The Game of Life

Step 400 of the acorn methuselah.  Gliders exit at 12:30 and 5:00.

Despite a lifetime of overlap, and many opportunities to cross paths, I am only now learning of the Game of Life, introduced by John Conway in 1970—the same year that I was introduced to the world. After a few exploratory hours, I knew I’d have to share the insights I gained from it relating to how we react to the present predicament.

What is it that I mean by The Game of Life? Aren’t we all caught up in it? I refer, here, to a zero-player “game” taking place on a two-dimensional grid of square cells (pixels) that can either be alive (on; black) or dead (off; white). Only two simple rules govern whether a cell will live or die in the next time step of the game:

  1. Only “live” squares having two or three adjacent live neighbors (out of 8 possible) survive to the next step.
  2. An empty (“dead”) square having exactly three live neighbors will be “born” into living status on the next step.

The first rule captures a sense of balanced population density: too sparse and survival is hard; too dense and overshoot kills. The second rule is a sexy three-way approach to procreation. Settle down, people.

A surprising degree of complexity emerges from these simple rules, whose deterministic prescription nonetheless leads to essentially unpredictable outcomes. The lessons and parallels to real life—and even breakdowns in the comparison—are highly instructive, I find.

Emergent Complexity

The first point of amazement is how cool the outcomes to such simple rules can be. Small sets of a dozen or fewer “starter” cells can cascade into surprisingly intricate and dynamic patterns—leaving behind persistent structures, evolving over impressively long times (sometimes indefinite), and producing a menagerie of different “animals” of varying complexity. One can even create logic gates and thus a computer—and then run the Game of Life on this computer!

Many of the emerging structures have acquired names in a Game of Life lexicon. Stable, static structures (“still lifes”) called blocks, beehives, loafs, ponds, boats, and ships are common products. Periodically repeating patterns (oscillators) called blinkers, toads, beacons, pulsars, and the penta-decathlon run through a sequence of morphologies before returning to the original state and starting over again. A really cool class called “spaceships” execute a repeating pattern that also shifts its position in the process so that it translates across the space as it goes. The simplest of these is a ubiquitous 5-cell arrangement called a “glider” (figure below) that moves like a bishop on a chess board and appears as a spontaneous creation in many runs. A few special arrangements called “glider guns” have been found that spit out a steady barrage of gliders. Some spaceships, called “puffers” leave a trail of debris that could be any of these previously named structures. In fact, one design leaves—of all things—glider guns in its wake so that the growth of live cells is quadratic as an ever-expanding armada of gliders sets off to nowhere.

Five gliders in a line traveling to lower right, showing the four evolutionary steps before repeating.  Each one will land on the next one’s position 17 steps later.

I encourage you to take a detour and play around, if this “world” is new to you. A number of sites provide a means to launch graphical simulations, including an extensive Wiki page and a Lexicon page. Don’t forget to come back to this article three hours later.

That all of these phenomena can arise from such simple rules in limited dimensionality is an important lesson. It doesn’t take much.

Unpredictability

The emergent complexity is significant enough that one cannot look at a random assortment of even a half-dozen live cells (close enough to each other not to trivially die out) and decide how long it might run or what sorts of structures might emerge. The only way to know is to execute the sequence. This fact makes the outcome seem random, even if it is not.

Industrious explorers made a game out of finding long-lived runs based on compact, simple beginnings (called “methuselahs”; see a long-lived example and this table summarizing the properties of various instances). For example, the rabbits and bunnies can go for over 17,000 steps, resulting in over 1,700 populated static/oscillating cells (and a few dozen escaping gliders) all starting from fewer than a dozen “live” cells.

Slight variations produce dramatic differences: one of these dies completely in 3 steps, while another goes 1103 steps and releases six gliders.

To illustrate the difficulty of prediction, one of the above shapes (in isolation) goes on for 1,103 steps and releases six gliders, while others die after three steps, enter oscillation after 6, or become a static loaf after 5. They only differ in the placement of one cell. You can try them in an interactive tool using, for instance, the R-pentomino as a starting point (launch via the box in the upper right and use the draw tool to toggle pixels; run/play or do one step at a time). This degree of unpredictability is just from a five-cell example (results from left to right: dead in 3; loaf in 5; methuselah; 4 blinkers in 6). In our real world of vastly more richly-interacting agents, how could one possibly predict the future even under perfectly deterministic rules? I encourage you not to think about your own brain at this point. Save that for later.

Determinism

The Game of Life is strictly deterministic: the same initial pattern will always produce the same result, time after time. After all, the rules are simple and operate without exception.

That said, the unpredictability (without actually running the full simulation) makes it feel open-ended. Deterministic behavior can therefore have every appearance of being wild and untamed—a very important lesson.

