How large would a mega structure have to be to host 1 billion people indefinitely?

Question

Ruins of buildings, abandoned cities and suburbs dot the coasts of Earth's landmasses, inhabited by wild animals and the pieces of a once vibrant civilization. The oceans are relatively empty, the soil and dirt sapped of all its nutrients, the vegetation on earth fighting the strong winds clinging to the loose ground. Barren. Humanity seems all but gone, the planet destroyed by ignorance of its most feral inhabitants.

The inhabitants did not leave though; they retreated into their shelters to wait out the storm. The shock and fear that came with the last harvest prompted the citizens of the world for answers. Out of the chaos came Arthur Hawking, a architecture and food scientist from ICBC/Coke/Pepsi. He brought the idea of a self sustaining structure to the masses, the possibility of having everything a human needed in one building. Agriculture would occupy the majority of the structures space, but automated farming would allow a consistent flow of product to the population.

Yuma Arizona was chosen to host the structures, a small town in the south west corner of what was Arizona. This city experience the most sunlight in a year, ideal for the arrays of solar panels that would be used to supplement the nuclear power generators build into the structures. The buildings were originally going to be just for some united states citizens who could afford it, but mass hysteria after "the last harvest" made these structures seem like the only resort for all people in the world so the interest skyrocketed and the world flocked to Yuma, 10 billion people in 2081.

How large would one of these towers have to be to host 1 billion people? My story takes place in 2081 (just read this book), and I am looking to accurately scale the buildings in my head and story. Ideally 10 of these would fit into Yuma, so they would scale vertically more then horizontally. They need to account for agriculture space, living space, and some general use space that would likely be equal to the living space (schools, research facilities, clothing fabricators, restaurants etc.). The plan is to wait out the storm, which likely would be about 1000 years (for a healthy topsoil and native animal population). Once everyone is in the structures, each one will be sealed from outside physical contact to allow nature to develop undisturbed by humans, and to restrict any potential crop blight to a single structure.

EDIT: These structures are too large to fit exclusively in Yuma, so for the sake of future answers I will allow the structures to exist in the entire Yuma County.

EDIT 2: Thanks for the help, along the way I was shown this book/site and now feel like I have to recommend it. If this interested you check this out.

Answer 1

Oh, the pain...

Bedrock is your first problem. You're not building an itsy-bitsy building like the Burj Khalifa or the Tower of Pisa, your'e building the building, the biggest, honkingest, Oorah-est building on the planet. And you're guaranteed to crack the foundation if we don't go all the way down to bedrock, grind the bedrock flat, drill in a bazillion holes for metal rods, and lay the thickest foundation the world has ever seen ... because you want to build up, not out (in a area that's just begging for more urban sprawl), which is probably a horrible idea, but let's roll with it.

So, let's examine a geological map of Yuma County, AZ. We'd need someone like Arkenstein XII to give us the low-down, but I'm betting the word "Granite" is good (My thanks to Ash for pointing this out), which is a good thing, because there's more of it than basalt. Unfortunately, what you'll quickly discover looking at that map is that while you'll have good sun coverage for your solar panels, it's pretty much the worst possible place on the planet for a foundation the size of what you're looking for. The situation isn't as bad as I feared, but it's still inconvenient. But, there's spots out there we can make this work because we're about to calculate our footprint size.

If I remember my geology classes in high school correctly, the areas of the world that would be best for a honking huge foundation are pretty much the worst for solar panels. I could be wrong, though. High school was sometime during the Cretaceous Period. But it's worth thinking about.

Water is your second problem. You want to be self-sufficient for a billion people. This is actually a really ugly calculation. People need to drink, dispose of waste, experience hygiene (in a sealed system w/1B people... oh, yeah), then there's crops, industry, humidity control, yada-yada-yada. I'm going to take a complete pull-it-out-of-the-air guess and suggest calculated per-person you need something like 100 gallons of water per day. Let's pretend we have 100% efficient recycling and that the recycling is capable of processing every drop of water each day (an oversimplification that literally will make angels weep). That's 100 billion gallons of water or a storage tank 13.4 billion cubic feet in size. If it's just half the height of the aforementioned Burj Khalifa (1,358 feet) it covers a little over a third of a square mile. Let's call it a third for convenience and just build a little higher.

