Snowball Effects

An ancient fable describes a king who offered a humble man a reward. The man asked for a chessboard and for a single grain of rice (or sometimes wheat) on the first square, for two on the second, for four on the third, and for each successive square to double the previous amount. The total pile of rice by the time the board is completed is larger than Mount Everest and exceeds annual global production by thousands of times. This is a classic demonstration of exponential growth - in which each change builds upon the one before. Here we take a look at what such growth scenarios in science fiction can tell us about this area.

 Multiplication

Exponential (or snowball or geometric) growth occurs when in each successive stage, the starting number is multiplied by a constant [1]. As for example:

More generally, instead of doubling, the growth could be by a factor of 5 or 1.5 or 1.05 or indeed any number (call it r) greater than 1.00 (i.e. more than simple replacement of the original) and the result will still grow rapidly with each successive step, reaching a level of r^(n-1) after n steps (or generations).

Perhaps the most common and striking uses of this kind of multiplication in science fiction occurs in the context of reproduction.

“The Trouble with Tribbles” (1967) is one of the best known episodes of the original series of Star Trek (TV series, 1966-1969). In this narrative, an itinerant merchant called Cyrano Jones persuades the Enterprise’s Lieutenant Uhura to purchase a tribble - a small animal that looks like a ball of fur and makes a pleasing noise and vibration akin to purring. To Uhura’s surprise, the tribble soon delivers a dozen offspring which she distributes to her friends. However these new tribbles also multiply within a few hours, as do their offspring, and soon the Enterprise is in the midst of an exponential growth situation - the population of tribbles not only becomes large enough to threaten shipboard operations but also consume a shipment of grain (quadrotriticale) which had been intended to help colonise a strategically important world. The tribble population fills both the Enterprise and Space Station K-7, amounting, as Spock tells his captain, to:

“One million, seven hundred seventy-one thousand, five hundred sixty-one. That’s assuming one tribble, multiplying with an average litter of ten, producing a new generation every twelve hours, over a period of three days.”
“That’s assuming they got here three days ago?”
“And allowing for the amount of grain consumed and the volume of the storage compartment”

Fortunately for the Enterprise, the tribble plague also uncovers a hitherto unsuspected threat from the Klingon Empire.

The episode included a memorable scene which showed Captain Kirk buried in an avalanche of tribbles spilling from an overhead locker, and also provoked sequels in the form of Star Trek Animated episode “More Tribbles, More Trouble” (1973) and the Star Trek: Deep Space Nine time travel story “Trials and Tribblations” (1996) which placed the DS9 crew into the original episode, in a parallel plot line to the original. While “Trials and Tribbleations” focussed on the Klingon threat, “More Tribbles, More Trouble” returned to the issue of the tribbles’ breeding rate, with a new predator and a supposed gene modification suppressant for their reproduction both proving inadequate to the task of controlling it.

The atmosphere of all three episodes was humorous, and included some of the worst puns in the series, but they nonetheless make a serious point about explosive population growth, the challenge it presents to food supplies, and the difficulty of introducing predators without unexpected consequences.

Star Trek’s tribbles were reminiscent of an earlier science fiction story - in fact sufficiently so that the television company’s lawyers wrote to the earlier author to apologise for any accidental perception of plagiarism, although scriptwriter David Gerrold had no memory of a conscious influence. The story in question was Robert A Heinlein’s The Rolling Stones (novel, 1952; aka Space Family Stone), in which adolescent space-travelling twins Castor and Pollux Stone hope to make their fortune through small scale import/export. Arriving on a Mars which fits the harsh-but-habitable norms of contemporary science-fiction, they look around for opportunities, and come across a native creature a shopkeeper describes as a ‘flat cat’:

“It has a latin name but I never bothered to learn it.” Angelo tickled it with a forefinger; it began to purr like a high-pitched buzzer. It had no discernable features, being merely a pie-shaped mass of sleek red fur, a little darker than Castor’s own hair. “They’re affectionate little things, and many of the sandrats keep them as pets - a man has to have something to talk to when he’s out prospecting and a flat cat is better than a wife because it can’t talk back. It just purrs and snuggles up to you.”

