# How and when could a Dyson sphere civilization figure out the shape and size of their sphere?

Kinda inspired by the non-disappearing ship question, but I actually asked this particular question on an unrelated forum a few years ago. Reposting it here (well, paraphrasing, really, since I don't actually recall much of the original specifics) because I'm interested in the opinion of this particular community.

Consider a typical science-fiction Dyson sphere - a few hundred million miles (or, if you prefer, a few AU) across - with a human (or essentially human) civilization living on it. Many different such civilizations, actually (since the sphere is very big).
They've been there since an awful long time ago; long enough to have developed all of their technology beyond the early stone age while already on the sphere, and long enough that not even legends preserve any mention of the original arrival (I'm assuming, for clarity, that such an arrival did indeed happen, perhaps 100,000 or so years earlier, but it doesn't really matter).

With that in mind, at which point in tech level (assuming an advance roughly along an Earth-like technology tree) would they be able to figure out that they live on a giant sphere (as opposed to, say, a giant flat plane), and at which (presumably later) point would they be able to figure out the approximate size of their sphere? And how exactly (what sort of tools, methods, calculations...) would they be able to do that?
(Your choice of what counts as "approximate"; if you have several different scenarios for different levels of precision, I'd be happy to see all of them.)

There are really two different questions here (plus an addendum), for different definitions of "Dyson sphere", which I'm putting together because I'm not sure I could write enough detail for both separately. (I'd be happy to make a short separate question for one of the versions if the mods think that it would be better.)

1. A regular (for science fiction, at least) Dyson sphere, with an inner-facing habitable surface. Light and heat is probably provided by a star in the middle; gravity must be artificial, because the natural gravity balances itself. Your choice on how the day/night cycle works (or whether there even is one) - this might matter for the specifics, obviously.
2. An inverted Dyson sphere - a sphere of similar size (several hundred million miles, i.e. a few AU, in diameter) with a habitable outer surface. Will probably have stars orbiting around it for light and heat (might or might not, depending on the specifics, also work as a day/night cycle). Natural gravity is easy in this case - just make the sphere sufficiently thick (a few thousand kilometers); but in this case there's not much difference whether it's natural or artificial gravity.
3. The addendum version - same question (when and how will the inhabitants any other hyper-large space habitat (such as a Ringworld like the one in the Niven series, or a Culture-style orbital).

Just for the record - I'm looking for reasonably realistic technological solutions (aside from the existence of the sphere itself), not fantasy.
If you happen to think (I personally doubt it) that there's no real way to figure out the size (at least, within the current level of Earth technology) - say that, and explain why.

...If this question is off-topic, sorry. (You can also try to fix the tags, if I chose them incorrectly.) This is my first question on Worldbuilding SE (and I hadn't asked very many questions on other SE sites before either).

• Hello, and welcome to the site. Your question is a little too broad, and falls out of scope on WB SE. Try asking a single pointed question, then posting the others separately when you've established the answer to the first. Learn more about this site's scope and purpose at meta.worldbuilding.stackexchange.com/questions/3206/… – AndreiROM Mar 28 '16 at 17:17
• So the question is essentially: "What experiment could someone living in a dyson sphere conduct to demonstrate that they live in a dyson sphere as opposed to a giant plane?"? – Saidoro Mar 28 '16 at 17:41
• @Saidoro - and what tech level they would need to conduct such an experiment, yes. That was basically what I was going for. – January First-of-May Mar 28 '16 at 17:44
• Your question(s) is written in a confusing manner. Please focus on one question with a solid direction. Also, the fact that the society is "inner-facing" seemed like a passing thought. You said they live "on" the sphere. I and someone else already misinterpreted your question and the fact they're inner-facing is more than a passing thought - it changes the dynamics of any experiment a citizen of this sphere could conduct and changes the nature of this question entirely. So are they on it, or inside of it, facing the planet? – The Anathema Mar 28 '16 at 17:44
• @TheAnathema - I didn't want to make two nearly-identical questions (even with radically different expected answers) about inner-facing and outer-facing Dyson spheres (plus another for ringworlds). And I thought that the word "on" could refer to either surface (but I would have used "in" for the inner-facing sphere question if I did make two different questions). – January First-of-May Mar 28 '16 at 17:53

Let's get this out of the way: classic, rigid Dyson spheres are not stable and neither is a rigid ringworld around a star.

