Conway's Game of Hope

Alphabet created by users. Surrounding oscillators from Golly.

A fable of emergence.

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Pandora sat in the living room of her home, staring at the box.

It was gold plated and decorated with ancient Greek motifs. A latch kept the box’s lid sealed, and Pandora had been instructed not to open it. But there is no better way to stoke curiosity about an object than to forbid access to it, and Pandora’s curiosity burned deep.

Eventually, the urge overcame her, and she opened the lid to peek inside. But just as quickly as she had opened the box, she understood why it had been sealed, for a violent stream of terrible ghosts erupted out. The spirits of death, illness, grief, and all other evils which had been imprisoned within now escaped into the world, cursing humanity with their awful powers, and Pandora had acted as their unwitting accomplice. She slammed the lid closed, but it was too late.

As she did, she glimpsed one entity still remaining in the box: the spirit of hope. Pandora remembered what she had learned about hope in her reading of philosophy. Aristotle had linked hope with courage, writing:

A coward is a pessimistic sort of fellow, for he fears everything. But a courageous man is the very opposite, because confidence implies hopefulness.

Pandora saw an opportunity for redemption. She could release hope from its prison, spreading courage to withstand the pain and suffering she had wrought via the evil spirits. She reached for the lid, eager to now become humanity’s benefactor.

She paused. Pandora remembered that Plato had a less optimistic view of hope than Aristotle. He had written that false hope could make people gullible and prone to ignorance, in the worst cases causing those with hope-fueled delusions to commit serious wrongdoing. Commenting on those led astray by false hope, Plato had written:

It is among these men that we find the ones who do the greatest evils to cities and individuals.

According to Plato’s reasoning, she should contain the spirit to prevent a curse of false hope from infecting the world. She withdrew her hand and sat in anxious contemplation.

A knock at the door broke the silence. Pandora cracked it open. An older man with pale complexion and swept back, disheveled graying hair stood there, his brow furrowed with concern.

“Quite sorry to bother you. I happened to be passing by your home when I saw a wave of evil ghosts rushing from underneath your doorway, and I thought it prudent to investigate.”

Pandora related to him the events of the past few minutes.

“This is indeed a terrible accident, although you certainly can’t be blamed for simple curiosity. Concerning the issue of whether to release hope—I do not envy your position in making such a decision. You bring up good points from both Aristotle and Plato. I believe the Danish philosopher Søren Kierkegard agreed with Aristotle, saying ‘a person’s whole life should be the time of hope.’ But then again, the German philosopher Friedrich Neitzche wrote that hope is ‘the worst of all evils because it prolongs the torments of man.’”

“It seems there is much debate on whether hope is a blessing or a curse.”

The man nodded. “I should introduce myself. I am Dr. John Conway. I have encountered the spirit of hope in my own journeys, and if it would aid in your decision, I can show you what I learned from that encounter.”

“My name is Pandora. I would gladly learn from your experience.”

“Excellent. I will bring you to the place where I met hope.”

Conway led Pandora to a patch of dirt near her house. He gathered some stones and a small stick from the ground. Pandora observed with puzzlement as he used the stick to draw a grid in the dirt.

“Excuse my unusual method, but I ask that you humor me. I said I would bring you to the place where I met hope, but the path we must take is hidden. If you permit me, I will explain to you the rules of a particular game, which when played properly will reveal the path.”

“Please, continue.”

“The game in question is played on a two dimensional grid of square cells, like the grid I have drawn here in the dirt. Each cell of the grid can either be ‘alive’ or ‘dead’. If a cell is empty, it is dead; if a cell is filled with a stone, it is alive.

“The game is divided into discrete turns called ‘generations’, and each generation depends on the previous one.

"To compute the next generation of the game, we look at each cell of the grid in turn, and count all of its neighboring living cells

“Depending on the number of living neighbors, we compute the cell’s next state according to two rules: the ‘birth rule’ and the ‘death rule’

“The birth rule specifies that dead cells with exactly 3 living neighbors become alive. This rule promotes the propagation of living cells across the grid, like the spread of a mold across the surface of an old piece of fruit. It allows patterns and structures to grow in the presence of other structures, expanding to fill available space.

