A Dyson Sphere takes the shape of a hollow sphere, with a sun at the center. The People lived in the inner shell of such a construct, known as the Worldsphere. (NA: The Also People, BNA: Walking to Babylon)
Most fictional depictions of a Dyson Sphere -- including those in Doctor Who -- are, in fact, a Dyson Shell, something generally considered to be an engineering impossibility. An actual Dyson Sphere is a swarm or bubble of energy-collecting satellites around a star; the "sphere" is the shape of that swarm or bubble, not a solid object. The Star Trek: The Next Generation episode "Relics" was also based on the concept of a Dyson Shell/Sphere.
A Dyson Sphere is a construct whose inner surface is approximately 1 AU from a star (the entire structure being a little more than 2 AU in diameter). This allows for a very large surface area capable of supporting life. Gravity (through the mass of the structure, or by artificial mechanics) would essentially pull everyone 'down' towards the hull and away from the star. Topsoil, and atmosphere would most likely be present between the hull of the sphere and the star. (TNG episode: "Relics")
Such a structure as a Dyson Sphere would have an internal surface many times the size of all 24th century Federation planets combined. The Federation did not have the technology at the time to create such a construction.
The Dyson Sphere was first theorized by Freeman Dyson.
Miles O'Brien underwent a telepathic "engineer's nightmare" in which he wandered along a sandy desert surface inside one. As for what seemed months, if not years. This illusion was brought on by the pah-wraith aliens that targeted his fragile, vulnerable psyche from his previous Argrathi incarceration. (Star Trek: Deep Space Nine Millennium; DS9 episode: "Hard Time")
One type of Dyson Sphere would appear to be of different dimensions, as when one falls from the 'top' of the inside of this sphere, gravity pulls that person to the 'bottom' of the sphere rather than to the hull. Also, in this case, a Romulan cloaking device may have been used to hide the sphere, but the gravity well it created would be quite noticeable. (NF novel: Double or Nothing)
The USS Titan stumbled upon the planet of New Erigol, which the inhabitants had attempted to hide using a Dyson sphere like construct around both the planet and the star. Although the covering eliminated visible output from either, it still allowed high energy pulses to escape, allowing for its detection. (ST - Destiny novel: Gods of Night)
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A Dyson Sphere/Shield World, or "shell" (as it appeared in the original paper by Freeman Dyson), is a hypothetical megastructure that was originally described as a system of orbiting solar-powered satellites meant to completely encompass a star and capture its entire energy output. Dyson speculated that such structures would be the logical consequence of the long-term survival of technological civilizations, and proposed that searching for evidence of the existence of such structures might lead to the detection of advanced intelligent extraterrestrial life. A three meter thick Dyson Sphere with an Earth orbit radius would require an estimated 2.12057325 x 1023 cubic meters of material.
Since then, other variant designs involving building an artificial structure — or a series of structures — to encompass a star have been proposed in exploratory engineering or described in science fiction under the name Dyson Sphere. Most fictional depictions of a Dyson Sphere describe a solid shell of matter enclosing a star, which is considered the least plausible variant of the idea.
While it is believed that some of these design variants are impractical, if not physically impossible, some designs do not require any major breakthroughs in our basic scientific understanding for their construction.
In the book Halo: Ghosts of Onyx, it is revealed that at the core of Onyx is a Slipspace rift that shunts those that pass through it to a structured Dyson Sphere, dubbed beforehand by the Forerunners as a Shield World Installation. It is separated from a normal space-time symmetry by advanced Forerunner Slipspace technology. Although Dr. Halsey calls it the Micro Dyson Sphere, the Sphere is 300 million kilometers (or approximately two astronomical units [1 AU is 149,598,000 km]) in diameter on the inside. The Micro Dyson Sphere's star has a spectra and energy output slightly smaller than the dimensions of Sol. This was going to be the haven where the Forerunners were to seek refuge when Halo activated, but no one knows what went wrong with their alleged plan, as the expanse appeared to be completely deserted when UNSC forces arrived. In Halo 3, these circumstances were revealed to be the result of the betrayal of the Forerunner AI Mendicant Bias, whose defection to the Flood forced the Forerunners to abandon their plans to evacuate into the shield installations.
Basically, the structure fulfills its function as a "Shield World" perfectly. It protects the occupants of it and gives them a huge "world" to live in. Potentially, an entire new civilization could be constructed in a Dyson Sphere.
After their arrival in the Forerunner-constructed Dyson Sphere, Dr. Halsey explains the structure. She labels it a Micro Dyson Sphere because it is in fact very small, or perceived to be small when viewed from normal space. Linda also describes it as being in the bottom of a large bowl.
The Micro Dyson sphere is located in a factually compressed Slipstream space. In other terms, any object inside might appear to be a certain size, but inside the Slipstream bubble it is much, much larger. Consequently, they never left Onyx, but simply entered a Slipspace bubble.
In Halo Wars, there is also a type of Micro Dyson Sphere. However, this Shield World, as referred to in the game, is vastly different from the Onyx installation. Unlike Onyx, which was made up of trillions of Sentinels over which a planetary surface had formed, this world was built in a method similar to a Halo Installation. It was a hollow structure with planetary surfaces on the interior and exterior being supported by numerous pillars. Inside this was a more familiar construct of ship-sized portals allowing access from an opening on the outside to the interior world. The interior was somewhat similar to Onyx's Dyson Sphere, possessing an artificial sun, though it and the entire structure are on a far smaller scale. However, unlike Onyx, it was not designed as a bomb shelter, but as some sort of harbor, shipyard or hiding place for ships, as it contains The Apex, which had dozens of Forerunner ships stored within it. It is unknown whether the same is true for the Onyx Installation, but if it is, then it may have served as a possible means of escape for the Spartans trapped there.