A rigid dyson sphere is not stable, since there is no net attraction between a spherical shell and a point mass inside. If the shell is pushed slightly, for example by a meteor hit, the shell will gradually drift off and eventually hit the star. This is a classic problem in elementary mechanics and is usually solved in introductory textbooks.

This is because of the shell theorem which states that the net gravitational force on an object inside a hollow sphere is zero regardless of the object's location within the shell. This seems non-intuitive, if you dig a hole through the center of the Earth you'll float in the center, but it's exactly the same thing. Instead of the "center" being a point, it's the entire hollow interior. There's just a lot more "center".

There's many other problems. Here's Fraser Cain, publisher of Universe Today, on the subject.

But let's assume whomever built this figured out a way to solve all that, and it's still working, and the stabilization mechanisms aren't immediately obvious like giant thrusters.

With that out of the way...

...at which point in tech level (assuming an advance roughly along an Earth-like technology tree) would they be able to figure out...

1. they live on a giant sphere (as opposed to, say, a giant flat plane)?
2. the approximate size of their sphere?
3. ...and how would they be able to do that?

For consistency I'll take "a few AU" to mean 3 AU.

3 AU rigid Dyson sphere, outside

I'm starting with the outside inhabitants because it's the most Earth-like. Ancient humans figured out the Earth was round by a number of careful observations. The people on the Dyson sphere could do the same, but the size of the sphere makes this much harder so they'll figure it out much later.

The distance to the horizon depends on your height above the surface and the radius of the sphere. In particular it's the ratio of your height and the radius of the sphere. It's not a simple equation, so I'll let Wolfram Alpha take care of it.

A 1.8 m tall person standing at sea level on Earth will see the horizon at about 5km. We'll use that as our benchmark for what can be figured out without instruments. On a 3 AU sphere, the horizon is 1300 km away. The inhabitants are trying to view it through 1300 km of thick atmosphere. This means, to even a careful observer, the world will behave flat. The resolution of the eye and atmospheric effects will prevent people from seeing the horizon or measuring it accurately without precise instruments.

The other way the ancients figured out the world was round was by observing the shadow of the Earth on the Moon. The Dyson Sphere is at the center of its system, so it won't do a traditional eclipse. The orbiting stars will shine on other orbiting bodies. As it slips below the very distant horizon there will be a shadow, but this shadow will appear so straight and go by so fast that ancients will likely not observe it.

What they can't do is make the ancient observation that different stars are visible at different latitudes. Changing latitude on the Earth is a matter of traveling north or south an inconvenient but doable distance, about 111 km per degree (pi * radius / 180). But the Dyson sphere is too big. They'd have to travel 7.5 million km to see one degree of latitude change.

And so it goes. Eratosthenes observed that the Sun casts a different shadow at noon at the same time of year at different latitudes. You can't do that on your Dyson sphere because you'd have to travel too far to be in a different latitude. Circumnavigation is also out. Everything is just too big for ancient observations.

Since the Dyson sphere is not rotating (or rotating only very, very, very slowly) they will see a fixed field of unmoving stars in the sky, plus the objects which are orbiting the sphere, which includes a few small stars. They will observe these set and rise, and other bodies do so as well. They can figure out, at least, that their planet is not an infinite plane.

The first suggestions that their world is a sphere will come when the laws of gravity and material science are discovered. They will quickly discover that a finite flat plane is not stable, gravity will want to pull it into a sphere and the material of their planet will not be able to resist. They'll also notice that the orbits of their orbiting stars and moons are wrong for a plane. Eventually someone will work out, via their moon's orbits, that their planet must be a giant sphere.

It likely won't be until fast and sustained means of travel, long distance radio communications, and precision instruments that people can get enough distance between two points in a single lifetime to actually measure the curvature of the sphere. Say, early 20th century technology. Once they hit that point, they can use this data to approximate the size of their sphere.

It might take even longer because there won't be this slow build up of casual observations suggesting a sphere: to all observers the world truly is flat. The discovery that the world is round would be a scientific curiosity, like determining the age of the Earth, but which will have far, far reaching metaphysical implications.