“The death rule balances out the birth rule. It states that cells can only stay alive in the next generation if they have 2 or 3 living neighbors — no more, no less — otherwise they die. We can think of this rule as discouraging living cells from spreading too fast, overpopulating the board.”

“I can’t imagine how the game plays out in practice. How does one win?” said Pandora.

“The game does not have a winner. In fact, I should clarify that the game does not really have ‘players’ in the usual sense at all. Our interaction with the game is rather constrained: we are responsible for setting up an initial configuration of stones and then carrying out the rules of the game deterministically from there. Let's run through some examples to get a feel for how the rules are applied.”

Will the highlighted cell below be alive or dead in the next generation?



“You are a quick learner. I will leave you now with some harder puzzles. Compute the next generation for each starting configuration, and the path I spoke of will be revealed to you. I will meet you at the place where the path ends.”

Conway drew a series of grids in the dirt, placing stones in each grid. He walked away, leaving Pandora to solve the puzzles he had left her.

Given the starting configuration, draw the subsequent generation of the game by toggling cells in the lower grid.

Not quite.


Tap the grid above to toggle cells.

Conway’s game grew on Pandora the more she played. Although the rules of the game were relatively straightforward, it was surprisingly difficult to predict the next generation from the previous. Something was hidden within this deceptively simple format. The rules formed a subterranean structure of which she could only see the surface.

She glanced up and saw a path now existed where there had been none before, leading into a nearby forest. It had appeared with no fanfare—less of a grand apparition and more like a shift in perspective, as if the path had always been there but had only now become noticeable. She started into the forest.

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Pandora followed the path as it wound through the dense wood and emerged into a great clearing. A monumentally large building made of marble and limestone blocks stood in the center of the grass. A wide staircase led up to a row of columns lining the front of the building. Pandora climbed the stairs and passed through the row of columns, behind which a set of grand doors revealed themselves. Stone letters carved into the top of the doorway announced the building’s name: “The Library of Complexity”.

Conway, who had been standing to the side of the doors, greeted her as she approached.

“Your presence before these doors proves that you have seen a glimpse of emergent complexity within the game I showed you, as the Library only reveals itself to those who have seen. I must warn you that entering this place is not without risk. For some, complexity’s siren song is transfixing, and they are unable to navigate back out of the Library.”

Pandora was not one to let danger stifle her curiosity. “I accept the risk.”

They entered through the doors into a great hall. A vaulted ceiling hung impossibly high above them, and the hall stretched into the distance so far that the end was not visible. The sides were lined with stories of balcony walkways, and on each walkway spaced at regular intervals were wooden doors. The building’s exterior, although impressive, had not prepared Pandora for the shocking dimension of its interior.

Conway led them to the left side of the hall, up multiple flights of stairs, and along a walkway. Each door leading off the walkway was labeled using a system of symbols, letters and numbers unfamiliar to Pandora. They stopped at a door with the label "B3/S23”. Scrawled beneath the label was a handwritten note: “Game of Life”.

Through this door they entered into another great hall. Bookshelves made of dark wood lined the sides, each densely packed with leather bound books of various sizes, stacked tightly to the ceiling and extending far into the distance. Ladders afforded access to the upper bookshelves. What body of knowledge did this collection contain? Pandora picked a book off the closest shelf to investigate. Its cover featured a grid pattern of cells, resembling Conway’s game.

“The cover shows the starting configuration, and the pages of the book show the subsequent generations, each on its own page,” said Conway.

Pandora cracked the book open to the middle, but found blank pages there.

“The books don’t work quite like that. In order for the generations to appear, you must turn each page, one after the other, starting at the very first page. No pages can be skipped.”

Pandora flipped to the first page and saw a grid pattern. She turned the next few pages one at a time as Conway had instructed, and each subsequent page showed the next generation of the game. She drew another book from the shelf and paged through it, and then another. Each book had a unique starting configuration on its cover, and each showed a different evolution of the game. She stepped back from the shelf, pondering. There were at least a hundred books on the shelf immediately in front of her. Was each book unique? How many books could there be in total? She eyed the expanse of the book-filled hallway.