They buy the creature for their younger sibling. Unfortunately, the flat cats show the same geometric population growth as the tribbles. In a chapter called “Flat cats factorial” Heinlein describes how after thirty-seven days in space, the flat cat has eight cute offspring, then…

Sixty-four days later the kittens had kittens, eight each. Sixty-four days after that, the hundred and forty-sixth day after Phobos departure, the kitten’s kittens had kittens; that made five hundred and thirteen.
“This,” said Captain Stone, “has got to stop!”

Fortunately for the Stone family, given their location in deep space and limited resources, the solution here is relatively straightforward. Low temperatures and a lack of food put the Martian-native creatures into a state of dormancy, allowing them to be rounded up and kept safely in an empty hold until they are able to monetize the now copious supply (with suitable warnings) in the asteroid belt.

While both tribbles and flat cats invoke the cute and funny side of multiplicative reproduction [2], science fiction has also explored its darker aspects. The literature of human overpopulation is extensive. More ominous still is the spectre of autonomous self-replicating machines capable not just of replacing themselves at the end of their lives but also building further copies - which will themselves do the same, resulting in exponential population growth.

Amongst scientists and futurists who have considered the consequences of such devices are physicist John von Neumann and technologist Eric Drexler, who suggested their use for interstellar probes on the macro scale, and the risk of a ‘grey goo’ apocalypse on the nanoscale, respectively.

Von Neumann probes are interstellar spacecraft capable of repairing or reproducing themselves from raw materials in any solar system they encounter and were proposed as a relatively inexpensive and efficient way for any species to explore its galactic environment before sending a crewed mission to promising targets. Each probe could send its progeny to several star systems instead of having to pick one destination at each stop. However others, including notably science fiction author Fred Saberhagen in his Berserker series, have noted that such probes would also be extremely effective ways to identify and destroy any species showing evidence of threatening their originators.

Nanotechnology is a different proposition: the human effort involved in manufacturing a device too small to see is enormous, and yet their applications en masse to medicine, production, computing and endless other areas could be incredibly valuable. Hence the most logical approach is to construct a handful of nano-scale devices which are themselves designed to construct others in an exponentially growing self-replicating system. While recognising the revolutionary value of such devices, Drexler warned that if such devices malfunctioned (perhaps due to cosmic ray strikes, solar radiation or programming errors) then they could continue self-replicating from available resources until the entire surface of the planet was coated in a thick layer of nanobots - and nothing else. Again, science fiction writers have written extensively about the potential and risks of nanotech, including, for example, several Doctor Who serials.

A science fictional example of machine replication which effectively illustrates the dangers of snowballing population growth can be found in the universe of Stargate: SG-1 (TV series, ??). Here the SG-1 exploration team encounter a form of machine intelligence that consists of large numbers of self-replicating and self-organising units. While each unit or block looks like a small tablet, they usually take the collective form of a mechanical spider or insect, converting any and all materials at hand into more of themselves. Thus a handful of blocks can grow their population exponentially into a plague. Aboard any spaceship, with its essential hull and life support systems vulnerable to conversion, this presents an obvious problem, but on a planetary scale, with unlimited raw material available for replication, the impact can be one of mass extinction. The Replicators present a recurring problem through several series of SG-1, in the course of which the team discovers the child-like Android who initially created them for companionship, and also more complex Replicator forms capable of combining into human-like androids. Indeed, as the series progresses it becomes clear that an advanced alien race, the Asgard, are fighting a losing war against the Replicators’ advance. Crucially, unlike in the ‘grey goo’ scenario, the Replicators are capable of learning from, and assimilating technologies from, those they encounter.