3 AU rigid Dyson sphere, inside

I'm not going to do a full treatment of this one, but I am going to dispel a few common misconceptions. First, because of the shell theorem, there's no net gravity from the sphere on the inside surface. There's the opposite problem, you'll fall into the central star.

The sphere would have to be spun to produce centrifugal force. To spin a 3 AU sphere to produce 1g at the equator (the gravity of the central star can be ignored) requires solving the centripetal acceleration equation (a = v^2/r) for velocity (sqrt(a*r) = v). Plugging in the numbers gives us 2.1 million m/s, a noticeable fraction of the speed of light. In contrast, the Earth rotates at the equator at 465 m/s. This would likely tear the sphere apart.

I'm going to handwave the gravity and rotation problems, I think they're unsolvable with known physics, and just say the builders figured it out somehow. My sphere is not rotating and magically has 1 g everyone on the interior surface.

Like with the last answer, the problem is the scale: the "horizon" (which I'm using as a proxy for "the distance you'd have to see before you notice the land is curving up") is 1300 km away and 1 degree of latitude is 7.5 million km. Most of the ancient techniques won't work, or the geometry isn't correct. The critical clue is the day/night cycle, if it has one.

The first thought is that you'll see the "horizon" curving up. The horizon would be 1,300 km away. Viewed through an atmosphere, you won't see it. The "horizon" will be a muddy, ground colored blur going all the way overhead. Pointing surface telescopes at the "horizon" won't help, you'll be looking through lots and lots of atmosphere.

Next you'd think "you'll look up and see the other side", but that's 6 AU away. You won't be able to resolve anything, the sky will be a ground-shaped blur. If there is no night there will be a star always at the zenith preventing good observations for a very long time. If there is a night cycle, eventually features the other side will be resolved, but it will take some very good optics.

Jupiter is closer to the Earth than the other side of the Dyson sphere, and it's a tiny dot in the sky. 17th century telescopes were able to make out Jovian moons and the Great Red Spot, so I'll put it somewhere between the 17th and 19th century when people begin to see features in the sky.

On a sphere where it's always day, the inhabitants won't even know their world is finite. There will be no hints to casual observers that they live on anything but a flat, infinite plane. But if there's a day/night cycle, that provides the critical clue.

A night cycle provided by Sun shields orbiting around the star would provide the information necessary to work out not just that they're living inside a sphere, but also its size. The day/night terminator shadow would be seen moving across the land and eventually up the interior of the sphere, across the sky, and back down again. This would be clear evidence you're living inside a sphere. If the Sun shields are vertical bands, you would see a band of light lit up across the sky. If they're rectangular, you'd see rectangular shadows. You can work out the distance to the other side by measuring the apparent width of the band at different times, and how fast it appears to travel across the sky. This could be done in ancient times as soon as you have geometry, as it was done on the Earth.

3 AU Ringworld orbiting a star, interior

A 3 AU Ringworld has most of the same problems as living on the inside of a 3 AU Dyson sphere. Scale and atmosphere prevents casual observers from seeing the rising horizon or the ring on the other side. There are two major differences which allow ancient observers to see that they're living on a giant ring.

If they are in perpetual daylight they will never see stars, and thus won't see the ring obscuring them. But if there is a day/night cycle they will see stars, and they will see a great, towering arch of blackness stretching from one side, overhead, and back down behind them. Only geometry is necessary to work out that they're living on a ring and the size of the ring. At some points on the ring it will be eclipsed by their star, and they will be able to work out the size of and distance to their star.

The other clue is the edge. Presumably this edge will be very high, 100 km at least, to keep the atmosphere from spilling over the side. It could even be made to look like a natural mountain range. The distance to this edge depends on how wide the ring is, but someone will live near the edge, or travel to the edge, and stories will come back. The edge will be explored and, eventually, climbed. Then the nature of their world will be obvious.

This will likely take mid to late 20th century technology. Everest wasn't ascended until 1953, and that's only 9 km. Above 8 km your brain does not get enough oxygen and you eventually die. Climbers must carry oxygen above this point. Our ringworld edge explorers will have to carry their oxygen for another 91 km. A dedicated expedition with a large logistics train to supply food and oxygen, setting up chains of base camps and supply lines, a kin to modern Antarctic exploration, would be necessary.