So, a third of a square mile, doesn't sound too bad, and we should be able to find a nice, big chunk of basalt even near Yuma we can work with. Let's pour that sucker a solid 30 feet deep, fully reinforced and tensioned, of course, and another 2,000 feet (easy) of honeycombed water tank so we can build a support structure on top of it. It's the honeycombs needed to support the rest of the building that causes the height (which is a round guess) because they consume volume not previously accounted for.

And it's worth noting that if you didn't have a crisis driving this process before, the diversion of water to fill that tank will cause it. OK, maybe not really, but the resulting fight between California and Arizona over water rights would likely start a second civil war in the U.S.

Food is your next problem. This fascinating article boils the issue down to a simple number: one acre per person if you do it the "natural" way (growing crops, grazing cows, etc.). That's (*cough*) one billion acres under your roof. To give you a sense of size, The United States is 2.3 billion acres. To make things interesting, just over 1 billion of those acres are devoted to crops and grazing. Considering the potential for U.S. food production, that casts a shadow on that previous article's conclusion that 1-acre-per-person was needed, but we'll roll with it. So, you need to build a building (now it has to be vertical) that houses the entire food-production land usage of the United States. That third-of-a-square-mile is looking a bit small, as is my 100-gallon-per-person estimate.

And something must be wrong with my sources at this point. But not too wrong. People have been complaining about too-many-people vs. too-little-arable-land for a long time now. This might simply be drawing a huge underline below the problem. In other words, unless you can compress food production something awful, your premise isn't believable as you need a planet to feed a planet worth of people.

Let's assume that you've improved food production technologically such that, somehow, you need only 1/1,000^th the space to feed a person. Thus, one billion becomes one million ... acres (sigh). A third of a square mile is about 211 acres, so you need 4,740 floors for food production at about ten feet per floor (which is incredibly unrealistic) so now your building is working on 50,000 feet of altitude (which is well above the 35,000 foot average cruising altitude of a Boeing 737) and we haven't even begun to address industry and infrastructure — much less housing.

I did say building up, not out, was a horrible idea....

And it's time to stop

Yes, you can lower the height of the building by spreading out the foundation, but that causes problems with a believable foundation (remember that geological map of Yuma County). Obviously we can't conceive of a believable way to do this today. So I'm going to go out on a limb and suggest...

Your building's foundation is the size of Connecticut and the building is approximately 20,000 feet (6,100 meters) tall.

And we won't talk about what the weather around this sucker would be like.

Answer 2

Putting a billion people in Yuma, AZ, including food production, industry, and commerce, would require building the entire area to a height of about 2km.

The math:

Based on the answers on this question, we should be able to feed people using about 25$m^2$ of space, per person, using reasonable near-future assumptions about aeroponic food production. Based on this discussion of living space in modern Japan, the minimum living space allocated per-person is about 25$m^2$ of additional space. For modern cities, about 50% of city space is residential, with the rest being industrial/commercial. (Proportionally, if we include agricultural allocation in residential areas, this will increase to about 66% of our space.) Additionally, modern cities are around 30% roads. For a megastructure, we'll also need a lot of space to allocate for atmospheric control. Let's assume that's about the same amount of space as roads.

As a per person estimate, we'd then need $50 \times 2\times1.6=160m^2$of space to account for transportation, food, housing, industry commerce, and atmospheric infrastructure. A billion people would then need a total of 160 billion square meters of space.

Using the Burj Khalifa as an estimate for floor height, each floor will be about 4 meters. Yuma AZ covers around 300 square kilometers of space. Assuming we stay roughly within its boundaries, we'll need $\frac{160\text{ billion }m^2}{300\text{ million }m^2/\text{floor}}$, for a total of around 540 floors, at a total of 2160 m tall buildings.

This, notably, does not include the area you'll need for solar panels, if that's what you're using for power. Solar panels, notably, cannot be stacked. Sunlight provides around 1kW/square meter of energy. Assuming our crops need about the same energy input as the sun provides, and that crop production is about half of our total energy needs, we'll need 50kW/person, for a total of around 50 billion square meters, or 50,000 square kilometers of area for solar panels. That's assuming maximum efficiency: modern solar panels are something like 10%-20% efficient, so you'd need 250-500,000 square km of solar panels. (Cuba, for reference, covers around 100,000 square km.)

If everyone is to fit in Yuma County, rather than the current city limits of Yuma, the available area increases by about a factor of 50. Under that assumption, you'd cover the entire area to a height of about 50 meters to house a billion people, or 500 meters to house ten billion. 500 meters is about the height of the inhabited portion of the Burj Khalifa, though unlike that building, the megastructure would cover an area of 14000 square kilometers or so. Solar power will require the same area, which will likely be infeasible without either extensive automation to do maintenance or a substantially more dispersed population.