Division

A different form of population growth (albeit one with a similar effect!) can be found not through multiplication but by division. Veering momentarily into fantasy, this was effectively illustrated by Goethe’s 1797 poem The Sorcerer’s Apprentice, which was memorably animated by Disney in their film Fantasia. Tasked with fetching water, the apprentice instead enchants a broom to do so. When the broom fails to stop when the container is full, the apprentice cleaves it with an axe. Both halves of the broom resume their full form, and both carry yet more water. In the Fantasia version, this repeats, and the brooms start spontaneously dividing, resulting in a dangerous flood, before the sorcerer returns to break the spell.

The comedic animated television series Star Trek: Lower Decks played on this theme in a slightly ridiculous episode, “An Embarrassment of Dooplers” (2021). Here the crew of the starfleet support vessel USS Cerritos are escorting an alien emissary to a trade negotiation. As we’re told:

“Dooplers are a species which involuntarily duplicate as an emotional defence mechanism.” 

Unsurprisingly the chronically insecure emissary begins to duplicate, gets anxious about that fact and each of the duplicates themselves duplicate until soon the entire ship is swamped in Dooplers. It takes the usual chaotically creative approach of the Cerritos team to come up with a solution which causes the Dooplers to recombine. As in Fantasia, and unlike in many of the other narratives discussed here, no attempt is made to explain just how the energy and matter required for the duplication comes from.

In more serious science fictional contexts population growth through fission of one thing into two is most often seen in the context of the cellular division that results in disease spread. In the post-apocalyptic television drama Survivors, for example, the disease which defines the scenario is caused by a single vial of infectious material, shown in the opening credits. Not only do the disease organisms (whether viral or bacterial) increase their population by doubling at each cell division, but the spread through the population is also exponential. The few people infected by the disease therein each infect others, who infect others, spreading geometrically through the travel network until the whole world is infected from divisions of those original viruses - in a global pandemic pattern all too familiar in the post-covid world, but with a much higher mortality than currently known infections.

Similar snowballing spread of diseases from initial infections are common in science fiction, appearing in a variety of medical science fiction, including, for example, in James White’s Sector General series about an interspecies medical station or Murray Leinster’s Med Service series, as well as vast numbers of post-apocalyptic or even zombie movies.

An interestingly different medical SF take, which blurs the line between multiplication and division can be found in Contamination Crew by Alan E Nourse (short story, Worlds of IF, Feb 1952). In this narrative, humanity has found its place in the Universe by providing medical services to other planets. A medical team investigating a reportedly indestructible alien menace, the hlorg, accidentally brings a contaminant aboard its mothership, which rapidly begins to consume material from around itself. A quick investigation proves unhelpful:

“Your hlorg is an ideal anamorph. A nothing. Protoplasm, just protoplasm.”
Jenkins looked up sharply. “What about his cellular organization?”
“No cells,” said Bowman. “Unless they’re sub-microscopic, and I’d need an electron-peeker to tell you that.”

Unfortunately, while the hlorg is not undergoing cell-division and not breeding, it is growing - as we’re told - at a geometric rate, doubling in size repeatedly in a matter of hours. When forcibly divided, the protoplasmic creature’s halves attempt to rejoin, and if they can’t do so, develop into separate entities. Exposure to freezing vacuum just causes the creature to fracture into multiple parts - each capable of consuming more - and nothing the experienced medical crew tries seems to kill it… until they discover that it is susceptible to the weak acid of the human digestive system, leaving them with only one option if they want to get back to Earth:

“with liberal use of Happy-O we can occasionally convince ourselves that it is rare beefsteak, and the Green Doctor, our pro-tem cook, has concocted several very tasty sauces, such as mushroom, onion, etc. We reduce the hlorg to half its size each day, and if thoroughly heated the chunks lie still on the plate for quite some time.”

… after all, the hlorg had consumed their own food supplies and turnabout is fair play.

Similarly able to process any material or other source of energy into more of its own form is the alien organism which falls to Earth in The Leech by Phillips Barbee (short story, Galaxy, Dec 1952). Arriving as a dormant spore, it doubles in diameter (so quadruples in area) every day, consuming anything that touches it until it’s three miles across and

The leech was growing at a geometric rate. It could cover the United States in a few months.