As for flying over the top, this will take 1960s technology. Balloons are still not capable of going that high, the modern record is only 40 km. In 1961 the record for an aircraft was 35 km. It was only in 1962 when the X-15 finally reached 100 km.

Other megastructures

As I mentioned at the beginning, rigid rings and spheres around a star are unstable. Feasible megastructures are variations on a Dyson swarm, a ring of independent, probably spherical, structures orbiting around a star.

The inhabitants of these structures would be able to determine the nature of their world, and the swarm, as quickly as we did on Earth. The curiously regular lights in the sky that remain in the same point in the sky, but move against the fixed background of stars, would be a great focus of curiosity. Their relative closeness to their own habitat would allow observation with even the most basic 17th century optics.

They would likely develop space travel earlier. Instead of a political or military space race, they'd have a clear economic incentive to visit other habitats, to trade with them, and colonize them. I'd imagine they'd reach another world in the 1930s or 40s, just as the technology is becoming available.

• Light pressure from the central star on the inside of the Dyson sphere should push it back on center when it randomly drifts off-center. I don't think that's going to be a problem. A Dyson sphere isn't a ringworld. – Lensman Mar 29 '16 at 7:34
• Maybe Volcanoes are giant thrusters to keep Dyson Sphere Earth in place... – GolezTrol Mar 29 '16 at 8:07
• @Lensman: Light pressure is only useful as a stabilising mechanism when dealing with a statite cloud where there is no mechanical connection all the way around the shell, and even then the density must be tremendously low for light pressure to counteract gravity, to a degree which precludes any kind of earth-like surface for habitation. If a complete shell, though radiation pressure may be greater at an individual point if it is closer to the star, remember that the overall force applied to the shell in all directions is the same - the star is not outputting more radiation on the near side. – Carcer Mar 29 '16 at 8:35
• Dyson sphere, inside: There is no horizon (1300km, or otherwise). A horizon is based on the curvature of the surface you are standing on curving away from observer. That curvature results in there eventually not being a straight line form the observation point (eyes) to the distance surface (i.e. such line is blocked by the surface). On the inside of a Dyson sphere, the surface curves toward the observation point. There is no blockage of line of sight: thus, no horizon. You see it all until blocked by the star, or other obstruction (level of detail depends on optics quality, etc.) – Makyen Mar 29 '16 at 20:14
• @Makyen For the inner sphere "horizon distance" is an approximation for "how far would you have to see before you noticed there's a curve". I've edited it to make that clear. At 1,300 km I believe atmospheric haze and distortions would not let you see anything but a blur. I'd love to see someone tackle the optical equations. – Schwern Mar 29 '16 at 20:23

Living inside a "regular" Dyson sphere, the horizon would not turn down but up. You could see that you weren't on a flat plane. The curvature would be slight but visible. You wouldn't be able to see across the sphere, but you could see something for quite a distance.

The ancient Greeks were able to calculate the distance to the moon and sun prior to the Roman empire. Also the sizes of those and the Earth. It's just trigonometry and observations at different altitudes. That knowledge was rediscovered during the Renaissance but didn't become common until science education did (within the last couple centuries). Calculating the curvature of the outer sphere should be of similar difficulty.

• While the basic idea is correct, the horizon of a 3 AU sphere is too distant for this observation. – Schwern Mar 28 '16 at 19:47
• Wouldn't that depend on how far along the sphere you went to make different observations, or how high the mountains were? Maybe not. Dyson spheres are too large for my imagination to process human-scale interaction with them. – J.D. Ray Mar 28 '16 at 21:58
• @J.D.Ray On a 3 AU radius sphere, 1 degree of latitude is 7.5 million km, making it very difficult to travel far enough to make a noticeable difference to anything but the most precise instruments. The eye-level horizon is 1,300 km (Earth's is 5km). Climbing a 2 km mountain pushes the horizon out to 42,000 km (160km on Earth). This would seem to be how you'd figure it out, but I don't think you can see through 1,300 km of atmospheric haze to notice the difference. Someone who knows optics could work that out. – Schwern Mar 28 '16 at 23:31

Early to late antiquity, if they really thought about. Features on the sun and the lack of parallax could give it away.