Sure, we might peek into sub-units and understand the complex emergence of, say, a glider and be able to describe its workings, thus predicting that an isolated glider goes on “forever.” But outcomes of interactions are much harder to predict. And guess what: the real world is full of interactions (unlike GoL that starts in the middle of an infinite void).

Invalidities

To be sure, our world is not well represented by The Game of Life (GoL), which possesses the merest shadow of the degree of complexity found in our real world. The table below highlights some of the key differences.

Attribute Game of Life Real World
Dimensionality 2 3
Space discrete squares continuous
Time discrete steps continuous
Range adjacent only infinite
Reactions binary: live/die continuous
States 2 multitude
Rules 2, simple many, nuanced
Surroundings empty void connected, interactive
Determinism strict quantum probabilities

In every way, the GoL is simpler than the real world: fewer dimensions; more restricted interaction range; binary states and reactions; and only two rules (vs. what is found in a physics textbook). Many GoL simulations produce gliders heading off into the void, predictably lasting “forever” as they cruise into empty space. Life on Earth affords no such escape from interaction.

To me, the myriad shortcomings of the GoL model in representing the real world only serve to emphasize how much richer emergent complexity can be under a richer contextual framework. I mean, if the unpredictable swirling conflagrations of activity one sees in GoL runs can arise in such a restricted space, it boggles the mind to consider what might emerge on a far more elaborate stage.

Indeed, the emergence of even single-cellular life boggles our minds: we cannot fully connect the dots from first principles as to how such magnificent beasts arose. The emergence of consciousness in animals is literally mind-boggling—and always will be, in all likelihood. Yet, at least as challenging is coming up with some clever mechanism that could act to defeat the emergence of complexity in our world. Just try to stop it! We simply can’t rule out that the world we see is emergent from basic relationships between particles—even if we can’t comprehend the scope of the complexity using our own limited brains.

“Breeder 1” puffer that generates glider guns in its wake.

Think about it this way: imagine looking at the glider gun puffer (image above; a “breeder”) in the context of the simple rules of GoL. How easy is it to understand what each part does and how its morphology cyclicly transforms without explicitly executing the algorithmic sequence of steps? It’s nearly impossible, and keep in mind that this “device”—along with its rule set—is pathetically simple compared to a microbe. Even so, the GoL is already complex enough that the only way to appreciate outcomes is to just run the sim.  I invite you to think on that.

Determinism in Reality

Now shift, if you can, to consideration of our physical universe involving quarks, electrons, gluons, photons—all interacting via four fundamental forces (two of which are infinite range)—based on masses, electric charges, “color charges,” spin states, relative positions, momenta (velocities)…all in three dimensions and all in relationship (not just nearest-neighbor as in GoL). Can funky complexity emerge, even if everything is following a finite rule set of physics? First: who are we to say no? What hubris would accompany such a claim? Second: look around! Gaze at your own navel. At this point it’s okay to think about your brain.

The physical universe is roiling with interactions, and even though the rules have been elucidated, the consequences have not. One can imagine the real world as being a much more elaborate version of the Game of Life, applying the rules one step at a time (except time is now continuous, inviting the use of calculus). However, the universe throws in a monkey wrench, in that every interaction at the quantum level has a probabilistic nature—ruining perfect clockwork determinism. But even these probabilities operate within unbreakable rules, in a quasi-deterministic fashion.

The universe, therefore, appears to simply unfold by a set of rules—quantum indeterminacy included. Many find this proposition to be extremely distasteful, and reject it out-of-hand based on no evidence that things behave otherwise (while we also lack evidence that the rules of physics are ever broken). Indeed, nothing about life feels deterministic. It feels open-ended and subject to our impulses/control. Likewise, watching a GoL run, one is struck by a sense of unpredictability and seeming open-endedness, even though we know intellectually that it is even more deterministic than the actual world (no probabilistic element thrown in).

Your Role in Life

Just as the only way to ascertain the outcome of a non-trivial GoL setup is to run the damn thing, the only way to appreciate the future states of our universe is to experience the outcome of our own “simulation.” Thus, we are operationally open-ended, even if living on a quasi-deterministic foundation.

A common reaction to the prospect of determinism is aversion to the notion that we are slaves to a script—merely executing a known future. Life really doesn’t feel like that! Yet, we are indeed slaves to physics (take that as a dare, if you wish), but “script” is taking it a bit far. There is not, and never has been a script. The future is not, and cannot be known. Just as one might create a GoL initial setup and define rules, this act is far short of scripting outcomes (and the inverse problem of coming up with initial conditions to produce any desired complex outcome is essentially impossible, unless learning from prior “forward” run behavior). In the GoL, we can’t know what “script” will emerge (from non-trivial starting points), just as no earthly being can possibly know in every detail how the universe—and life and individual choices—will evolve. The universe can dutifully follow rules without exception and without being scripted. It’s a self-writing play that goes where it must, following its nose.

