If the Russians want in, the lake idea might still work:
I still think doing it underwater is easier and cheaper than in orbit, even if you have to hire a separate scuba diver to patrol every 1000 square meters of the surface, picking off barnacles by hand.
I don’t see how this could possibly work, for the reason Chronos suggests. The very violent transition from orbit to atmosphere looks as if it must be catastrophic, most especially for a large and flimsy structure.
It would be like trying to de-orbit a blimp (only much worse due to the much larger size).
Band name!
So, geodesic domes can be made stable - what fraction of a geodesic do you need before it can essentially support itself? Could you, for instance, take a smallish panel, connect all the end with cable, and have a relatively sturdy piece, that could be airlifted?
Bore a mile-wide, mile-deep hole into the Antarctic bedrock. Start construction from the bottom up, gradually embedding the completed portions in a solid matrix of some low-melting-point material like Wood’s Metal. When construction is complete, focus your array of orbital lasers on the metal block to melt it and pump out of the interior of the sphere. Use the residual heat to terraform Antarctica.
The problem with constructing it underwater is the scale of the thing; even at half a mile deep (the minimum size for such a structure) the depth is beyond the depth of commercial offshore diving using atmospheric diving suits. You could assemble the structure at near surface depth and then let it descend at neutral buoyancy as you add layers, but your ability to inspect and repair the structure (which will inevitably be required) is extremely limited at depths of >600 feet, and essentially impossible once you get down to 2000 ft or below (the maximum depth for sustained active personnel diving , James Cameron movies notwithstanding) you just can’t effectively inspect or maintain the structure, which would be disastrous for a construction project of this size. Things will also corrode in fresh water, albeit not quite as aggressively as in sea water.
That’s because you are thinking of a descent like that performed by a space capsule or the Shuttle, where it is at orbital speed which is many times faster than the rotation speed of the Earth. The heating and pressure from re-entry is actually the craft’s way of shedding excess momentum. Certainly a large structure like a “Cloud Nine” could not survive such an aggressive return. However, if you build it at a might higher orbit, say near geostationary orbit (where it is orbiting at the same rate as the ground beneath it) and then slowly deorbit it straight down in a powered involute you can cause it to re-enter with effectively no differential tangential speed and a modest amount of radial speed. This would take a large amount of energy and propellant, of course, but since we’re assuming space-borne construction anyway, there’s no problem filling it with some kind of low molecular weight gas and placing ion thrusters all over the structure to keep the re-entry speed reasonable. Once it hits the upper atmosphere the building pressure will apply buoyancy forces to slow it down without excessive heating; the basic problem is really keeping the force differential across the structure even enough that it doesn’t get overstressed in a small area. Heating is a non-issue because it just isn’t going to be moving very fast relative to the atmosphere.
The other option is to use some kind of unobtainium material with a very high tensile strength and lower it down from an asteroid in geostationary orbit. This way the speed can be controlled very easily, and you gently lower it down into the atmosphere at precisely the Earth’s rotational speed.
Admittedly, both of these approaches require technologies that are nascent or non-existent (from an applied stantpoint, at any rate) but still probably more feasible than attempting to build the structure at ground level.
A hemisphere is self-supporting; short of that you have to oblique the structure to get an even force balance, otherwise the elements closer to the base see progressively higher stresses. However, even if you build a panel this way, precisely how are you going to airlift a panel in position and orientation that is several hundred feet wide? It’s not feasible by crane or helicopter, and I would not want to attempt it with an airship. In this case, gravity is definitely a harsh mistress.
Stranger
If you’re in Antarctica and looking for a low-melting-point material, why would you think of a metal at all?
The problem with using water is its absurdly high heat of fusion would require a heck of a lot of energy to remove the ice, and its relative hardness would make it difficult to mechanically removed interstitial sections. There are also the problems of thermallly-induced stresses on the structure. Paraffin would probably be better, albeit very, very messy.
Stranger
Maglev could do it but the city will only be a fraction of a inch or so above the ground.
How big of a wick would you need to burn a mile-wide paraffin candle?
I was thinking (perhaps wrongly) that the changes in volume from liquid water to ice would introduce forces that could disrupt or distort the structure. Also, changes in ice crystal structure under increasing pressure could do the same. Metal or wax or some other substance might not have those drawbacks.
The bouyancy of ice in water might also be a significant problem; if the potting compound melts unevenly, is it better to have the frozen parts exerting upward forces or downward forces? Ideally I suppose the frozen and molten phases should have the same density, and the heat of fusion would be as low as possible.
A tensegrity sphere will support itself once it’s built, but how do you build it? Interesting. Could you possibly do it this way: Build it first as a giant balloon with the geodesic members as cables held under tension by pressure. Then gradually thicken and stiffen the support structure until internal pressure isn’t necessary anymore.