“Every possible configuration of the game lives here, each inside its own book.” said Conway, sensing her wonder.

“For what size of grid?”

Conway chuckled. “A perceptive question. You’re wondering if the starting configurations are limited to a finite size. They are not, in fact. Every possible starting configuration of the game lives here, for every possible size of grid.”

Pandora understood, although what she understood was impossible. The hall did not only appear endless; it literally had no end. If every possible size of grid was represented, there would be an infinite number of starting configurations, and therefore an infinite number of books were stored in the hall. Pandora felt dizzy at the vastness.

“Who built this place?”

Conway smiled. “No one. It has always existed.”

“As you can imagine, the quantity of books here can make it difficult to find the one you’re looking for. In my youth, I was a regular visitor to this section of the Library, and I found it necessary to develop a navigational system.” Conway gestured to a large piece of equipment near the entrance. “I believe it is still in working order.”

The equipment resembled an array of cabinets. The doors of the cabinets were a muted blue color, except for the center cabinet, which had no doors, exposing an assembly of beige panels. The panels were packed with technical affordances: dials, gauges, switches, knobs, rows of small lights. Printed on the topmost of these panels was large text that read “DIGITAL EQUIPMENT CORPORATION DATA PROCESSOR”. In front of the assembly was a white desk, above which a console with a circular, domed screen was mounted. On the desk was a small metal box with a flat button and inset ball. Pandora surmised that this must be a controller for the whole apparatus.

Conway confirmed her suspicion as he seated himself at a chair in front of the desk and placed his hand atop the box, fingers resting on the inset ball. With his other hand, he pressed a toggle button on one of the cabinet panels, and the circular screen in front of him flickered faintly to life. The domed screen showed a grid. Conway expertly manipulated the input device to move a cursor on the screen, and used the flat button to toggle cells of the virtual grid on and off, drawing a pattern.

Satisfied with his design, he pressed another button on the panel in the cabinet. A fine glowing strand of light emanated from a port on the side and marched down the floor of the hall, forging a path into the distance. They followed the light-path until it made a right turn towards a bookshelf and terminated at a book there. Conway took the indicated book from the shelf and handed it to Pandora. The same design Conway had drawn on the circular screen was on the book’s cover, and beneath it, a scribbled title: “R-Pentomino”.

“This section of the shelf contains all the ‘Pentominos’, named as such because they consist of five living cells in their starting configuration (pento being Greek for five). I labeled them by letter: A, B, C, and so forth. Of all the pentominoes, ‘R’ is special. I expect the contents of this book will intrigue you.”

Pandora sat on the floor, leaning her back against the bookshelf, and began flipping through the book. The generations of the game blurred together like frames of a movie. In the first few pages, the starting configuration dissolved into waves of static. As the noise sloshed around, it left small clumps of stable cells in its wake, like fungal fruiting bodies bursting from the soil after a storm. One body in particular was not only stable, but moved, crawling across the page like an ant. More of the crawling ants emerged from the chaos until six of the blocky creatures were scampering along the grid. The patterns entranced Pandora.

She turned the last page of the book and frowned. “Why does it end?”

“The book ends when the pattern either dies out or begins to repeat itself.”

Pandora closed the book slowly. She sat in silence for some time before speaking.

“When you first taught me the game of stones, I could not predict the dynamic behavior that would emerge from such a constrained rule set—a whole that is more than the some of its parts. Why does that happen?”

Conway shrugged. “I can't explain it, any more than I can explain why atoms form molecules or stars form galaxies. They just do.”

The parts created the whole spontaneously. Nestled in-between the parts and the whole lived an explanatory gap, a blind spot that reason could not penetrate.

“In this R-Pentomino book, the rules unfold into a mesmerizing embroidered tapestry. The ants that crawls across the page—”

“I call them ‘gliders’.”

Gliders. They appeared to be real, differentiated objects with position, speed, and direction; in every reasonable sense, the glider could be said to exist. Yet, although the game’s rules supported the glider, they did not predetermine it. The glider was composed of grid cells, but its thingness sat on top of the grid substrate; it asserted its own existence.

“Are there other such entities hidden within the books here?” said Pandora.