Unfortunately, the military approach, including atomic bombs, just gives it an energy-rich meal. Eventually it is lured from Earth and directed towards the Sun, leaving the scientists to wonder at the last minute whether that was really a good idea...

Switching from factors greater than one, another brief detour into fantasy gives an interesting example in Convergent Series, a short story by Larry Niven (aka The Long Night, F&SF, March 1967). This involves a lazy anthropology student who comes up with a new idea in the field of demon summoning and tests it, finding rather to his surprise that he appears to have succeeded. Thinking fast, he sums up the rules as the demon explains them:

“And if I do use the wish, you have to remain in the pentagram until my wish is granted, or until twenty-four hours are up. Then you teleport to Hell to report same, and come back for me immediately, reappearing in the pentagram.”

With the demon agreeing, the student hurriedly makes a wish and then redraws the pentagon. Reappearing, the demon senses that it is too small and changes size…

He was back, spread-eagled before me, two feet tall and three feet off the ground. His black know-it-all grin disappeared when he saw the pentagram wasn't there. Then — he was seven inches tall, eyes bugged in surprise, yelling in a contralto voice. “Whereinhell's the — ”
He was two inches of bright red toy solider.
“ — Pentagram?” he squealed.

By drawing the pentagram on the demon’s own stomach, the rules of magic have forced the entity to reduce his size to a fraction of the original to fit in the shape, only to find that reduces the pentagon and he needs to be smaller still - an exponential decrease in size that will never end.

Interest

While most geometric growth scenarios have considered doubling or higher rates, an interesting case can be found in situations where the increment is much smaller, perhaps just a few percent, but repeated many times over a long period of time, leading to a slower but nonetheless exponentially growth which can have a dramatic effect if continued for long enough - the scenario found in banking in compound interest.

John Jones’s Dollar by Henry Stephen Keeler (Amazing, April 1956) is a short story which describes a visiphone lecture delivered at the University of Terra in the 33rd Century, looking back at a historical and economical effect which spanned a millennium. In 1921, a man called John Jones deposited a silver dollar in the bank at a three percent interest rate in the name of his fortieth-generation descendant along the line of the eldest child in each generation, with compound interest paid annually.

The lecture goes through the calculation. After ten years, the deposit is worth $1.34, after a hundred it was worth $19.10. By two hundred years the value was $364, and after three hundred years it was $6920. With the original Chicago bank now the global bank of Terra, the exponential growth continued. After five hundred years, the investment was worth $2,520,000. In the next century, a medical breakthrough increased the human lifespan to two centuries, and:

“I referred to the bank account of John Jones the fortieth. It, gentlemen, had grown to such a prodigious sum that a special bank and board of directors had to be created in order to care for, and reinvest it. By scanning the following notation, you will perceive the truth of my statement:
2521 600 years $47,900,000”

As humanity spreads through the Solar system, the compound interest continues:

 “2821 900 years $332,000,000,000

"The meaning of those figures, gentlemen, as stated in simple language, was that the John Jones Dollar now comprised practically all the wealth on Earth, Mars, and Venus—with the exception of one university site on each planet, which was, of course, school property.”

 By 2900, with the thirty-ninth generation of John Jones alive, a valuation showed that the amount due to their future child exceeded the total value of all wealth, land and energy in the Solar System, including that of the Sun itself. Fortunately for humanity, the man dies childless.

“As a result, there was no one to turn the Solar System over to. Immediately, the Interplanetary Government stepped in and took possession of it. At that instant, of course, private property ceased. In the twinkling of an eye almost, we reached the true socialistic and democratic condition for which man had futilely hoped throughout the ages.”