Your “role” in the giant act is, therefore, unknown to you or to anybody. Let’s say that in reaction to information you receive about the predicament we’re in (e.g., from this blog and other sources) you are convinced to embrace a different vision for the future that you would like to help realize. That’s great! Maybe that’s your role. But is it truly your own choice if you’re a compulsory actor in a quasi-deterministic evolution? Who cares? Sure? Why make this about ownership for the sake of vanity? Why not focus on the outcome, and be pleased to play any role in its achievement? If that’s the way the universe happens to unfold, and you played a part in reaction to the full history of stimuli presented to you, then why protest?

Of course, if you’re the sort that refuses to go with the flow in this way, then that too is part of how the universe unfolds—not ruled out—and your role will be what it is.  Determinism easily accommodates (crafts?) those whose brains reject it.

Consider that this post could, in principle, persuade someone to relinquish non-productive concern about their existential specialness, appreciating that they are simply a magnificent cog in an intricate universe of emergent complexity, humbly accepting a role as a part of nature (not its master, steward, underlying purpose, owner, etc.), rolling up sleeves and starting to do something to change the course of modernity at the local level—however modest. Such motivation would not operate outside of a tangled deterministic history of stimuli and responses, and Ill take what we can get!

Reconciling My Role

How is it that I keep pressing for change when I call free will an illusion and give myself to the quasi-deterministic flow of nature? Why isn’t my reaction to absolve myself of any responsibility and spend my days trying to set records for Ding Dong consumption?

Well, apparently I’m not wired that way—although the Ding Dong option is tempting. I am a human being capable of absorbing and reacting to situational stimuli in the context of my evolutionary history as a social animal. Just as every fiber of my being reacts in vehement opposition to the thought of ripping a leg off a baby, a coyote, a newt, or even an ant, I react in horror to the knowledge that modernity has initiated the sixth mass extinction—quite senselessly. Call me crazy, but my impulse is to try to reverse this practice and avert a tragic outcome. Stimulus—response. No intervention to “deterministic” physics is necessary, and I’m 100% okay with that (not that it’s for me to say, anyway).

I guess the reason I keep returning to this theme of emergent complexity, seemingly open-ended determinism, and free-will-dismissal is that I see removal of pretense as a crucial part of the journey. Human supremacism is perhaps the core flaw in the operating system of modernity, and self-elevation to something special/transcendent is part and parcel of this attitude. Humility is better realized by accepting our existence as nothing more than incredible products of evolution, united in kinship to all other life, responding to troublesome inputs in a staggeringly sophisticated way.

Humans have demonstrated quite convincingly the capacity to inflict great harm to ecological health at a global scale. What happens next depends on the unpredictable outcome of the battle between concerns for self (needs, desires, drives, competition, procreation, the present and vested near-future) and not-self (social groups, cooperation, the community of life, biodiversity, the far future). I have argued that we don’t even have the option to continue satisfying selfish desires in the face of ecological collapse: we depend on the entire community of life in ways we do not appreciate (and never will, fully). So, human self-focus becomes self-defeating in the full context. This puts me strongly in the camp of advocating for not-self concern. Again, it’s part of my wiring as a social animal. I seem to care.

In the end, it comes down to whether this creature who evolved a set of skills so exceptional that it could destroy the world has also developed a sufficiently broad social sense to value the more-than-human world enough to protect it from itself. There are no guarantees at all that evolution contrived to make restraint and wisdom powerful enough to prevail over the exercise of power. At present, the outcome is not looking so great—perhaps one of many experimental blunders of evolution. Yet, I am bolstered by knowledge that some (Indigenous) cultures have prioritized the community of life and exercised the wisdom to within bounds. That’s huge!

Whatever the case, I will continue to play the role it seems I am set out to play, and hope that I happen to be on the “right side of history.” In the game of life, the only way to know is to keep playing.

Postscript: I found this article to be an informative history and commentary on The Game of Life.

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10 thoughts on “The Game of Life

  1. (A couple of points about the table – is the real world 3 dimensional? There might be other dimensions imperceptible to humans which, therefore, can never be known about. Are space and time truly discrete, or do they become granular at quantum distances/time fractions? Idk.)

    Although we may not be able to rule out that the world we see is emergent from basic relationships between particles, the corollary of that is: we cannot definitively say that it is.
    You say the universe appears to unfold by a set of rules, and that many find this proposition to be extremely distasteful. That's like saying "It's turtles all the way down. Many find this proposition to be extremely distasteful."