Addendum: I misread the question, and though you were only putting a billion people into Yuma. For 10 billion, these numbers would increase by a factor of 10. Using current or near-future technology, I'm not sure a 20km building is anywhere close to feasible. Also, your solar panels will cover an area close to the size of the EU, so you'll probably want to spread your towers out through an area about that size, anyways.

Answer 3

Very. Very. Very large.

Just for some comparison, have a look at a project of this type being proposed for Tokyo bay. Remember, you're not just building a giant apartment block. You need to account for places people work, create food, eat food, most of the stuff you'd have in a regular city.

The structure would house 1,000,000 people. The structure would be 730 meters (2395 feet) high, including five stacked trusses, each with similar dimensions to that of the Great Pyramid of Giza.

So you'd need something a THOUSAND times larger than this.

Let's try some math here... The greatest density of human occupation right now on a large scale is Manila. 1.6 million people occupying 38.5 square km. It tops out in District 6 with around 70k people per square km. (This is almost three times the population density of Manhattan, btw...).

Now, trying to turn SQUARE meterage into CUBIC meterage is a tricky problem, but I've played with before. I wound up with a rough approximation of ~2.7 cubic km for Manhattan (~1.6MM population) and ~7.75 cubic km (8.6MM population) for all of NYC.

So your starting point is around 10MM people in a cubic building that's about 2km on a side. Manila triples that population density without TOO much trouble, so let's say we can go up by a factor of 5. Now we're up to 50MM in a 2x2x2 cube. Now we get rid of all the roads, all of our transportation of people and goods uses conveyer belts and elevators a la a la Caves of Steel (Asimov, 1953). Let's say that gives us another factor of 2 in terms of space efficiency.

That's 100MM people in a 2km cube. You need at least ten of these.

Answer 4

I can't tell you how big the whole structure is going to need to be but I can give you a couple of figures that will help you get your head around the scale:

• Farmland, you'll need the equivalent of 25000 Hectares (roughly 62500 Acres) of farmland to feed that kind of population, hybrid Aeroponic/Aquaponic systems may reduce the overall volume this requires but that's the land equivalent. Certain fungi or a suggested technology that could assemble basic carbohydrates and proteins from atmospherically sourced oxygen, hydrogen and nitrogen (not sure how viable that is by the way) will get this number down but it's still going to be huge.

• Accommodation, based on tiny houses you'll need at least 50m³ per person to carry out the basics of life; eating, sleeping, and washing. You can probably half that, or better, using communal facilities and assigned shifts but quality of life will suffer badly.

That doesn't cover anything but the absolute bare essentials either, so no luxury foods, i.e. anything that grows on a tree or needs terrestrial living space (so very little fruit and nuts, no red meat, dairy, poultry, or eggs), and no physical recreation space, shared or otherwise. You also need to factor in "migration space", the corridors, elevators, stairwells etc... that people need in order to move around the space you create, and "utility space", the ducting, wiring, and piping to carry the necessaries of air, water, and power. Plus you need to generate the power and source and clean the water and air. Power for farming will need to be sufficient to supply at least 5Wm^-1 of tuned red-infrared lighting for maximum photosynthetic efficiency. Human inhabited spaces will need full spectrum lighting to keep people healthy.

Technical note: a Megastructure is usually at least 1000km in at least one dimension so the current footprint of Yuma is too small to put even one into.

Answer 5

Well, you're powering this by solar. So you need enough space for a billion people's worth of solar. Elon Musk calculated this for the population of the United States, which is roughly a third of what you have. From here, we have

If you wanted to power the entire U.S. with solar panels, it would take a fairly small corner of Nevada or Texas or Utah; you only need about 100 miles by 100 miles of solar panels to power the entire United States. The batteries you need to store the energy, to make sure you have 24/7 power, is 1 mile by 1 mile. One square-mile. That’s it.

So figure on thirty thousand or so square miles. That's a square about 180 miles on a side. Or 300 km. You want to put the people inside though. That's going to crank up the power requirements. You need artificial light for crops and air circulation. So let's double that to 360 miles or 600 km on a side. That's how big your solar arrays are. Assuming we put those on the roof of your megastructure, it needs to be at least that big. That's 360,000 square kilometers or 130,000 square miles.

