A Primordial Particle System in three dimensions
AA Primordial Particle System in threedimensions T homas S chmickl * and M artin S tefanec Artificial Life Lab of the Institute of Biology, Karl-Franzens-University Graz, AustriaUniversitätsplatz 2, A-8010 Graz, Austriasubmitted 27 th January 2019
Abstract
This article describes the conversion of the two-dimensional Primordial Particle System into a three-dimensional model that exhibits comparable features. We present the transformed model here in the formof a pseudocode implementation and detail the modifications required for this conversion.Keywords: Self-organization, Emergent pattern formation, Self-replication, Third Dimension, ArtificialLife, Emergence of Life, Morphogenesis
I Introduction
A very simple model called PPS (Primordial Particle System) demonstrates that very simplerules (described by a simple motion law) can generate complex structures that exhibit manyproperties also found in life forms (growth, reproduction, physiology, nutrient cycle, life cycle,behaviour, information-processing, and finally also death) [1], [2]. In addition populations of theseindividual emergent structures show similar behavior that is also found in animal populations(emergent ecosystems following density-dependent growth). This model differentiates itself fromother, seemingly similar, self-propelled particle systems that show collective dynamics [3] by itssimplicity. The original article and its associated demonstration movie triggered significant intereston the web (youtube , reddit , etc.) with several people starting to re-implement and modifyit. A question often asked by the community is “does it also work in 3D?”. Here our aim is todemonstrate that it is in fact trivial to extend the model into three dimensions and explain thiswith the amount of information that is needed to conduct this transition on the new spin-off codethat was generated recently, in order to replicate and be able to check our observations. II The mathematical model
The original 2D model of the PPS assumes that a certain number of particles are initially positionedat randomized spots, with randomized orientation (uniform random distribution). For a 3D model * Corresponding author: [email protected] https : // . youtube . com / watch ? v = makaJpLvbow https : // . reddit . com / search ? q = %22 primordial %20 particle %20 system %22 a r X i v : . [ n li n . PS ] J a n mplementation, these starting conditions are identical, just that the random positions are extendedto < x i ( t ) , y i ( t ) , z i ( t ) > .In our simulation we distributed 770 particles in a cubic space of 31 x 31 space units. Eachparticle i is orientated with two angles < ϕ xy i ( t ) , ϕ tilt i ( t ) > whereby ϕ xy denotes the angle in thex-y plane (like in the original 2D version) and ϕ tilt denotes the angle in the x-y plane (like in theoriginal 2D version) and ϕ tilt denotes the angle in which the particle rotates its front heading upor down. There is no third rotation axis used (“roll”), only “pane” (left-right) and “tilt” (up-down).In order to translate the motion from 2D to 3D the same motion law is applied twice, once forcalculating the change in ϕ xy and then also for the rotation of the tilt angle ϕ tilt . The followingpseudocode describes the algorithm. The function “in-cone” reports all other agents in the 3D conein front of the particle, facing into the direction of its front heading, that is the direction it movesto with its forward motion. As the opening angle of the cone is °, it is in fact a half-sphere. Inorder to get the particle in the right, left, up and down half-sphere, each individual particle alwaysrotates into this direction, senses the number of particles in the half-sphere and then rotates backto its initial position. Given the fact that four half-spheres around the particle are processed, everyneighboring particle is counted twice, once for the left-right rotational decision and another timefor the up-down rotational decision. This gives the follow main loop for the simulation in 3D:(Please note that we use the signum function here, which returns − for all negative inputnumbers, + for all positive ones and for an input of zero. All angles are given in degrees2nd not in radians.) Most parameters remain unchanged in the transition from the 2D to the 3Dversion. In order to allow the simulation to run faster (in 3D much more particles have to becalculated for the same density) we reduced the interaction radius to r = space units. Thisreduced radius is then compensated by a slightly increased density-dependent rotation amount,thus β = degrees in our 3D simulations. The fixed rotation α was kept on °. The speed v = space-units/step was also slightly decreased to fit to the reduced interaction radius. Thedensity-dependent color scheme was adapted to fit the higher number of particles that one particlecan encounter now:In order to allow to make the interesting denser (non-green) structures well visible and notbeing occluded by the free green particles we made all green particles semi-transparent. III Results
The following Figure 1 exemplary shows an ecosystem of PPS structures that emerged from arandom distribution: Figure 1:
Exemplary screenshots of the PPS in 3D.
IV Discussion and Conclusion
One of the first reactions to the model frequently expressed by other people is that it is veryreminiscent of the Game of Life [4]. But not just the very different nature of the model (particles3n continuous space that can move asynchronously) discriminates the two models. The fact thatit is rather trivial to transform the Primordial Particle System to the third dimension, as shownhere, indicates big differences. The changes to the original model are minimal, basically onlythe “sensing” of local interaction (forces) applied by neighboring particles is occurring now inhalf-spheres instead of half-circles as it was the case for the 2D version. The parameters actuallydid not require any change, we just reduced the radius r by a bit and compensated for thisby increasing β and decreasing v in order to let the system run at at feasible computationalspeed. This way fewer particle-to-particle interactions had to be accounted for but the achievablemaximum rotation potential was kept on the same level as it was the case in the 2D version. Thefact that parameters had to change only minimally (if at all necessary) and that the implementationis otherwise in fact identical to the 2D model just applied twice for 2 rotational axes, showsanother important feature of the PPS model: it is simple and it stays simple even if extended tomore dimensions. V Acknowledgments
This work was supported by the COLIBRI initiative of the University of Graz.
References [1] Thomas Schmickl, Martin Stefanec, Karl Crailsheim: How a life-like systememerges from a simple particle motion law.
Scientific Reports https://arxiv.org , 1512.04478 (2015).,https://arxiv.org/abs/1512.04478[3] Hiroki Sayama: Swarm chemistry. Artificial life , 105-114 (2009).[4] Martin Gardner: Mathematical games: The fantastic combinations of John Conway's newsolitaire game “life”.
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