“When I discovered this place, I paged through many books in order to answer that very question. I brought colleagues to help search. We found that the glider is but one of many such entities, each with its own characteristics. Some are small, others are made of hundreds of cells. We created a taxonomy of patterns: ‘still-lifes’ are cell-groups that stay the same from generation to generation, while ‘oscillators’ repeat themselves over some period. There are ‘beehives’, ‘snakes’, ‘blinkers’, ‘spaceships’... the list goes on. We became naturalists of the patterns in the books, documenting and organizing the creatures we found inside. I called the game ‘The Game of Life’.”

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“I want to explore more books of your so-called Game of Life.”

“Please do—but remember, this place can be consuming. Be careful not to lose yourself.”

As Conway settled into the chair of the DIGITAL EQUIPMENT CORPORATION DATA PROCESSOR, Pandora set off into the depths of the hall. She found that many starting configurations did not lead to interesting behavior, but some did: patterns that repeated themselves in short loops (“oscillators”, Conway had called them), or locomotive creatures like the glider that lumbered across the page, or even small machines that created gliders at regular intervals. Her understanding and infatuation with the game heightened.

Pandora found herself in a section of the hallway where the books were larger than before. The grid patterns on the pages of these books were dense, each grid cell barely a dot on the page. In these larger books, streams of interacting gliders composed larger structures—circuit-like networks implementing signal communication and primitive information processing. These configurations were like intricate schematics for machines, and each turn of the page was a tick of the machine’s delicate clockwork. Pandora theorized that given a large enough book and a precise enough starting configuration, Game of Life machines could be designed to manipulate numbers, perform math and logic—maybe even understand human language. Were there books that could read and write philosophy?

Page after page, volume after volume, Pandora excavated the subterranean structure of Conway’s game, revealing a system inextricably woven with the richness of nature. The books told a story of complexity.

Pandora wondered what was behind the other doors of the Library. Were there other types of books with entirely different patterns and emergent machines? She returned to the entrance of the Game of Life hall to find Conway asleep in his chair. Sliding discreetly past, she exited into the outer hall. She inspected the neighboring door along the walkway and found it had a similar label to the Game of Life door: “B2/S23”. While the Game of Life’s birth rule applied to cells with three neighbors (“B3”), this door evidently led to a hall for a different game in which the birth rule applied to cells with two neighbors (“B2”).

Pandora entered to find another book-filled hall, but the books there disappointed her. The grids tended to overpopulate with live cells and then halt. She grew bored and ventured into the next hall, only to find a similarly uninteresting game. Hall after hall yielded the same results. The Game of Life’s emergent complexity was evidently rare among games in the Library. Given the apparent extent of the Library, finding an interesting game was like finding a needle in an infinite haystack.

Exiting into the main hall, she noticed that far down the hall on the opposite side was a door with a red banner above it. The evidence of a human presence stood out against the monotony of the Library’s repetitiveness, and Pandora eagerly made her way down the stairs, across the hall, and up again to the walkway where the red banner was. It read “Unified Physics Project”. She opened the door beneath and entered.

This was a book-filled hall like the others, but Pandora could make out a large form in the center some distance from the entrance. As she drew closer, the form resolved into a giant nest-like structure constructed of piled up books. A man sat in the center of the nest, muttering to himself.

“... started infinitely dimensional ... as the universe evolved ... the dimensionality cooled to three ...”

Pandora found the chant-like quality of his speech enticing, although she was mystified by its technical contents. She approached the nest, but he was oblivious to her presence—a book in his lap consumed his attention. Pandora watched as he systematically paged through it, jotted down notes, and tossed it aside, contributing to the book nest accumulating around him.

“Hello!” she said.

He ceased his muttering and glanced up, a look of surprise painted on his round face. He peered at her with wide eyes through thin framed glasses before starting to speak again. “The principle of computational irreducibility ... boundedness of observers ... the gauges you specify to describe the structure of spacetime ...”

Pandora gave a nervous laugh. “I’m sorry?”

“... with those constraints you can derive relativity ... mass is the permanence of hypergraph rewriting ...”

He opened a book from the nest, showing Pandora the first page. Rather than the Game of Life’s grid based patterns, the pages of this book contained intricate networks of connected lines, dense spider webs that evolved and grew with each page.

