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This is my first question in this site, so please don't bo too hostile to me. If I made any mistakes, please let me know.

In my story, there is an island inhabited by people, numbering enough to sustain reproduction. There are continents (and other islands) on the planet, but something prevents them from exiting the island ("exiting the island" here is defined as becoming further than a kilometre from the nearest point of the island).

What is the minimum area of the island such that it is able to sustain technology advance?

It's better if you can provide miminum sizes for specific technology levels. I think resource availability is the main problem.

Edit:

The people in this scenario had inhabited (mostly) the entire planet (like humans do today), but someday they were attacked by an imperialist nation. All (or virtually all) the other nations formed an alliance, but despite that the empire which managed to conquer the world. Most of them (the alliance people) died and the rest (not more than a million) was forcibly moved to the island by the imperials (so they didn't destroy themselves with technology). They didn't have time to bring the technology itself but the scientists managed to arrange a "tech tree".

So just consider they can't exit the island because if they do, the imperial ships will destroy the people which exit. The empire doesn't send people to check directly though, it's part of the story so please don't ask.

The tech tree is shaped like a map (in paper form, of course). There is a book which accompanies the tech tree, containing some information about the technology. Therefore, they don't have to figure out the next step in the technological advance. And yes, safeguarding the tech tree and the book and creating backups are (semi-?)major plot elements.

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Well, the first thing you need to worry about is sustaining life, that means farms, agriculture, and given trade will be impossible(cannot leave island), you need to make enough to have a buffer for bad times.

So, what was the basic rule? An acre of farmland per person. And that's just survival. Given trade is impossible, you may want to invest more in backups--so I'll go with an acre and a half per person.

So, now we have the basic calculation of how much per person. How many people are needed to ensure inbreeding doesn't become a problem? That's a bit more problematic, given it's more about monitoring who makes babies with who. But as a rule of thumb, let's go with an upward of 3,000 people (check around on this site, I'm possitive I remember a question that deals with that, something about a generational mothership meant to find a new habitable planet)

So, 1.5 acres per person. 3,000 people. 4,500 acres just to sustain the people. Now, you need resources beyond survival. You need building materials, you need metal ore, you need industrial terrain to process this, and you need space to train people to do so. You also need government to guide and govern the people, and you need space to do research. (yeah, this isn't quite as simple as your originally worded question makes it)

So, given resources are limited, you cannot just throw anything away. Once someone finds a mining hotspot, and you start mining for your metals, you're going to have a LOT of useless dirt and rock and slag and other junk you have no use for. Find a use, or your people will suffer in the long run. Use it to build houses, for all I care. But given you need a metaphorical global economy on an island? Waste is your enemy.

And that includes the waste of the people! Both sewage and garbage, you'd better have something to combat that, or your people will be facing their own Black Death. Avoid that at all costs. Doubling the fact that these people will be dependant on fisheries, unless there are enough farm animals for meat and the like, you'll need them to be super careful with overfishing (given the limited range of where they can fish). That means they cannot pollute the water, and they cannot chase away the fish they need to come back. This might sound like a joke, but check Moana for island-locked life and what it takes to ruin that forever. Nothing can or should go to waste. Nothing.

So, with this additional information, we need a basic idea of how much space for the extra needs. Let's round it off at 3 acres per person, for all the entertainment venues, government, research facilities, and the like. Keep in mind that once technology starts becoming more predominant they can start building up. Or just start building smarter! Harappan civilization might help you there. They were excellent builders, and city planning seems to be at the forefront of their planning.

Combine all this, and you'll have a civilization that might go the distance. But you're not likely to get any further than European middle ages, simply for their lack of trade (personal opinion, and treat that as such).

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  • $\begingroup$ Unless this is a gift from the gods, or a civilization building game, that's unrealistic. You don't start off knowing where you're going, not in real life. You start with a question, and you use a means to find an answer. Sometimes you find something, sometimes you don't. Sometimes the question doesn't have an answer, but along the way you find answers for questions you didn't ask yet. Hope this helps $\endgroup$ – Fayth85 Mar 5 '17 at 14:11
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    $\begingroup$ You're solving a problem with a problem. Why this island? What caused their deaths? Why were they able to get to this island, but are now unable to leave? Furthermore, if they kept the schematics, did they keep the theory behind how it works? Are there any engineers, scientists, doctors, etc still alive to keep the people on the right track? Or are we facing an post-apocalyptic scenario where you should just be grateful to be alive? These are all factors that can alter the playing field--not to mention the plausible PTSD your people will face on top of everything else they have going on. $\endgroup$ – Fayth85 Mar 5 '17 at 14:25
  • $\begingroup$ You might also want to put some thought into what technological level they were at the height of their previous civilization, and what effect this had on their downfall. There might be a group within the survivors that develop techno-phobia, just look at the world of Spira in Final Fantasy X, and how that affected them. $\endgroup$ – Fayth85 Mar 5 '17 at 14:27
  • $\begingroup$ @user_194421 and please don't forget to factor in that people need to the time to work on this. Meaning they cannot be working the land to be able to eat, and still have time to ponder how a steam engine work. Nobles were always considered the thinkers of their time, because they had servants handling the day-to-day stuff. $\endgroup$ – Fayth85 Mar 5 '17 at 14:36
  • $\begingroup$ Population can grow right? After a while you will have many people which don't have to work the fields or fish. And yes, there are engineers, scientists, doctors etc. among the survivors; after they got the farm established they arranged the tech tree along with some of the concepts needed to achieve technology in the tree. $\endgroup$ – user_194421 Mar 5 '17 at 23:00
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Smallest real-life examples

This is a list of mostly isolated 'islands' and how far along they got technologically. It is relevant that we don't ultimately know how far these islands would have progressed if those silly Europeans hadn't shown up with their guns, germs, and steel and made everyone wear pants. But this post isn't about speculation, its about history.