From the interior, the inhabitants could deduce they live inside of sphere by observing the, figuring out its a sphere and from the lack of any apparent parallax for the sun, deduce they must live on the inside of a large curved surface.

To produce 1-g of centrifugal gravity would at the equator would require IIRC that the sphere rotate faster than the sun's equator. The sun's surface is far from featureless, in addition to sunspots, there are also light patches and filament like structures. As the only astronomical object, we can presume it would be closely observed. It's rotation would be the only time keeping mechanism possible.

The domed churches in Christendom of the dark and early medieval ages were built at large camera obscura for time keeping by tracking the sun. The same could built on a Dyson-sphere but with even higher resolution as the apparent position of the sun would never change.

The sun observers would see the various features of the sun appearing at the rim, then moving across the sun so they appear flat and then disappearing around then other rim and then reappearing. Also, features near the equator would appear to move slower than those towards the poles. The easiest shape that would give theses sorts of behavior would be a rotating sphere.

They could also place multivalent boundaries on the size and distance of the sun by calculating the combinations of size and distance that would allow humans to observe it as anything other than a featureless dot.

But a spherical rotating sun will cause problems. Owing the lack of apparent curvature on the inside of the Dysonsphere, the inhabitants will start with the assumption that the world is flat. However, once people travel a few hundred or few thousand km, they will notice that the sun is always directly overhead and even the tracked features on the sun, don't show any parallax.

If the "Earth" is flat and the sun's shows no parallax, then sun must be very far away and extremely large for humans to be able to resolve features on it. But at that size its speed of rotation would have to be incredible to move features over such a fast surface in such a short time.

Conversely, if the "Earth" is flat and the sun nearer, then the features at least, should show some parallax, especially as they move.

Eventually, it will occur to someone that if the sun is of a certain size, at a certain distance, then only way to explain the lack of parallax would be to assume the "Earth" itself is curved such that each point on the "Earth" is the same distance from from the sun. From there, they could perform some multi-varient calculations to give possible ranges of the size and curvature of the "Earth".

It wouldn't exactly be proof, in the modern sense but it might be accepted on the basis of Occam's razor.

From the exterior: The exterior will have some rotation, however minimal. It would next to impossible to create such a huge object that has zero angular momentum. You'd probably want some just to stabilize the thing.

Even if it rotated very slowly, the rise, transit and eventually setting of the constellations would eventually give away the existence of a curved surface.

• The basic idea is correct, but the problem again is scale. Observers cannot travel far enough to do the necessary observations at different enough latitudes. On a 3 AU sphere, 1 degree of latitude is 7.5 million km. Also a 3 AU Dyson sphere rotating at 1 g would tear itself apart, but I'm happy to hand-wave that. – Schwern Mar 29 '16 at 19:39

I'll assert that the thickness of the mass that makes up the sphere will create enough gravity so that "down" happens when standing on the inside.

Second, I'll assert that the diameter of the sphere is properly balanced with the radiation output of the star, as well as the thickness of the atmosphere, to provide a habitable environment (this probably doesn't have anything to do with the question, since you stated that someone is living there).

In a Dyson sphere, there are no stars in the sky. There may, however, be inner planets orbiting the star (or artificial satellites, asteroids, even asteroid belts). I believe the only way to determine that they're living inside such a sphere is by observing things that orbit the star between the inner surface of the sphere and the star itself. Copernicus-level technology and math would be able to work it out, I think.

