One of the more seriously bad-ass concepts in modern science fiction is the “Dyson Sphere.” The idea as generally portrayed (such as in the “Relics” episode of Star Trek: The Next Generation”) is a hollow spherical shell a hundred million miles or so in radius, with a star at the center. The purpose of this vast effort would be to capture all of the output of the sun, not just a small fraction, and to create a place to live with trillions of times the surface area of an Earth-like world.
A few small problems, not least of which being that it probably could not be made to work. For starters, gravity. No matter how massive the shell is, if it’s even remotely evenly thick and dense, there will be no net gravitational force inside the sphere. If a person (represented in the detailed schematic below by the little purple figure) happens to be near the wall of a hollow spherical shell, the mass in “Area 1” will obviously attract him. But subtending the same solid angle is “Area 2” on the opposite side of the shell. And while “Area 2” is vastly further away from the person than “Area 1” (say, 200,000,000 miles, as compared to six feet), “Area 2” compensates by being vastly more massive than “Area 1.” The end result is that the two areas cancel each other out. A consequence of this is that the only gravitational force felt within the sphere is that cause by the central sun. Typically, in science fiction hand-wavy “gravity generators” are built into the wall so that someone could stand on the inner surface.

Several alternatives to “gravity generators” have been dreamed up to counter this little problem:
1) Stand on the outside of the sphere. Duh. (but it’s cold and dark…)
2) Build a transparent sphere inside the larger opaque one, smaller in radius by a hundred miles or so. This will trap a hundred-mile-thick blanket of air, and will keep stuff from falling into the sun. The solar gravity, and the gravity felt by being on the *outside* of the transparent sphere would be minimal; for all intents and purposes, it’d be freefall. But it’s also be *weird,* and over time everything – dirt, trees, water, dead folk, bird crap – would cover the transparent sphere.
3) Spin the sphere to generate centrifugal force. But this will cause everything to fall downhill to the equator, leaving most of the surface area barren, uninhabitable and at an unfortunate slope. More, all the air will pool at the equator, so good luck with those climbs…
4) Build the Dyson Sphere not as a solid spherical shell, but as a stacked set of Ringworlds. Each Ringworld would be a ribbon of material a million miles or so wide, with rims several hundred miles high. Each would spin to create centrifugal force. Stack Ringworlds of various radii together to form a sphere. There would need to be frictionless (magnetic, most likely) bearings between the worlds, and the closer to the “poles” you get, the further down towards the horizon the sun would get. But this would seem to maximize the total livable area; plus you could have rings with different gravities. Below is a crude schematice of a cross-section of such a sphere. The individual rings are shown *way* bigger than in reality. Rather than a dozen or so, you’d probably want a hundred or two.

And… there’s:
5) Build the Dyson Sphere the way physicist Freeman Dyson actually wanted. Not as a solid sphere, but as a collection of separate, individual artificial space habitats. Instead of a solid wall intercepting all the energy of the sun, there’d simply be so many colonies that they would blot out the sun. This would seem to be not only the easiest way to make a sphere that traps all of a star’s energy, it would also seem to be damn near inevitable, assuming we can get off this rock.
There has been a long arguement over exactly what Dyson meant when he first described the concept, whether solid shell or cloud. Well, rather than hash that out, here are the letters (from Science magazine, all from 1960) where he first describes his concept. There are three letters of note: his letter of June 3, where he describes what seems to be a solid sphere, a rebuttal letter from July 22, where a reader takes him to task for the concept, and a reply from Dyson somewhat later where he clarifies his position to that of the “cloud.” The scans are made from crappy photocopies I made… yikes, about 18 years ago.
