How long would it a self-replicating gelatinous cube to take over the entirety of planet Earth?

Question

Context:

Experiment 87171 is a gelatinous cube that takes up a space of exactly 27 cubic centimeters of volume (meaning that each dimension is exactly 3 cm in length).

It was created by the Helix corporation, an extremely corrupt business that exists in my world. The irony about them is that, when they make things that are supposed to be harmless, they end up being deadly. When they make things that should be deadly, they end up harmless. They wanted to make their own brand of gelatin. Instead, they made gelatin monsters.

Basic facts about the cube are the following:

1. Hydrophilia-Despite having no sentience of their own, the cubes naturally seek out any water source or source of moisture that is within a three-meter radius. Once they make contact with that source, they drain it until there is no more moisture left in that body or object.

Because of their love of water, they are also pyrophobic. Extreme sources of heat are their only known weakness and fire is the only thing that kills them.

2. Mitosis-Once they absorb approximately 30 milliliters worth of water, the cubes will then split in half and become two cubes with the exact same dimensions and abilities as the others. This process is not instantaneous. One doubling occurs every 30 minutes exactly.

3. Shape Changing-Since they are gelatinous, these beings are able to change their shape at will. They are able to squeeze through any opening provided it is 3mm in thickness or more. Anything less than that and they will be unable to get through the opening.

At the start of the story, the scientists start out with approximately 100 of these cubes in a scientific facility near the Atlantic Ocean. The cubes got loose and sought out the nearest and largest body of water that they could find. All 100 of them found the ocean and jumped right in.

I am assuming that this would be enough to cause an extinction-level event in the world at large. But, in order for the drama to be realistic, I need to know long it would take before things got to that point. I need a baseline for what will happen if the heroes do nothing.

Thus assume no one knows what to do, every effort to stop the cubes is fruitless, and these things just multiply indefinitely. The only thing people can do is run and hide.

Cubes can move approximately 1 meter every 3 seconds and keep that up for 1 hour. Then they stop for 3 hours and start moving again for an hour, so on in an endless cycle.

Question

With the following information about the cubes, how long would it take before their multiplications end up killing every last human on the planet, either directly or indirectly?

• This question is about human deaths. Animal deaths would not count in this scenario.

• This question is regarding the human death toll. Specifically when the human population would reach 0 due to these cubes.

My Current Ideas

1. Considering that there is doubling involved, I know that it would be an exponential growth problem, so if we assume constant growth it would be 100 cubes at hour 0, 200 30 minutes later. 400 a whole hour later and so on, but the math would get significantly more complicated after that.

2. Killing off the fish supply would definitely speed up the deaths of people, though I have no idea how much it would increase the rate.

Answer 1

This isn't actually an extinction level event - at least, assuming no one does anything stupid

I ran some calculations. It take 30ml of water to create one of these, which is only 27 cubic centimeters, meaning that these gelatinous cubes are more dense than water. 111% more dense, to be precise. To give you an idea of how dense this is, water at 1000 bar (or about 10km) is only 104% as dense as surface water. This means that the cubes will slowly sink the bottom of the ocean, duplicating along the way. The cubes will come from the bottom of the Atlantic Ocean, swallow it whole, and then the water level will have dropped. The Atlantic Ocean boasts an average depth of 11,962, which means the water level will drop by 1,196.2 ft.

Unfortunately, that's not enough to save the rest of the oceans. The gelatin will spread through the Indian Ocean to the Pacific Ocean for the most part, though waves will be able to skirt by going north and south of the Americas. All in all, all the oceans become gelatin.

A few back of envelopes calculations indicate that this will be done in a few days, given the limitations that the cubes will experience as previously discussed. Ocean is ~ 4.46x10^22 30ml units. 4.46x10^22 ~ 2^75.2 , meaning 75 doubling units, knock off 7, because we start at 100, so we're down to 68 doubling units. Each doubling unit is 30 minutes, so we have 34 hours, but it will be chokepointed at some aspects, as mentioned earlier, so a realistic estimate probably wouldn't exceed double that - around 2.8 days, but no earlier than 1.4.

However, all inland water will still be fine. So the world isn't doomed, unless some decides to expose that water to these cubes. There is still sufficient water, especially in the short term.

Of course, weather pattern shifts and the fight over the remaining water will probably cause apocalyptic level events. And, yes, the reduced ecosystem will probably trigger mass extinction events for a lot of species, certainly. But no humanity. When the dust settles, it's likely that a lot of humanity will die, but the remaining survivors should be able to comfortably live far inland and not have to worry about it, as long as they're smart about water usage and carefully harvest gel when they need more. (This actually seems like a good setting for a post-apocalyptic novel.) Of course, this is assuming someone doesn't do anything stupid like try to recklessly nuke the ocean back into water because that would, ah, be problematic. It's unlikely to work, and if it does, the heat requirement might boil the entire ocean and that would certainly be bad. Don't nuke the ocean.

On one final note, there's a Kurt Vonnegut novel which discusses a very similar topic - Cat's Cradle, if you are interested.

Answer 2

To predict the time-to-extinction for humans, we'll need to know what will kill them.

I think there are three primary possibilities:

• cubes kill humans by extracting water from their bodies

  This one will be hard to calculate without doing a serious simulation. Let's pray that something else kills them first!

• humans starve

  The last human will die ~1-3 weeks after the last food source is exhausted. (My guess is closer to 1 week, given the circumstances.) The pace of cube spread will determine whether food shortages come into play.