• The first paragraph is something often suggested by science fiction writers, but for an inner-facing sphere, sadly, wrong (the gravity of said thickness from the other parts of the sphere will exactly balance, so that net gravity from the sphere itself would essentially be zero, leaving a slight net force towards the central objects, if any). For an inner-facing sphere to be workable, we do need artificial gravity (and, depending on how it works, even that might not be enough). But yes, the "habitable envirnment" part is assumed (note that I never said which kind of star is in the middle). – January First-of-May Mar 28 '16 at 18:43
• Basically, this comes from Poisson's equation, which applies to both gravitational and electric potentials. This is because $\nabla^{2}\phi(\mathbf{r})=4\pi G \rho(\mathbf{r})$, and in a region with no mass density, this becomes Laplace's equation. If we then integrate over the region inside the hollow object, use the divergence theorem, we will see that the surface integral of $\nabla \phi$ is proportional to the mass inside the surface. We can take $\nabla \phi = \mathbf{g}(\mathbf{r}) = \mathbf{F}/m$ out of the integral, leaving just the surface area. – Obie 2.0 Mar 28 '16 at 20:32
• To the final paragraph: there would be no observations to connect the orbits and spherical nature of celestial bodies to their own situation. In fact it would be wrong to do so: the Dyson sphere is unlike the rest of them. – Schwern Mar 28 '16 at 20:43
• Or more briefly: $\nabla^{2}\phi(\mathbf{r})=4\pi G \rho(\mathbf{r}) = 0$. $\int \nabla^{2}\phi(\mathbf{r}) dV = 0$. $\int \nabla \phi(\mathbf{r})\cdot \mathbf{\hat r} dS = 0$. $\phi(\mathbf{r}) = \phi(r)$. $\mathbf{g}(r) = g(r) \mathbf{\hat r}= \nabla \phi(r)$. $\int g(r) \mathbf{\hat r} \cdot \mathbf{\hat r} dS = 0$. $g(r) \int \mathbf{\hat r} \cdot \mathbf{\hat r} dS = 0$ Thus $g(r) = 0$. Source: I am a physicist. – Obie 2.0 Mar 28 '16 at 20:44
• You are correct in thinking that the piece of sphere pulling directly opposite the piece our test mass is sitting on exerts very little force. But that is not the only source of opposing force. The piece right next to the location of the test mass will pull very slightly "up". A much smaller component of this force is up, but on the other hand it is also much closer to the test mass. In fact, all the pieces of the sphere besides the portion right under the test mass must be pulling "up," if only slightly. – Obie 2.0 Mar 28 '16 at 20:54

I have some thoughts for your questions, which I'll address in series. But the tl;dr is that it's definitely possible to figure it out.

1. The inner-facing habitable surface. I hope this works by anti-gravity generators rather than rotation, because otherwise it will have weird atmospheric effects. The simplest way to figure out that you're on a sphere is to just look at the part of the sky that isn't star. The night that I'm imagining is like the day/night cycle in Ringworld. However you do night time, if it blocks out light from the star, then the sky should be mostly black. This should allow the civilization to see other parts of the inside of the sphere at essentially the very beginning of life, in particular because the other parts might still be bright. How long it takes them to figure out that it means they're on the inside of a sphere is another matter, and getting the size of it is something else entirely. The size would require a reference point quite a distance away, which might require the civilization to be into the Europe-naval-colonization era from earth's history. Depending on some atmospheric calculations that I can't do at the moment, they might be able to figure it out as soon as they see ships move "up" the horizon, but that might not be visible (ie. the ship could be too small).

A small addendum: if they were on a plane, they could use shadows between cities to figure out the distance between them and the star. The fact that they can't either means that the star is infinitely far away, or they are in a dyson sphere. If they can figure out the distance to the star (ie. by more advanced techniques like time between solar flares / northern lights & using some particle physics) then they can figure out the radius of the sphere too.

2. The outer facing habitable surface. Assuming this has the fairly easy gravity specifications that you stated. Let's suppose there are some stars orbiting the sphere in a ring. We can label that ring east-west. The first thing that the civilization may notice is that the stars are (I hope) not all perfectly identical. If one is different, then you can use it to figure out how long it takes them to orbit around the sphere. If the stars are relatively close to the surface of the sphere, then the habitants can use a surface-plane approximation + parallax (ie. shadows) to figure out how high up they are. With that, you can easily get speed from more shadow / parallax tricks, and combining that with the time to orbit gives you the circumference of the sphere (and radius). If the stars are close, this can all be done well before you have a telescope.

If the stars are far away, it's a little trickier. When you travel east and west, you'll notice that the time gets a little bit off (as you find exactly the same as the earth) except the same difference requires more distance. If you can measure this, and have good enough telescopes to see ships disappearing below the horizon, and can do any shadow measurements, then you can get the size of sphere as well as the distances to the outer suns.

