If this short excursion strikes a spark of your interest, we will consider its purpose to be fulfilled. Reaching after, in a sense, life, for now these realms require someone already there to give them a shade of liveness, and you can provide that for a while. In return, if anything else, they may make some aesthetical gift:

(Symmetry does not substitute for self-organisation though; only $\frac{1}{8}$ of such space, in a sense, exists, the rest is hollow reflection. As soon as asymmetry is introduced at single location, to say nothing of any noise, it propagates everywhere.)

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