Earth Day 2009

Spherical Cellular Automaton

by JJ Ventrella

According to
Gaia Theory
, the Earth is not just
a complex ecosystem, it is a
super-organism
. Earth
has a protective atmosphere with an unlikely
balance of gases which can only be explained by
way of some self-regulating system. Earth's
biosphere is
self-organizing and works in such
a way as to keep its systems in some kind of meta-
equilibrium, broadly conducive to life.
We humans are recently-developed cells within the super-organism, and we have caused major disruption in the equilibrium. The fever called Global Warming is a call for us to reduce our footprint and find a more integrated and balanced relationship with the other cells of Earth. Old paradigms got us into this mess. New paradigms will help bring humanity to a new era of harmonious cohabitation on Planet Earth. |
Buckminster Fuller
expressed the necessary shift
in paradigm with the
Spaceship Earth
metaphor. And today,
William McDonough
reminds us
that the concept of away has gone (away). We used to be able to throw things away, but where
is that now?
The human brain evolved in what appeared to our species as an infinite flatland. And while we now know for certain that the world is round, we still carry with us old habits, which are starting to come back around and bite us from behind. In a round biosphere, everything is inter-connected on the global scale. Earth's biosphere is the largest ecosystem, as well as the only spherical ecosystem that we know of. |

Cellular Earth Animation
The animation above was created as a celebration of Earth Day. It is updated from a version created in 2006. It is a Java applet. On a fast computer it should run at about 30 fames per second. This is a spherical grid of Cellular Automata . Cellular automata are typically arranged on a Cartesian grid in 1, 2, or 3 dimensions, which makes it easy to compute neighbor-cell interactions. This animation uses a spherical model, which is a non-Euclidean. surface. The propagation of dynamical patterns happens in-the-round, including the back-side of the sphere. The cell positions are arranged on the sphere as follows: First, twelve points are arranged in a regular, icosahedral pattern. Then a method for increasing the geodesic frequency is applied, whereby every pair of neighboring points gives birth to a new point lying in-between the pair. This increases the number of points to 42. This process is repeated five times to make a total of 10242 points. This is equivalent to the recursive subdivision of the triangles of an icosahedron to generate a geodesic dome. |
Unlike on a flat surface, points on a sphere do not tessellate with perfect regularity. This is illustrated by the geometry of geodesic domes, in which some vertices have five connecting neighbor vertices, while all others have six. The resulting regions of five-fold symmetry correspond to the twelve vertices of the icosahedron. Dymaxion map Buckminster Fuller's Dymaxion Map provides a way for a flat image of Earth's continents to be mapped onto a sphere with less distortion than standard mappings such as the Mercator Projection There are of course other ways to construct a geodesic dome, based on the octahedron, tetrahedron, cube, etc. But geodesic subdivision which is based on icosahedral/dodecahedral symmetry provides the least distortion in the regions of the "pinch points", where spherical curvature is manifested, and where critical cells have smaller neighborhoods (see next section). |

Neighborhoods
The Game of Life is the most popular cellular automaton, which uses 9-cell neighborhoods. But hexagonal grids can also be used in cellular automata, as well as many others. In this spherical model, every neighborhood has either 5 or 6 cells - each cell corresponding to a point (vertex) on the geodesic surface. Neighborhoods are calculated based on proximity of points after each step of increasing the geodesic frequency. When the final geodesic subdivision is finished, the local neighborhood of each cell is stored in memory. This is used to apply the cellular automata rules. |
The Rules
The interesting thing about cellular automata is the rules - the manner in which cells change their states as a result of their neighborhood states. The rules in this technique are similar to those described in "Gliders and Riders", a chapter in the book, Stigmergic Optimization. Here is a Java applet showing this technique. More details on this rule system are published in the Artificial Life X conference proceedings in 2006. |

Evolving Rules
The rules are represented as a set of parameters - "genes" - and evolved using a genetic algorithm. To find rules which create interesting space-time patterns, the genetic algorithm has an interactive component, such that a human can provide the fitness metric. A random population of individuals (represented as cellular automata rules) are generated. Then the results of randomly-selected rules are viewed and ranked as either good or bad. While the cellular automaton is running, at semi-regular intervals, randomly chosen cells have their states randomly changed. This provides some stimulation, so that interesting dynamics can emerge. The individuals with a good rating have a higher chance of mating with other high-ranking individuals, using crossover and mutation. The fitness values of the entire population are decayed over time, with each viewing, so that aesthetic trends that may have been ranked high in the past are allowed to fade (if they have been given a good ranking, then their genes have already propagated into the present population anyway). This technique was also used to evolve 2D Cellular automata, and also Mandelbrot imagery to approximate faces. |
Artistic Motivation
Cellular automata have an appeal to the mathematical mind. But also the dynamics can be visually striking, and evocative of natural processes. Small narratives emerge from the primordial soup. Global warming is a concern to many of us. Solutions to our status as Earth's skin cancer must be sought, and they must be deep solutions - by way of a paradigm-shift in all humans in terms of our impact and role in the health of the planet. Humans evolved thinking that land was infinite. But we live on a sphere. What goes around, comes around - the impact of human civilization is global. This understanding must happen on many fronts - scientifically, politically, and aesthetically. The Gaia Hypothesis of James Lovelock and Lynn Margulis can be a starting point. If nothing else, it is a poetic expression that provides a meaningful narrative. The creation of this sphere of cellular automata - rendered in Earth tones - is a way for me to share my vision of Earth as an exquisite super-organism - a thin biosphere of interacting cells. - Jeffrey Ventrella |

See the Video showing Glider Dynamics on a Geodesic Grid

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