    As far as lacking evidence that the rules of physics are ever broken… what about the 'Big Bang'? What caused it? What caused its entropy to be so low? If everything evolves, then the 'rules' of physics are no exception. They only appear to us as set-in-stone rules, due to our relatively short lives.

    "[Is] it truly your own choice [to embrace a different vision for the future] if you’re a compulsory actor in a quasi-deterministic evolution? Who cares?" – but you obviously *do* care! Otherwise you'd just be eating Ding Dongs, whatever they are, and not writing entertaining and informative blog articles.

    "Consider that this post could […] persuade someone to relinquish non-productive concern about their existential specialness, appreciating that they are simply a magnificent cog in an intricate universe". This is a loaded statement, as it projects on anyone who might disagree with any of your post, that they have "concern about their existential specialness".
    *Obviously*, such concern, were it to exist, would be non-productive and ridiculous. Also, while I happily accept I am a nobody, I do not accept that I am a "cog", however magnificent, in however intricate a universe. I am an animal. Alive. Not inanimate, like a cog.

    Finally, you say "I seem to care". Well do you or don't you? If you feel it, then it's real. Like free will.

    • As with other exchangess between us, I'll keep this to a single unproductive volley.

      Turtles: false equivalency! We actually have loads and loads and loads of evidence for physics, while not knowing of a single instance of "not physics" in our lives or of turtles supporting the earth, for that matter. I'm not having it.

      Big Bang: our knowledge of physics in extreme conditions is incomplete (may always be so), and indeed the rules can evolve (would never say otherwise). I have not, however, opened the door for rule-breaking. Whatever the rules are (have been), they dictate what's possible.

      Ding Dongs: people of one worldview are likely incapable of inhabiting the head of those holding another. The fact that I can not care about whether my choices are independent of "deterministic" physics and do not find myself giving up in protest is proof enough for me (my worldview does not demand ownership, so nothing "breaks"). Rather than insinuate that I am lying or have not managed to grasp my own worldview, I encourage you to explore how one could adopt this stance. And yes, I know my initial statement here may be symmetric. In this case, I believe I can see how some would see the cube "flip" to the other view, but I do not find that perspective as satisfying.

      Cogs: Again, "cog" and "magnificent life" are not incompatible to me. Animate entities are constructed of the same matter as everything else obeying the same rules, in astounding complexity. I accept that. My concern is the idea that we (or life) somehow is thought to override the laws everything else in the universe must obey (specialness). Life itself is certainly special. The relationships are incredibly rich. But why must atoms in relationship be inert? Emergent complexity allows marvelous, mind blowing lifeforms to present themselves. Just because we lack the cognitive capacity to grasp it all (why would we possess that capacity?) does not render it invalid out-of-hand.

      Finally, physics: string theory wants compactified extra dimensions, in which case quarks, electrons, and our laws of physics are themselves emergent properties of the topology of these dimensions: no change to the emergent phenomenon of a quark moving in macroscopic 3-D space. Quantized space/time: sure, at the Planck scale maybe, but far too finely diced to alter the immense parameter space available, which is the key point, here.

      • An unsolicited opinion: I find so much value from each of your perspectives. I hope neither of you become entrenched in your respective positions. Sometimes, I wonder if the "real thing" is some rich amalgam of your individual views. There is a lot of "mystery" in the origin of life and consciousness.

  2. > To be sure, our world is not well represented by The Game of Life

    I know the above statement is not the main point of your article but there is a Nobel prize-winning physicist who thinks that at the deepest level physics may be well represented by a cellular automata of some sort. https://arxiv.org/abs/1405.1548

    Also Stephen Wolfram has been exploring similar ideas recently. https://www.wolframphysics.org/

  3. The Game of Life is the way it is because somebody external to the game created the board it is played on and imposed the set of rules it is played by.

    So who or what created the universe and imposed the laws of nature that physicists are learning to understand?

    • I think you give humans too much credit, here. Did we invent 2+2=4, or other rules of math that just are? I would say those things are true whether humans stumble on them or not. Likewise, an infinite sea of abstract "games" with rules exist, whether explored or not by human animals. Any alien species can realize that 2+2=4 (we'll all get the same answer: never 5), just as anyone who explores this particular game will see the same exact emergent phenomena. It is not ours to own: it has been there all along.

  4. Thanks for that posting on the GoL and the reminder of John Conway (may he RIP).

    Here's one thought that I had as I read your table of attributes of the RealWorld vs. GoL: time and space in the real world may not be continuous! I'm sure you know of this conjecture…

  5. If evolution did not make "restraint and wisdom" powerful enough on its own, then perhaps we need to help it out a little by becoming its final instrument, and using some of that technology we use to destroy nature so cruelly instead to correct our own flawed nature.

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