Television sitcom Red Dwarf explored the effects of compound interest in the episode “Me Squared” in an exchange between ship’s computer Holly and Dave Lister, the last human alive after being revived from three million years in suspended animation. Dave is understandably startled when Holly announces they are being pursued by two fighters from the North Western Electricity Board, NorWEB, for Dave’s crimes against humanity:

“It seems, when you left Earth three million years ago, you left two half-eaten German sausages on a plate in your kitchen.”

“Did I?”

“Do you know what happens to sausages left unattended for three million years?”

“Yeah, they go mouldy.”

“Your sausages, Dave, now cover seven eighths of the world’s surface. Also you left £17.50 in your bank account. Thanks to compound interest you now own 98% of all the world’s wealth. Because you hoarded it for three million years nobody’s got any money except for you and NorWEB.”

“Why NorWEB?”

“You left a light on in the bathroom. I’ve got a final demand here for 180 billion pounds.”

“180 billion pounds! You’re kidding!”

“...April Fool.”

 While this is clearly absurd for many reasons, it is not the only comedic SF to make use of compound interest. In Douglas Adams’ The Restaurant at the End of the Universe (novel, ??; based on BBC radio series The Hitchhiker’s Guide to the Galaxy), the eponymous eatery Milliways, exists in a time-bubble which swings it backwards and forwards across the final singularity once each meal sitting. Time travellers from throughout history visit it for fantastic meals:

All you have to do is deposit one penny in a savings account in your own era, and when you arrive at the End of Time the operation of compound interest means that the fabulous cost of your meal has been paid for.

Of course, the assumptions in all three examples here - that the bank remains solvent and continues paying interest at a constant rate; that its accounts are passed onwards to other organisations as civilisations rise and fall, stars died and planets are destroyed; that currency is not devalued in the process (whether by government or simply by the constant erosion of inflation); and that the account does not pass to (and be spent by) heirs along the way (except for John Jones’s Dollar, where this is deliberately averted), amongst many others - are a long way from realistic. In fact, the final demand for John Jones’s Dollar after a thousand years was $6310 trillion - a ludicrous sum in 1956, let alone 1921, but far less so now where Apple and Nvidia are both valued at more than $4T each, a mere century after Jones’ investment.

These stories nonetheless demonstrate the power of exponential growth, even if the scaling factor exceeds one by no more than a few percent.

While the compound interest in these cases is financial, Star Trek: The Next Generation explored a more serious effect of minute changes which compound over time. In the episode “Up the Long Ladder” (1989), the Enterprise crew come across two worlds settled from the same long-since stranded spaceship. One has returned to agrarian life. The other maintains a stable civilisation by repeated cloning from the five original survivors of a crash. When Dr Pulaski asks how they have overcome ‘replicative fading’ the answer proves to be simpler than expected: they haven’t. Each generation of clones accumulates minor mutations which compound those acquired by the generation before. The result is that the population is rapidly becoming unviable and in desperate need of new genetic material. Even with a success rate of 99.9% of cloned genes, in a dozen generations more than two percent of the DNA might have degraded or mutated, which is significantly more than the differences between the human genome and that of chimpanzees [3]. Here the multiplying factor being compounded is only marginally less than one, but its effect is still growing exponentially.

Explosions

While most of the examples above have focussed on individuals (or their property), the same form of exponential growth is also found in explosions. Most famously, in a nuclear reaction, each atomic fission event releases two neutrons, each capable of triggering another fission event, which each trigger two more, and so on in an ultimately catastrophic release of energy. This is known as a runaway chain reaction.

Much of the mid-twentieth century science fiction of nuclear disaster played on these themes. A tense example is Boyd Ellanby’s short story Chain Reaction (Galaxy, September 1956 [4]). This describes a small group of physicists visiting one of their former colleagues in a mental hospital, playing poker as they wait anxiously to learn whether he was wrong in asserting that a new weapon being tested will set off a chain reaction that will destroy the Earth… or not.

