The idea of a technological civilization re-engineering its planetary system in pursuit of energy goes back to science fiction stories of the 1930's. But the term 'Dyson sphere' is what sticks with us, and it has its origins in a paper written by Freeman Dyson in 1960.
The paper, published in Science, has the moderately innocent sounding title of "Search for Artificial Stellar Sources of Infrared Radiation". But you only have to read the Abstract to catch a whiff of the marvelous piece of inventive, but grounded speculation that unfurls in this single page article.
Specifically, this is what that summary paragraph says:
"If extraterrestrial intelligent beings exist and have reached a high level of technical development, one by-product of their energy metabolism is likely to be the large-scale conversion of starlight into far-infrared radiation. It is proposed that a search for sources of infrared radiation should accompany the recently initiated search for interstellar radio communications."
The rest of the material is a light-handed, but quantitative argument for a plausible trajectory for a growing civilization - using Earth and the solar system as a basic template (without suggesting that this is what we'll end up doing, but rather that this is the kind of thing that something might end up doing).
I'm talking to a class I teach about this, and so I tried to rephrase the argument into a few simple bullet points (Dyson's own explanation is equally sparse and probably more elegant), that I'll share here:
1) If a civilization grows at 1% per year, it only takes about 3,000 years to grow by a factor of 1012
2) The material resources presently available for us to do stuff with amount to about 1020 grams and Earth receives about 1020 ergs per second of energy - so a 3000 yr growing civilization will run out of energy or the matter to construct better means of acquiring (stellar) energy
3) Using our solar system as a template: Jupiter contains about 2x1030 grams of ‘available’ mass, and the TOTAL rate of energy output of the Sun is 4x1033 ergs per second
4) It ‘only’ takes about 1044 ergs to disassemble and re-arrange the matter in Jupiter - or about 800 years worth of total stellar output
5) Therefore there would be strong motivation to re-purpose Jupiter’s mass into a 2-3 meter thick spherical shell at Earth’s present (habitable zone) orbit, surrounding the Sun, capturing all energy and habitable on the interior surface…
He then goes on to point out that this sphere must maintain thermodynamic equilibrium with the enclosed star and the rest of the cosmos, so from the outside it must glow with the luminosity of the star, but with a spectrum shifted to the infrared - consistent with the balmy temperatures of its surface. And he suggests that one avenue to pursue in the hunt for such structures might be to study binary star systems where one star is easily visible, but the other is an intense infrared source.
It's a terrifically calm presentation of an idea that more or less makes one's head explode, even almost 60 years later. It's also perhaps a useful reminder that as a species we've yet to provide good evidence that we will manage to place ourselves on any such technological trajectory.