Cloud Nine is the name Buckminster Fuller gave to his proposed airborne habitats created from giant geodesic spheres, which might be made to levitate by slightly heating the air inside above the ambient temperature.1
Geodesic spheres (structures of triangular components arranged to make a sphere) become stronger as they become bigger, because of how they distribute stress over their surfaces. As a sphere gets bigger, the volume it encloses grows much faster than the mass of the enclosing structure itself. Fuller suggested that the mass of a mile-wide geodesic sphere would be negligible compared to the mass of the air trapped within it. He suggested that if the air inside such a sphere were heated even by one degree higher than the ambient temperature of its surroundings, the sphere could become airborne. He calculated that such a balloon could lift a considerable mass, and hence that 'mini-cities' or airborne towns of thousands of people could be built in this way.
A Cloud Nine could be tethered, or free-floating, or maneuverable so that it could migrate in response to climatic and environmental conditions, such as providing emergency shelters.2
See answers to this question:
Can Cloud Nine be built?
What you need is to find some calculations relating to hypothetical "Cloud Nine" structures.
One problem I see with "cloud nine" structures is their spherical shape. If one loses lift due to leaks or something, and descends to the ground all its weight will be held up by a small section of the bottom which probably won't be strong enough and will probably break.
Landing in water would enable it to submerge a large proportion of its volume and then float supported by water over a larger area, less likely to break the sphere. But water pressure increases rapidly with depth. If a mile wide sphere extends hundreds or thousands of feet below the waterline, the water pressure might break the lower sections of the sphere.
I also wonder what would keep a sphere from tipping over and floating upside down in the air, while the people all plummet to their deaths.
So perhaps the floating city would be designed as two hemispheres side by side instead of one full sphere. Maybe the living spaces of the city could be in a series of large but relatively small hemispheres arranged around the perimeter of a giant geodesic sphere for lift.
For maximum surface area within a giant sphere, the floor area should be a circle at the "equator" of the sphere. If the buildings are tall, that will put more weight in the upper half of the sphere and make it more likely to be top heavy and tip over.
Perhaps the buildings would all be one story above the "ground", with multiple stories of basements below the "ground", and with water storage in tanks below the "ground" level, etc. Thus the lower hemisphere may be heaver than the upper hemisphere to avoid tipping.
How would the "ground", a circular disc a mile in diameter, be supported? By cables from above, like a suspension bridge, or by supports from below? Either way should add a lot of weight to the town, which might make it too heavy to float.
Possibly There could be a relatively smaller circular area of "ground" right above the bottom of the sphere, and the tops of the buildings would support another and slightly wider layer of "ground" above it, and the buildings on the second layer could support a third, somewhat wider, layer of "ground", and so on.
And the highest levels of "round" might be ring shaped, going around the perimeter of the sphere and with open spaces in their centers. Each ring would have a greater diameter than the ring below it.
Thus the layout of the floating city would somewhat resemble that of Hell in Dante's Inferno, a set of concentric rings which get narrower and narrower at lower levels. The levels would be connected by ramps and elevators.
Or the layout of the floating city could be said to somewhat resemble the layout of A.E. Van Vogt's spaceship the Space Beagle, or the Death Star in Star Wars, except that most of the floating city would be empty space with the "decks" or "ground" filling only only small part of the total volume of the sphere.
To save weight, parts of a floating city, whether a "cloud nine" type geodesic sphere or some other type of design, might be made of aerogels.
Aerogels are a class of synthetic porous ultralight material derived from a gel, in which the liquid component for the gel has been replaced with a gas, without significant collapse of the gel structure.1 The result is a solid with extremely low density2 and extremely low thermal conductivity. Aerogels can be made from a variety of chemical compounds.4 Silica aerogels feel like fragile expanded polystyrene to the touch, while some polymer-based aerogels feel like rigid foams.
Despite the name, aerogels are solid, rigid, and dry materials that do not resemble a gel in their physical properties: the name comes from the fact that they are made from gels. Pressing softly on an aerogel typically does not leave even a minor mark; pressing more firmly will leave a permanent depression. Pressing extremely firmly will cause a catastrophic breakdown in the sparse structure, causing it to shatter like glass (a property known as friability), although more modern variations do not suffer from this. Despite the fact that it is prone to shattering, it is very strong structurally. Its impressive load-bearing abilities are due to the dendritic microstructure, in which spherical particles of average size 2–5 nm are fused together into clusters. These clusters form a three-dimensional highly porous structure of almost fractal chains, with pores just under 100 nm. The average size and density of the pores can be controlled during the manufacturing process.
So in a world where aerogels are cheap and plentiful, and where people build floating cities, aerogels would naturally be used to build the parts of a floating city which they were most suited for, to reduce the weight.
For example, I can imagine that aerogels might be useful for most of the parts in a building, so that the buildings in the floating town might be mostly made of aerogels.
And if the buildings in the floating city are mostly made of aerogels that would greatly reduce the total weight of buildings to be calculated.
Aerogel balloons have been suggested, Air filled aerogels are very light, but heavier than air. But if an aerogel could hold a vacuum, it would be lighter than air, and thus a solid but vacuum enclosing aerogel could be a lighter than air balloon, and maybe make a floating city.
Since aerogels are very strong, and also quite transparent, they could also be used for fake floating cities. Aerogel columns could support a city far above the ground. if someone could manage the lighting so that the columns couldn't be seen, it would look like the city was floating in the air, held in place by fake tethers to the ground.
So a conman might claim that the city was temporarily tethered in place, and that after it was settled by his customers, it would be set free to float in the wind.