That actually should be enough. You could make the whole thing one story and it would work. You could add a second story and increase your space by quite a bit beyond the bare minimum. With proper construction, they could do that after moving into the space. The important part being to push the foundation down farther than they'll have to dig. They might also put the batteries in that basement area as well, but those will only take up a fraction of the space.

If you're powering things by solar and want isolation from the planetary environment, you might consider moving off world. You could build a space station that could get solar power constantly with no breaks for night or weather. Then you don't need batteries. It would also be easier to build a cylindrical station in pieces so as to expand it after the beginning. Or just build new stations.

If off-world, you could terraform Mars, Venus, or even the Earth. That's much more difficult if you're stuck inside one big building on Earth.

Answer 6

Forget about volume - you need surface area

The answer by @Brythan answer goes into the right direction: identifying solar power, i.e. the sun-exposed surface of your building as the limiting factor.

However, the VAST majority of energy would be used to grow food - it's still the only sustainable way, and regardless of whether you do 'classic' agriculture or hydroponics or anything: you'll need the suns energy to somehow make your food plants grow, otherwise your population will starve.

I.e. you need enough surface to get the wattage by the sun to grow plants for 1B people. The order-of-magnitude calculation for this is 1 acre per person, so we're talking 1 Billion acres of needed surface area (4'046'856.422 km²). Now, this might be optimised on to some degree - but even if you get this down to 1% with the most efficient agricultural tech imaginable your building - i.e. some 40'000 km² - you'd still need to cover more than 10% of the entire area of Germany.

It doesn't matter if you build your building 1 or 10 or 100km high - if you don't get the surface area exposed to the sun you will not be able to produce enough food.

Answer 7

This is going to sound odd but here goes anyway. According to biblical prophecy there is a city that will descend to earth that is described as being 1500 miles square by 1500 miles high with 12 gates, 3 on each ordinate direction.

There are only two possible structural shapes that this could take: cuboid or pyramidoid. The trouble with cuboid shape at that scale is that it does not meet the "square" requirement due to curvature of the earth meaning that the only possible shape for an hyperstructure would be triangular in section such as conical or a polygonal based pyramid. The Kingdom from Heaven shape would be like the Giza pyramid.

This structural fact means that the ancients knew of the forces related to building on unstable soil /rock structure yet mimics a prophecy yet to come true.

By my calculation, that City from God would house up to 4 trillion people in their own square mile quarters of self sustaining agronomy with a river welling up from the centre of the base flowing eastwards out the city gate.

To get an idea of the scale, look at a map and describe a square perimeter from Morocco to the Shetland islands going east to the Black Sea then south to Ethiopia and back west to Morocco. 1500 Square miles. The Eastern central gate is where Jerusalem is today. Imagine for a second that walks of a city must ascend vertically for a cuboid on a sphere and very soon you recognise that 1500 above the earth the perimeter is no longer 1500 miles square. The structure would be sectional of a larger sphere and the sheer mass that the earth's cost would have to withstand would be dwarfed against such a structure.

Were it a pyramid shape, the forces would be better distributed across the earth's surface and capable of resisting such weight because the weight would be less than a third of the cuboid weight over the same area.

In your city, such a structure would be a good idea because the vast living area would be near the peak and agricultural area on the base below water storage facilities.

With a pyramid structure there are sufficient surface areas to capture rain and generate power by several means (hydroelectric, hydrostatic pressure, thermal differential, hydrothermal, solar hydroinsolation, wind and hydrogen dioxide reactors. The solar panel idea of sustainable energy is not sustainable as the materials are sourced from great distances to their final destination.

To understand how such large cities need to work we need to understand biotropic design and sustainable energy without accessible fossil fuels.

The shape of the building determines the surface area for solar power. To his 1 trillion people requires a quadrilateral pyramid 1500 miles square and 1500 miles high. Since the scale set by op is 10 bn, it is order of magnitude smaller and therefore 1 thousandth smaller meaning 15 miles square by 15 miles high. Since atmospheric oxygen is a limiting factor, the highest it could be built is 5000ft which means the remaining volume expands horizontally and requires a different shape based upon a triangular polygon.

An hexagonal based ring with a triangular section would reach 5000ft safely but the outward sprawl would be substantially larger and consume some natural features in its construction such as mountains and valleys. Such an undertaking would deplete vast water resources in the construction so technically, impossible to build in the given circumstances described

Hopefully this enlightens you on the structural issues limited by oxygen, materials, energy and water.