Start with a seaworthy vessel, with as much passenger and crew space as a typical pleasure cruise ship. Make it circular, and short (fewer levels high means a larger surface area).

Build stuff on the deck: pools, parks, a golf course, etc..

Cover the ship with a *large* geodesic dome, covered
with clear, airtight plastic. Partition the dome into two volumes of air, with a horizontal sheet of plastic, 30 feet above passengers' heads. Add controllable windows in the dome (both upper and lower parts) for ventilation. Heat the air in the upper portion dome using waste heat from the ship's engines.

If the dome is large enough, and the air in it hot enough, the ship can take off into the sky.

Takeoffs and landings would be done to/from water, in protected bays and harbours.

As an approximation, let us ask how tall a column of hot air
is required in order to lift a unit-height column of ship of
the same diameter.

Suppose first of all that, in cross-section, the ship is 2%
steel (in other words, if you drop a line from the deck
through the ship and out the bottom, on average it will
pass through steel for 2% of its length; this seems
reasonable, allowing for the floors and, in some places,
walls and bulkheads and furniture etc). Hence, on average
the ship has a density of 2% that of steel, or 0.16g/cm3.
This is about 130 times the density of air.

Now, how hot can we make the air? Let's say we can heat
it
to 300°C (or about 600K, which is very generous), from a
starting temperature of about 30°C (300K). Hence, its
density will fall to half of its original value.

Hence, to lift a "cylinder" of ship will require a heated air
cylinder about 260 times as tall.

Now imagine the ship has a height of, say, 10 metres (since
you wanted a fairly short ship); this is going to need an air
cylinder of 260 x 10 =2600 metres.

Of course we are talking about a dome, and the ship will
also be (crudely) an inverted dome. But, roughly speaking,
your plastic envelope is going to have to be about 2-3km
high. This naturally ignores the weight of the plastic dome
itself.

So, to answer the original question, very dumb indeed.
Not as dumb, though, as posting an idea without doing
even a little elementary arithmetic.

That's what I was thinking of in the first place, after all... Covering the ship with a large geodisic dome, and by large, I mean one whose diameter is *larger* than the diameter of the circular cruise ship, and where the center of the dome is *above* the center of the cruise ship.

Perhaps the misunderstanding was caused by you having a silly belief that "geodesic dome" means hemispherical or smaller. Here's the first line of Wikipedia's entry on geodesic dome: "A geodesic dome is a spherical or partial-spherical shell structure or lattice shell"

The way I figure it, a nearly spherical dome has close to the same volume as a spherical dome of the same radius.

And I'm also assuming that each degree of temperature increase gives the same increase in bouancy as the first degree of temperature increase.

So if a 1 degree increase on a 1 mile dome produces neutral bouancy, a 2 degree increase produces enough bouancy to send the dome upwards with the same accelleration as gravity would have been holding it down, had the air not been heated.

So a half mile dome, with a 8 degree increase above ambient temperature, should also be neutrally bouant. Replace a small slice from the bottom of this size dome with a circular cruise ship, and raise the temp another few degrees, and off it goes.

Take a look at the wiki article on "buoyancy". You will need to know the density of "air" (our atmosphere) at sea level, and at some altitude. You will also need to know the density of air at various temperatures.

try the equation for mb ("b" is subscript). It is the equation for buoyant mass. The "AVERAGE DENSITY" of the floating city is what you will use for the object density.

Solve for given altitude and see what you come up with. I will see if I can look at it when I get time. If the buck-myster says it can be done, well... maybe it can be done.

I *think* this will solve the problem. I believe Maxwell can correct me if I've mislead you.

A final note: Would life jackets come in the form of a parachute?

So a dome (1/2 sphere) 1km high will give you about 2*10^9 m³ of air. Every m³ of air at 1.3 kg and every 3°C being about 1% (13g ) mass, for a 5*10^7kg cruise ship you'd need a 2°C increase in temperature, but this still does not look at the 1.2 *10^6 m² of dome covering ...

loonquawl, "Dome" does not mean "half sphere." That would be "hemisphere."

I'm thinking of a shape more closely resembling a full sphere. So you can double the amount of air, and the amount of dome surface.

The total weight of the material of the (nearly spherical) geodesic dome will surely be less than the total weight of the cruise ship; thus, a mass of warm air which can lift a cruise ship can easily also lift the materials of the dome.

A much better mathematical question is, if we're willing to use higher temperatures (say, 10 or 20 degrees), how much smaller can the dome be made?