• humans die of dehydration

  The last human will die 3-10 days after the water is exhausted. Cubes compete directly with humans for water, and we need water very frequently, so my guess is that this is the most likely cause of death for most humans.

So, what will the pace look like?

You dropped the cubes right into the ocean, and I believe all Earth's oceans are connected.^{citation needed} The ocean floor is also topologically a big pit, so it seems like the first phase of cubageddon will be them multiplying without constraint until all the Earth's oceans (and connected waterways) are completely exhausted.

A cube needs 30ml to reproduce. Wikipedia says the combined volume of the Earth's ocean water is "approximately 1.335 billion cubic kilometers". Perhaps my dimensional analysis is wrong, but I calculate that is enough for 445 trillion cubes.

How many generations of cubes does it take to reach 445T?

You describe the growth as exponential, but I always thought that constant-doubling was geometric. You can find nice formulas (and even calculators) online for both curves, but I wanted to be sure I got the number right even if I had the wrong concept, so I wrote a little javascript to calculate the growth¹:

let count = 100
let generation = 0

do {
    count = count * 2
    generation = generation + 1
} while (count < 445000000000000000)

console.log(`count`, count)
console.log(`generation`, generation)

Spoiler: it's 52 generations (or maybe 53²). At 30 minutes per generation, 52 generations is just 26 hours.

That is so staggeringly rapid that I think the fudge factors won't make much difference. Neither would I expect cube movement speed to be a big factor, simply because the ocean will come to the cubes for most of phase 1.

Just to be safe, let's round up to 2 whole days before all ocean water is gone. That should also account for plenty of connected waterways and rivers that drain into the ocean.

Thus ends phase 1: the vast majority of all water is exhausted. On the bright side, there's still going to be plenty of freshwater and inland lakes, and Earth's food supply won't be much affected after just 2 days. (Yes, a massive crash is coming, but it may all be over before that hits, especially after the first wave of mass die-offs reduces the number of mouths that must be fed.)

Phase 2 is when most humans (and other animals) will die: from dehydration, and from violent conflict over Earth's remaining water and shelter from cubes.

My not-very-careful guess is that at least 95% of all humans will be dead from dehydration after no more than 12 days. So that's when phase 2 ends.

Phase 3 could last much longer. Small pockets of humans will be able to survive on stored food and water, particularly in places that are hard for cubes to reach. I think phase 3 will be dominated by 2 factors: the pace of the water cycle, and the speed of cube movement.

As we've seen, cubes multiply so rapidly that any neighborhood is ruined shortly after cubes arrive. So, the upper limit on every pocket of life being wrecked is: how long does it take for a cube to travel to the most remote place?

The most distant point from an ocean is the Eurasian Pole of Inaccessibility (or "EPIA") 46°17′N 86°40′E, in China's Xinjiang region near the border with Kazakhstan. Calculations have shown that this point, located in the Dzoosotoyn Elisen Desert, is 2,645 km (1,644 mi) from the nearest coastline.

As far as cube travel speed:

1 meter every 3 seconds and keep that up for 1 hour. Then they stop for 3 hours

I calculate average cube speed to be 300 meters per hour. At that rate, it takes a cube ~367 days to each EPIA.

Given that, I think the water cycle will give out first, recycling almost all free water into rain (i.e. often fed directly to the cubes). If I understand reservoir residence times correctly, most free water will evaporate and rain back down within ~6 months.

Some humans will probably still be alive, having managed to hide in places the cubes can't access. And 6 months is not unrealistic for food supplies stored in case of emergency.

Phase 3 ends once cubes have physically reached practically every place on Earth. The only humans left alive will be those who happen to be in safe shelters, i.e. impregnable bunkers.

Phase 4 is the final phase of human life. It lasts as long as humans can live in completely self-contained bunkers with closed water cycles.

Casual searching suggests that serious (non-government) bunkers could keep a small group of people alive for about 5 years. If the group is smaller, supplies will last longer but the people will go insane sooner. I'd expect a lot of suicides.

My guess is 6 or 7 years before the very last human dies. But I've honestly never been that interested in the problem space of bunker longevity, so this is just a guess.

---

^{1 One way to write a bug-free program is to make it so simple it's obvious there are no bugs.}

^{2 There are two hard problems in computer science: cache-invalidation, naming things, and off-by-one errors.}

Answer 3

For simplicity, let's just worry about the oceans for starters, the elimination of those would be catastrophic.

The volume of water in all the oceans together is approximately 1.335 billion cubic kilometers.*

Your cubes double every 30 minutes and we start out with 100 of them at 27 cubic centimeters. So to consume the entirety of Earths oceans, it would take 1.335*10^12 cc = 27cc * 2^(x/30) minutes. Solving for x we get roughly 1,066 minutes, or a little under 18 hours.

I don't have to worry too much about the cubes' movement rate as the ocean water will flow towards/into them as they absorb the water. Of course, some parts of the ocean are deeper than others, so they might have to move in order to find their way to those deeper areas, but for the most part, the oceans are going to be pretty empty in about 18 hours.

This would cause a massive disruption in our climate quite quickly. Rainfall would drop, temperatures in many parts of the world would change and become much more variable rapidly. If the water remained locked up like that, and the cubes continued to absorb what little rainwater fell, we'd probably have no more modern agriculture and billions of people would starve as soon as our food reserves started to run low.

This of course assumes no one does anything about it. A nation might try to nuke this rapidly-growing, planet-killing pile of cubes in the first few hours...

• https://en.wikipedia.org/wiki/Ocean