3. These methods all work for other mega-structures just as well. With them though, you get other benefits that make things easier like background stars at night being visible sometimes and sometimes occluded, or encountering the edge of the structure.

Of course, once you get to space travel all of these calculations can be nullified by direct observation.

• "...see other parts of the inside of the sphere at essentially the very beginning of life" // That makes no sense. Normal movement by animals spreads life faster than that, and wind blows spores and seeds even faster, even on a world millions of miles in diameter. – Lensman Mar 29 '16 at 7:52
• "...they could use shadows between cities to figure out the distance between them and the star. The fact that they can't either means that the star is infinitely far away, or they are in a dyson sphere." // No... all they can tell is that the sun is always directly overhead, and doesn't appear to move. They have no frame of reference to compare that to; they've never seen stars, or a sunrise or sunset, or lived on a normal planet. – Lensman Mar 29 '16 at 7:52
• @Lensman About the second point: they will know how simple ray optics work. They can figure out that light from a single source that always appears directly overhead is one of the cases that I said. This is basic geometry. About your first point, I don't see how that's relevant. Regardless of how fast life spreads, if you can see bright pieces of land in the sky then you might figure out you're in a sphere. – Lacklub Mar 29 '16 at 12:23
• @Lacklub If someone brought a glass bulb with air and wood to the moon, and lit a fire - how well would you be able to see that fire from the surface of the Earth. The other side of the sphere is far further away. Details on the other side will be completely invisible for the naked eye. – Taemyr Mar 29 '16 at 16:09
• @Taemyr Yes, but the other side is also very large. You don't need to see details (I think) to figure out some simple ideas. Once you get more advanced, you're going to even be able to do spectroscopy on the other side. But before then, a lot depends on things like day/night cycles and the characteristic size of features. If the oceans are the same size as the ones on earth, then it'll just be a blur. But oceans could be the size of the sun. There might be panels that give shade at night, rotating around the sphere. These variables make things easier, and would both be identifiable features. – Lacklub Mar 29 '16 at 18:13

Early. Antique age, or earlier.

The fact that Earth was round, not flat, was well-known and accepted in Greece during the 4th century BC (thanks to the Christians knowing the truth, namely Earth being flat, it later took almost another two thousand years before this was rediscovered).

Now, assuming your Dyson spere existed, the sun would be at the zenith, and would never move (from the observer's point of view) no matter where on the sphere you go. This follows from the fact that the sun is in the center of the sphere.
It does not take a lot to realize that an object viewed from different locations on a plane appears at different angles, and that the sun always staying at zenith is only possible if "the world" is the inside of a sphere.

To anyone who is approximately at the development level of Aristotle or Thales, it would be an immediately obvious fact.

• I don't understand the downvotes. This seems to be a perfectly reasonable answer. Can someone explain in what way this is wrong? – Tracy Cramer Mar 29 '16 at 18:32
• Doesn't the lack of (measurable) parallax just tell them that either their world curves up OR the sun is 'really far' away? Traveling at the speed of a trireme and using classical Greek instruments, how would they untangle those two? – Spike0xff Mar 29 '16 at 19:29
• @TracyCramer The problem, like most of the other answers, is scale. The surface will appear as a flat plane. Their sun being in a fixed position overhead can be explained because it's very large and very far away. On the Earth, 1 degree of latitude is 111 km. On a 3 AU sphere, 1 degree of latitude is 7.5 million km. That's a daunting distance even today. Observers can't get enough parallax to prove otherwise. – Schwern Mar 29 '16 at 19:45
• It would be obvious to someone with our knowledge at that tech level. The people living there who don't know about gravity and the like won't reach this conclusion, though. – Loren Pechtel Mar 30 '16 at 1:27
• @Schwern 7.5 million km isn't just daunting. I'm betting most people never even get close to that. Personally, I'm traveling about 15000 km per year by car and maybe the same using other transportation methods; so a total of about 30k km, rounding it up to 40k just in case I'm underestimating non-car transportation. At that speed, it would take me 187.5 years to reach 7.5 million km! Pretty much only people who travel a lot as part of the work will even get close to 7.5 million km in their lifetime. – Clearer Jun 7 '17 at 11:10