a really wide underground arcology to replace San Francisco

[Kragen Sitaker]

Suppose San Francisco's 750k people had 2000 square feet each, at 15
feet tall.  They'd fit in a cube 2823 feet on a side --- only half a
mile. (2823.1 = ((750000 * 2000) * 15) ^ (1/3)).  But that's almost
200 stories tall, the height of the World Trade Center.

Paolo Soleri's "arcologies" were planned along these lines.  They have
several benefits: everyplace is within easy walking distance, they
don't require any energy to heat, and they leave lots of land
unoccupied for natural uses.

Suppose you instead put these 22.5 billion square feet into a flattish
hexagon in which the objective is for no place to be more than about a
quarter mile from a subway station.

You can run the subway around a circle half a mile from the center,
dipping into the center between two stops, making a Pac-Man shape,
with six stops around the periphery and one stop in the center.  Now
you have a circle a mile and a half across, 3960 feet in radius, or 49
million square feet in area, which need only be 457 feet tall (30
stories). (456.71 = ((750000 * 2000) * 15) / (((5280 * (3 / 4)) *
(5280 * (3 / 4))) * 3.14159)) to hold everyone.  The distance from one
station to the next is half a mile, 805 meters; at 1 G, 9.8m/s/s, is
12.8 seconds (12.81 = (((805 / 2) / 9.8) ^ (1/2)) * 2) at a peak speed
of 40 MPH (42.76363636363637 = ((9.8 * 6.4) * 3600) / 5280).  Such a
subway would not have to consume any energy except in frictional
losses.

(1G of acceleration is unusually large by conventional transport
purposes; it's zero to 60 MPH in 2.73 seconds.  Accelerating a
one-half-ton small car at 1G at 40MPH requires 106 horsepower or 76kW,
given the GNU units.dat definition: 550 pound force foot / sec =
745.69987 kg m^2 / s^3.  If subway cars were mounted so that they
could swing freely from a pivot mounted above, and accelerated
smoothly so that they did not oscillate, this would probably be merely
uncomfortable rather than dangerous.)

Since the furthest you'd have to go on such a unidirectional subway
would be 6 stops, your longest trip would be 77 seconds of travel
time, plus however long the subway stopped at each station.  Since it
takes about 20 minutes to walk half a mile, it seems that perhaps a
subway that took up to ten minutes rather than two or three might be
acceptable.  That would allow you to spread out your city over a
larger area while remaining dense in three dimensions.

Arcologies suffer from a few potential problems:
- They probably require some form of centralized control to keep the
  heating and cooling systems working; if these fail, you're really in
  trouble.  But the Netherlands have dealt with a similar problem in
  the form of dikes for centuries.
- They would probably be somewhat dark, due to the unavoidable
  distribution of sunlight over dozens or hundreds of stories.
  High-albedo walls in places where people spend time could help with
  this.
- People may find them disturbing.
- Unavoidably, from the inside, it takes a long time to reach the
  outside.

Suppose we use a more conventional star or tree topology for our
subway, and we allow it to extend out three stops, or a mile and a
half, from the city center.  Now we need 18 stops around the hexagonal
periphery (each side of the hexagon is a mile and a half long, and
therefore needs three stops), but our travel time is the same as for
the Pac-Man topology, although we need 72 half-miles of track instead
of 7, and they must be bidirectional.  Our city is now a hexagon 3.5
miles across, with rounded corners; if we disregard the rounding of
the corners, it covers a surface area of 8 square miles.  (7.9566 =
(((3.5 / 2) ^ 2) * (((3 ^ (1/2)) / 2) / 2)) * 6).  This is only one
sixth the area of San Francisco, and it would only have to be 100
feet, or seven stories, tall to give people the 2000 square feet each
mentioned earlier.

(The height of an equilateral triangle of side 1 is sqrt(3)/2 = 0.866,
so its area is 0.433 of a square.  A hexagon contains 6 of them.)

(100 feet tall: 101.4 = ((750000 * 2000) * 15) / (((((3.5 / 2) ^ 2) *
(((3 ^ (1/2)) / 2) / 2)) * 6) * (5280 ^ 2)))

Now, this entire complex could be built underground, leaving the
surface undisturbed as a park, universally only six flights of stairs
away.  Perhaps parking lots could be built on the periphery.

Because it would only be 100 feet thick, it could be divided into
essentially independent climate-control districts, which would
normally fail independently.  It would be deep enough underground to
be immune to aboveground heat waves, in any case, with the primary
danger being the buildup of heat from energy dissipated within the
city.

It would require substantial excavation costs --- the same 22.5
billion cubic feet of dirt and rock, plus extra around the edges,
would have to be lifted out of the ground and transported elsewhere.
That's about 640 million cubic meters, and each cubic meter is around
1.5-2.5 tons, depending on the density of the rock.  Lifting 1.2
billion tons an average of 60 feet (18 m) against 1 G would require
6.4 x 10^14 joules, or 1.8 x 10^8 kWh --- about 18 million dollars'
worth of energy at current grid rates, about $25 per person.

I have the impression that building underground is actually much more
expensive than this, which suggests that excavation costs at present
are not primarily driven by the energy cost of lifting out rubble.