An interestingly oblique approach is taken in The Tale of Happiton by Douglas Hofstadter, which appeared in the 1987 anthology Mathenauts, edited by Rudy Rucker. This town is threatened by a demon with a yen for postcards, who sets up a device which means the town risks a 1 in 100,000 chance of annihilation every time the church bell rings (usually on the hour). However he will delay it by a factor of 1.00001 for each hand-written postcard he receives on a given day, i.e. by (1.00001)^n where n is the number of cards. This is an exponential scaling, so while a few people writing a few postcards will have virtually no effect, if everyone in the town is willing to spend an hour writing postcards, or support others to write full-time, the chances of their destruction go from as high as 1 in twelve per year to virtually nil in a century. Unfortunately everyone in town decides that they are too busy, or that their few postcards can’t make a significant difference and the town proceeds to ignore the threat.

Hofstadter leaves his moral unspoken, although it is relatively easy to interpret the looming threat as the cold war spectre of disaster through nuclear war or climate crisis, and the indifference of the populace to a commentary on the unwillingness of people to sacrifice even a small amount to make a real difference through collective action.

An explosion of a different kind can be found in The K Factor by Harry Harrison (Analog, Dec 1960). Here the concern is over social unrest but the spread of discontent is described through analogy to a controlled nuclear reaction:

“This balance of neutron generation and absorption is the k-factor of the reactor. Ideally 1,0000000.

That’s the ideal, though, impossible to attain in a dynamic system like a reactor. All you need is a few more neutrons around, giving you a k-factor of 1.00000001 and you are headed for trouble. Each extra neutron produces two and your production rate soars geometrically towards bang. On the other hand, a k-factor of 0.999999999 is just as bad, Your reaction is spiraling down in the other direction. To control a pile you watch your k-factor and make constant adjustments,””

The central characters are practitioners of societics - the manipulation of societies as a whole. As they describe it:

“If a society has a positive k-factor, even a slight one that stays positive, then you are going to have a war. Our planetary operators have two jobs. First to gather and interpret data. Secondly to keep the k-factor negative.”

Unfortunately, on the planet Himmel a rogue operative is manipulating the k-factor in their own interest and causing lethal social unrest. The imbalance leaves the narrator operative, Neel Sidorak, with a desperate struggle to establish where the equations have gone wrong and apply his societics theories before the society in question explodes.

Taking the issue less seriously but also looking at exponential effects in the manipulation of populations is The Snowball Effect by Kathleen MacLean (Galaxy, Sept 1952). When his subject’s utility is challenged by his university’s senior management, a sociology professor describes a snowball effect - process that once started grows exponentially [5].

As he explains, by giving members of a group an incentive to recruit and a penalty for leaving, a snowballing exponential growth can be achieved. In demonstration, of its effects in even the least likely settings, he offers the local Watashaw Sewing Circle a new set of bye-laws designed to ensure its exponential growth and then eventual collapse when it stops recruiting. Over the course of the next few months this plays out:

I placed some red stars on my graph for the first three months. They made a nice curve, rising more steeply as it reached the fourth month. They had picked up their first increase in membership simply by amalgamating with all the other types of charity organizations in Watashaw. changing the club name with each fusion, but keeping the same constitution—the constitution with the bright promise of advantages as long as there were always new members being brought in.

After six months, the organisation, with its forceful female leader, has swamped the town and expanded beyond its limits:

“They’ve opened a branch office in New York,” I said carefully Into the phone, a few weeks later. With my pencil, very carefully, I extended the membership curve from where it was then. After the next doubling, the curve went almost straight up and off the page.

Allowing for a lag of contagion from one nation to another, depending on how much their citizens intermingled, I’d give the rest of the world about twelve years.

Snowball effects

The exponential growth (or diminution) curves in these stories all follow the same mathematical law - whatever event occurs leads to the event occurring with still bigger impact next time. This is known as a positive feedback loop, and is usually as harmful in scientific and technical applications as its reverse. Positive feedback is responsible for the howl of microphones placed next to speakers (as tiny noises are detected, amplified, rebroadcast and detected again) and for the crowd mentality that magnifies small grievances into riots. As Harrison explained, they make the difference between a nuclear meltdown and a cold reactor shutdown. They are also fundamental to the threat of overpopulation and resource depletion.