“... the structure of the universe is an expression ... from an axiom ...”

Pandora wanted to understand what this man understood. Had he gone beyond the Game of Life, finding a hall of books that truly plumbed the depths of nature’s mysteries? Did the lace network of lines describe the dynamics of a miniature cosmos?

“... a single slice of space is a Hilbert space ... but the whole time evolution is a generalization ... we have this branchial space...”

She grasped at his words, hoping for a glimpse of the light, wanting it to lift her so that she could see.

“...progress through time in branchial space...”

A chance to understand the core of reality.

“...quantum entanglement...”

She lost herself in his chant.

“I know his hope.” Conway was beside her. Pandora blinked, waking from her reverie and finding herself again. The man in the nest was still muttering, his face buried in a book again, ignoring them. “It took me years of exploring the Library of Complexity to find the Game of Life, and it was hope that fueled that exploration—hope for a simple system that would give rise to emergent complexity.”

“It worked, didn’t it?” said Pandora. “You hoped for something, and you found it. Maybe he has found something too.”

“I was rewarded for my hope with the discovery of the Game of Life. I saw a reflection of the universe's emergent behavior in the grid cells of the game. But among some, the Game of Life fueled a desire for a more complex system that would not only mirror aspects of, but completely model all of reality. These explorers search the Library for a perfect simulacrum: a volume whose pages contain the complete story of our world, the genesis of life, of consciousness—of you and me! Perhaps this very conversation appears within its hypothetical pages.”

Pandora shuddered. The thought of opening a book and seeing a simulated copy of herself gave her goosebumps.

“Is it a false hope? I don’t know. Maybe such a book does exist. This man might believe that he has found the story of the world in this very hall—but I am skeptical. Those who yearn for such a discovery are owned by that desire. They believe in the Game of Life as if it were a creation myth, mistaking an interesting pattern for a vital spark. They have ventured too deep into the Library and forgotten the way out.”

The man in the nest was a prisoner to the manic theory that tumbled from his mouth, trapped by a hope for understanding that would never come.

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Pandora and Conway stood on the Library's steps in the warm sun.

“I am glad you brought me here. It has helped me understand the force contained within the box,” said Pandora.

“Excellent. I will leave you to your decision,” said Conway, bidding farewell before departing.

Pandora made her way back along the forest path. She drank in the lush plant and animal life that surrounded her in the cool forest. A peaceful energy radiated from the fractal leaves of ferns and carpet of moss. The environment was a welcome respite from the stark infinities of the Library of Complexity. She came upon a small opening in the trees where a lilac bush had claimed a patch of sunlight. Bees laden with pollen swarmed the pale violet flowers, buzzing with intense purpose in a choreographed dance. Further down the path, a growth of gold slime mold thrived on a weathered log, its porous network of cavities forming an intricate structure. Earthworms, mushrooms, birds, a woodland mouse hiding beneath a fern—the forest was a dense connected network: a mesh of parts, and of wholes made from those parts, and of gaps of mystery where the parts met the wholes.

She emerged from the forest, strode by the dirt patch where Conway had first taught her the Game of Life, and went into her home. The box sat where she had left it on the table.

Her encounter with the man in the nest caused Pandora to reflect on the wisdom of opening the box. She did not wish to spread the curse of false hope that kept him imprisoned in the Library of Complexity. However, the volumes of the Library were deserving of a different, softer hope, because they were beautiful stories, embroidered with complexity, generating hope from within themselves. The books would not unlock the secrets of the universe, but they could unfold those mysteries before you, splaying their incomprehensibility out onto the page, making legible the unknowable. This was not a deluded hope for a future “eureka!”, but a hope for the mystery of the now.

The glider was a sister to the lilac and the bee, the slime mold and the earthworm, blossoming forth, bubbling up from the generative froth of the universe, its spontaneous complexity cascading from page to page, filling the grid with evidence for a theory that could not be articulated and did not need to be articulated.

For some, she knew that hope would mutate into delusion and bring suffering. But for others, the spirit of hope could lift them so that they could see the joy beyond the walls of the world, even if they could not understand what they saw.

