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This scenario is based on three specific megastructures that are considered but never realized: Japan's Shimizu Mega-City Pyramid, 2004 meters tall and covering an area of eight square kilometers, planned to house one million people; Japan's X-Seed 4000, four kilometers tall and six kilometers wide and supposed to house 500,000 to one million people; and San Francisco's Ultima Tower, 1,828.8 meters tall, 140 square kilometers in area and having a capacity of one million people.

For this question, materials are not the focus here, as in an alternate history where the Industrial Revolution began in medieval China, they might have found time and materials to get around that problem. The real focus here is capacity because the postwar cities in this alternate Earth consist of only three different dimensions:

  • Pyramids 2004 meters tall and covering an area of eight square miles
  • Cylinders four miles tall and six miles wide
  • Cubes two miles tall and covering an area of 140 square miles

By 1950, there were 12.5 billion people, 90% of which lived urban. Eurasia, Africa, South America and Australia's populations could be found primarily in either of those three shapes listed above. (North America, meanwhile, is stuck in its steampunk past because it'd be more expensive to demolish and rebuild a building than building a new one from scratch.)

Now the question is--what would be the maximum capacity of humans in each of the three shapes listed above?

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    $\begingroup$ John I feel like this is some simple math. You need to define how much living space is afforded to each individual, once you do that you simply divide the volume of the structure by the space each person gets, taking into account probably 20% of the space for structural components, utility components, movement spaces (i.e. hallways) and common spaces. $\endgroup$ – James Feb 15 '17 at 18:43
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    $\begingroup$ @james And "maximum capacity" could be quite cozy since tiny houses are a thing and a small family of 3 people can live pretty comfortably as such. $\endgroup$ – rangerike1363 Feb 15 '17 at 18:47
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    $\begingroup$ @James: Bear in mind that just because someone has sufficient volume, does not mean they have sufficient living space. Living space has to be a certain shape as well as a certain volume. For example, someone cannot live in a cylinder that is so thin they can't fit in it, even though its height may make up for its ground area. $\endgroup$ – Piomicron Feb 15 '17 at 20:00
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    $\begingroup$ Also, how did this not get banned, even though it's clearly a mathematical question, despite the way it's framed, and not a worldbuilding question? I had so many of my worldbuilding questions closed despite actually being world building questions. $\endgroup$ – Piomicron Feb 15 '17 at 20:01
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    $\begingroup$ Big question is whether those superstructures need to provide only living quarters or also places to work, or maybe even food production facilities (i.e. farms). If living quarters only, who is going to take care of all of those people? $\endgroup$ – Alexander Feb 15 '17 at 20:21
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First lets look at their volumes:

  • Pyramids 2004 meters tall and covering an area of eight square miles (don't know if you meant to mix units but I'm going to assume you did).
    • $\frac{B*h}{3}=\frac{8*1.245}{3}=3.32 miles^{3}$
  • Cylinders four miles tall and six miles wide
    • $2 \pi r^{2} h = 2\pi 9 * 4 = 226.2miles^{3} $
  • Cubes two miles tall and covering an area of 140 square miles
    • $B*h = 2*140 = 280miles^{3}$

So your pyramid is the clear loser there (the floors have to get smaller as they go up too).

I'm not sure how much space your handwavium power generators, your shops and your places of work will take up. As far as people goes this concludes that 5.6-16.7 square meters is needed per person. Assuming they have rooms 2 meters tall, taking the average floor space of about 11 square meters so the volume per person is $V_{p}=22m^{3}$ in miles we have $V_{p}=5.3\times 10^{-9} miles^{3}$.

  • Pyramids: $\frac{3.32}{5.3\times 10^{-9}} = 626$ million people
  • Cylinder: $\frac{226.2}{5.3\times 10^{-9}} = 4.268$ billion people
  • Cube: $\frac{280}{5.3\times 10^{-9}} = 5.283$ billion people

Of course this can all be adapted for if you want to add in areas for other things, this assumes the whole building is living spaces. If you calculate the volume of space needed for that you just need to take it off the total volume and divide the remaining space by the volume you're giving one person (depending on how comfortable you want them to be).