Sustained hunter-gatherers:

  • Chatham Island (966 km$^2$) (smallest that I could find, there may be smaller)
  • The tragic story of the Moriori people, residents of Chatham island, is that they emigrated there from New Zealand in ~1500 AD. Since their crops wouldn't grow in the sandy soil they reverted to hunter-gatherers and lived in peace for the most part. 300 years later, a group of their Maori cousins from New Zealand decided to conquer the island. They kidnapped the first mate of a British brig, and made the ship take 900 of them to Chatham Island where they proceeded to massacre and enslave the ~2000 Moriori. In any case, this is an example of an island being small enough to enforce a hard limit on advancement. Easter Island is a similar case.

Sustained pre-metalworking chiefdoms:

  • Polynesia (~30,000 km$^2$)
  • I'm not counting New Zealand, since the chiefdoms had already developed on Tonga and the Societies before New Zealand was settled around 1280.

Sustained bronze-age culture:

  • The South America (17,840,000 km$^2$)
  • Of course, South America is connected to North America. But it is interesting to note how much North America brought to the table: maize. That's it. Pretty much all the rest of the developments in agriculture, husbandry, and metalworking came from South America, along with the most advanced construction and government systems. North America (the Maya) did make the first/only writing system, though there is some debate as to whether quipu is a writing system or not. In conclusion, South America probably would have gotten as advanced as it did by itself.

Sustained Age of Discovery culture:

  • Eurasia (54,759,000 km$^2$)
  • Again, while Africa is connected, and North Africa and Egypt were an integral part of technological development, I think it is a fair assumption to say that most agriculture and technology would have advanced at about the same rate if Africa wasn't around. It is also reasonable to assume that Eurasia could have made it to modern technology without the rest of the world, as well, though perhaps not as quickly.
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The most important factor for technological advancement is the understanding of underlying principles. You might have some advanced technology, but if you don't understand how to replicate, change and improve it, it's not much more than a gimmick.
The Romans had 'steam engines', but they had no useful understanding of air so it wasn't until we understood thermodynamics before we could design steam engines that were actually useful.

For understanding you need a lot of people that have nothing better to do than trying to figure out how stuff works. (If you already have a blue-print, depending on how well it teaches your the understanding and how well you can read it, that's a lot easier.) The speed of your technological advancement scales with the number of people 'doing science'. Which, if that's all you want to do, is the biggest number of people the society can afford to not produce food.

If you advance your technology, it'll require more education to reach the current level of understanding, so overhead (schools etc.) will become a bigger factor. On the flip side, many of the new technologies improve the productivity of the food producing people so you'll need fewer of those.

An island large enough to sustain technological advancement: one large enough to sustain a population. Bigger just means faster.

As per tech level: assuming you've figured out agriculture, until well into the iron age I don't see why an island of a few km in diameter won't suffice, given some luck in available iron deposits near the surface. Then with just iron and trace amounts of other minerals (copper, tin, etc.) you can develop well into the renaissance. Charcoal from wood can replace coal. Again, more area means more trees means faster development.

For more modern technologies you'll also need other more rare elements. Story-wise this is easy to fix, just have the island be a (former) volcanic island with plenty of metal and rare earth element veins. I am no geologist, but I would deem it somewhat plausible that an island the size of, say, Ireland could have enough minerals for more modern technologies.

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First of all, You should be branching out onto the waters making more space for your island anyways.
As you said, resource availability is the problem. For technological advance, you're going to need all sorts of resources; An island on its own cannot sustain itself.
If you are able to get the resources from an alternate place, then disregard the above. I think you'll need to define your technological advance for me; Are you building an underground bunker with a new type of nuclear bomb that takes up three entire city blocks? Are you advancing your tech towards some sort of reactor that only takes up a handfull of space?
A small island could be viable depending on the size or type of technology you're building; Do you want specific area's of the island? 300*300[]'d? It's really not easy to say.
Is your technology small or large? What kind of technology are you looking to create?
etc etc Military tech:
I can't imagine you would need that much space for building a new type of tank. If you have a place that you can use to create things, then it should be easy to build new tanks, new firearms, planes, etcetera.
A small island should be sufficient. The island itself could be any size, but the thing you need to worry about is resources. You can build off of the sides of the island no matter what size the island itself is and make more room and space for building.

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  • $\begingroup$ Not only military, but its focus is military. I think industrialisation (and electricity) must be researched (to mass-produce weapons and the like) so not pure military. $\endgroup$ – user_194421 Mar 5 '17 at 12:15
  • $\begingroup$ []'d area squared. $\endgroup$ – Nate Dukes Mar 5 '17 at 13:08

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