However many of the stories here, usually by construction, assume that the snowballing growth they consider will never be checked. In the fantasy examples I’ve cited, where physical laws are malleable, that might be true. However in science fiction it’s important to consider the physical problems with extending any function to infinity.

In the case of population growth through reproduction, the exponential growth is checked when the amount of food, houses or other resources becomes depleted. For organic reproduction, disease or predation are both encouraged by overcrowding and other stress factors, leading to lower survival rates amongst offspring, which may even drop below one per adult, and ultimately lead to a population crash.

Even self-replicating machines will be resource-limited. In Stargate: SG-1, when Replicators take over a starship, they can consume its decks and hull to some extent, but must limit their exponential growth and leave sufficient infrastructure intact to carry the newly-expanded population to a resource-rich solar system, or face being stranded in space.

In examples where division occurs, similar problems apply - each part of the original must either begin half the size of the original and will need resources to grow, or must consume those resources at the moment of division in order to create enough mass for a duplicate the same size as the original. The Dooplers in Star Trek: Lower Decks, for example, are utterly impossible as shown due to simple energetic and mass conservation arguments - their role in the episode is simply to show the challenge the crew faces in maintaining diplomatic relations with challenging species and despite mis-matched cultural expectations. Aliens which grow over time, as for example in the cases of the hlorg and the leach, are more energetically plausible but will themselves hit limits - the leach for example had destroyed a star system in the past, growing to enormous size, but then was diminished to the point of virtual destruction by the sheer emptiness of the void that needed to be crossed before another system could be reached. 

In fact, in the world as we know it, as well as in that of science fiction, most systems vulnerable to snowball effects are also self-regulating. As a snowball will stop growing when it reaches the bottom of a slope and come to rest, so other processes will also come to an end. An ecosystem liable to be overrun will see population booms and busts as either food is depleted or a predator takes advantage of the new food supply to engage in its own exponential growth. An organism dividing to fill an environment will either fill the available volume or, in the case of disease or parasites, kill their hosts, causing their own population to plummet. Even in constructed scenarios such as financial systems, self-correcting mechanisms come into play: compound interest can be negated by a high-inflation rate, by market booms and busts, by monetary devaluation or even by collapse of entire financial systems - as was seen in the bust of the tulip mania market in the Netherlands in the seventeenth century.

However an awareness of the potential for exponential growth, and ensuring that it can be negated is also part of the role of a scientist, statistician or engineer, as well as others in society. In MacLean’s The Snowball Effect, the sociologists created a society constitution deliberately designed to be unstoppable until recruitment ended - anticipating that it would die when reaching the limits of a small town. They did not anticipate the scope of a social construct to adapt and change in order to survive. Similarly, those who introduce alien species into ecospheres may not anticipate the ability of those species to enter an exponential population growth, or to adapt to and circumvent any attempt to prevent it - as many areas of Earth know to their costs.

When Chaos theory became well established with the growth of computer simulations in the 1970s, a new phrase came into common usage: the butterfly effect. The idea that the tiny ripple in the air caused by the beating of a butterfly’s wing, through a series of growth effects, might expand exponentially to cause a hurricane some months later shocked many. Where chaos theory differed from previous work on snowball effects was in pointing out that it was impossible to know which of the many butterfly wing-beats might have this effect, or precisely where and when the hurricane would occur. Non-linear growth is intrinsically unpredictable and sensitive to effects so small that they descend all the way into the fundamental quantum uncertainty that defines all matter. While a snowball effect, once established grows in a well-defined, predictable way, what might cause one is very different.

 Of course, most butterflies don’t cause hurricanes. Most populations or individual entities do not grow without checks. And most bank accounts will not take us to the Restaurant and the End of the Universe. The science fiction of snowball effects reminds us that some just might.