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  • $\begingroup$ Regarding the final paragraph, here's an idea: youtube.com/watch?v=zS_OiPO0LpM $\endgroup$ – JohnWDailey Feb 15 '17 at 20:24
  • $\begingroup$ Using indoor hydroponic farming I suspect you could keep north of a billion people in the cube city while also growing enough food for them and leaving room for corridors and public spaces. Assuming you had the energy. $\endgroup$ – Murphy Feb 17 '17 at 16:35
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Make an series of assumptions here. The current population density of Tokyo is around 6158 people per square kilometer. That's about as dense as you might want to get for the sake of sanity. Yes, there are more dense areas, but this allows space for public areas, manufacturing, etc. Given that your structure is made from handwavium, you only need to figure how high to make each level. The rest is just calculating the volume of the structure. To focus on your San Francisco tower, because the math is easy, you could guess a ceiling height at 3m. That gives you roughly 600 floors. given 6158 per sq kilometer, you get 862,120 people per level. That amounts to 517,272,000 total people living in the tower.

That's how I would calculate the population density of the structures, but with half a billion people in a San Francisco tower, you might want to dedicate some thinking to geological stability.

You mentioned in the question a population in the tower of 1,000,000. that puts your density around that of Salt lake City, Utah. Pretty roomy.

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    $\begingroup$ Tokyo's people per square kilometer assumes multi-story buildings, but your calculation later assumes that the city is built as single-story buildings. I don't have a solution for you, but assuming the population density of a 20 story building is representative of a single story building is going to greatly skew your calculations. $\endgroup$ – Ranger Feb 15 '17 at 19:06
  • $\begingroup$ There are places in Tokyo that are not dedicated to housing. As a result it should (sort of) average out. There are also places that are much more densely populated, so I chose the Tokyo figure as representative of a superstructure. Manilla has a population density of 41,000 per sq km, so I figure it could work out. $\endgroup$ – Paul TIKI Feb 15 '17 at 20:05
  • $\begingroup$ This was bugging me last night, so I had to take the math even further. My family of 5 lives in a typical north american suburban house. That house is a little over 1200 square feet. that works out roughly to 110 square meters. you could fit 9090 of my house in 1 square kilometer. If the population of Tokyo works out to 6000 (round numbers) per square km, then each individual would have 166 sq meters or 1760 or so sq feet. That's a good sized house per person and it's all on the flat. , no multistory needed $\endgroup$ – Paul TIKI Feb 16 '17 at 15:43
  • $\begingroup$ @PaulTIKI your house has an environment, namely roads, then there's shops, doctors etc in the vicinity. This factors in the tokio numbers, but not in your house. $\endgroup$ – Burki Feb 17 '17 at 15:54
  • $\begingroup$ My house also fits, comfortably, 5 people. The thing is that you are in an Arcology of sorts. That means that the support space per individual is much smaller. Not everything has to come from inside the structure. Also, the need for wide roads drops significantly because the structure itself is only about 11 km on a side. To go from 1 corner to the opposite would take a couple of hours on foot at most. You could efficiently use half of the dedicated square meters for the populations living quarters, still be fairly comfy, and have room for shops and the like even at 6000/ sqKm. $\endgroup$ – Paul TIKI Feb 17 '17 at 16:23
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To calculate how many people fit in your volume, you need to consider the following: How much personal space does the average person need? How much space must be reserved for infrastructure of all kind? A city requires not only housing, but also:
- roads or similar to get your population to and from the place they live
- a workplace for good part of your population
- shops
- hospitals
- schools
- roads and / or similar infrastructure to transport all the goods the people need and those the people make (if any, but there are bound to be some)

As you can see, practically everything that contributes is variable. That, among other things, is a reason why architects study for years.
If you come up with really clever ways of moving things around, you can cut back on your traffic infrastructure needs. If you design really smart apartments, you can have the inhabitants feeling comfortable with fewer square meters per person. So the best you could provide would be some ball park numbers.
My best guess would be: 20 square meters per person in personal space, plus triple that in infrastructure, times 3 meters height, you end up with 240 cubic meters per person, on average.

Depending on how you balance things, and how you imagine life to be for the individual and for society as a whole, you could end up anywhere between half that (think: galley slave) to any upper value you want: having empty space is always an option.

You can have synergies in a compact structure when it comes to moving people and goods around. But then you would need extra space to move air. A village has a whole planet to move air around enough to let everybody breathe, but a compact 3-dimensional city will not, so that needs help. And the denser you pack your inhabitants, the larger the infrastructure for air supply gets.

I assume the message is getting clear: You need to at least guesstimate on what life will be like, and what life needs to be supported, and how to cater for these needs. Only then can you answer how many people you could pack into your structures.

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