Pandora opened the box, sending hope scampering off into the world like the glider from the R-Pentomino.

The end.

Turing machine design originally created by Paul Rendell in 2000; reproduced from Golly.

This fable incorporates the following people, events and stories:

Pandora’s Box. In the Greek poet Hesiod’s original telling of the Pandora’s Box myth, the Greek goddess of hope, Elpis, stays behind after Pandora releases evil spirits from a jar (in later translations, the jar became a box).

The only thing that stayed within the unbreakable contours of the jar was Elpis [Hope].
It did not fly out.
Before it could, she put back the lid on top of the jar,
according to the plans of aegis-bearing Zeus, the cloud-gatherer.

The presence of Elpis left behind in the jar has caused much debate among interpreters. Is Elpis left behind to give us humans solace in the face of the evils unleashed upon the world by Pandora? Or is she trapped, not able to reach us? The debate is exacerbated by the ambiguous translation of the Greek word “elpis”, which can potentially have a negative connotation — something closer to “deluded expectation” rather than “hope”.

John Conway. The fictional character Dr. J R. Conway in the story is loosely inspired by the English mathematician John Conway (1937–2020). Conway discovered the Game of Life in 1970, and along with colleagues at Princeton, cataloged many patterns and emergent creatures, such as the R-Pentomino and the Glider. He sent a letter to his friend Martin Gardner detailing some early discoveries in the game. The Game of Life became one of his most celebrated achievements among the general public, which he eventually came to resent because it overshadowed his other more serious mathematical achievements. His biographer quoted him as saying, “I regard [Game of] Life as trash, frankly.” I have taken significant liberties in my dramatic fabrication of his personality and ideas.

Muttering man. The man trapped in the Library of Complexity is a stand-in for researchers of cellular automata (of which the Game of Life is a particular example of) who fantasize about grand applications of their simulations. In his book From Wheels, Life and Other Mathematical Amusements, Martin Gardner describes Conway’s own early fantasies along these lines:

Conway goes on to speculate that if you imagine a sufficiently large broth of randomly placed bits, one would expect that by pure chance self-replicating creatures would arise, and those best adapted to survive would live longer than the others. Interactions with the environment would produce mutations. As in organic evolution, most mutations would be harmful, but some would have survival value. "It's probable," Conway writes, "given a large enough 'Life' space, initially in a random state, that after a long time, intelligent self-reproducing animals will emerge and populate some parts of the space." (p. 254)

Researchers in the field of cellular automata, as well as the related fields of artificial life and complex systems, have received criticism for unsubstantiated or pseudoscientific theories. For a critical perspective on these fields, see: Cyberfeminism and Artificial Life by Sarah Kember; a 1995 article in Scientific American; a book review of a seminal work in the field of cellular automata; another book review of the same work.

DIGITAL EQUIPMENT CORPORATION DATA PROCESSOR. The computer that Cownay uses to navigate the Game of Life hall in the story is a reference to the real computer that his team programmed to run Game of Life simulations. A photograph taken in 1974 dramatically captures Conway sitting at this computer staring into a Game of Life pattern.

Tolkien on hope. In his essay On Fairy-stories, J.R.R. Tolkien talked of hope allowing one to see "beyond the walls of the world":

“The consolation of fairy-stories [...] is a sudden and miraculous grace: never to be counted on to recur. It does not deny the existence of dyscatastrophe, of sorrow and failure: the possibility of these is necessary to the joy of deliverance; it denies (in the face of much evidence, if you will) universal final defeat and in so far is evangelium, giving a fleeting glimpse of Joy, Joy beyond the walls of the world, poignant as grief.”

Game of Life patterns. The title pattern uses a still-life alphabet created by users on the forum. The forms that surround the title are composed from a library of oscillators packaged with the Game of Life simulation software Golly. The simulation at end of the story features a Turing machine design originally created by Paul Rendell in 2000.

Infinite library. In Jorge Borges’ Library of Babel short story, the universe consists of an infinite library that stores every possible book.

Made by Alexander Miller for The HTML Review 2024 / / twitter / instagram / mastodon

Thanks to Keith Hayes for information on the PDP7 computer, and to M & S for the feedback and patience.