Would it be technically feasible to build a “Minifig-Scale” Death Star?
How large would it be? Would it collapse under its own weight? What engineering challenges would need to be overcome?
As you might understand, the project would be unfathomably-coo-coo-for-cocoa-puffs-insane.
But I have a solution that takes us down a few notches to "stupidly ridiculous".
Bear with me, this may take a bit to explain. Let's dig in.
First, let's try and give it the best shot by making it as small as possible. The 1st Death Star was 120 km in diameter (with some sources stating higher numbers). And minifig scale (being variable) usually has a maximum value of about 1:44 (often it's around 1:38). That gives us a diameter of about 2,727.27 meters, or about 340,909 LEGO studs.
Just to give you an idea of how big that is, here's a rough scale comparison:
Would it collapse under its own weight? Yes and no.
A 2x2 LEGO brick can withstand a weight of about 375,000 bricks stacked on top of it, which is about 3,600 meters tall.
So... we're under that height, hooray! ... Except a few problems. First off, as shown in the image, the weight isn't evenly distributed-- it's all pushing down on that center point. In order to support it, we'd have to build a gigantic "bowl" to support the weight if it's on Earth. That'd prevent it from immediatelycollapsing.
But... there are still problems. Back in 2002, LEGO released a very successful collector-sized model of the Star Destroyer. It was pretty big.
I built mine and left it assembled for about 6 months or more. Until one fateful day, when I heard a report from someone online that had just taken their model apart. Apparently, the stress of the bricks started to warp the pieces, causing some of the internal support structure of the model to bend. I immediately took mine apart and noticed the same thing (you can see it here a little bit):
Uh oh... That means our Death Star won't collapse immediately to the point of the plastic failing... BUT... the plastic will warp if we let it sit long enough. And given the amount of weight involved, it'll probably happen very quickly-- within a matter of minutes or hours. It would probably start to sag right away, and then gradually moosh the bottom bricks.
That means we either have to build it very fast, or we have to give up the idea of building it on the Earth's surface like a normal creation. As other answerers have suggested, building it in outer space is great way to make it survive collapsing under its own weight.
Alright. Well, how much LEGO is this going to take? If it were solid 2x4 bricks, we'd need in the ballpark of 2,160,940,000,000,000 (2.1 quadrillion). But we don't need that much, because it'll be filled with hollow sections for the crew quarters, hangar bays, and other things. What's a good guess as to how much we can cut out? Probably quite a bit. I'd guess around 3/4 of it will be empty space.
Great! That gets us down to 540,235,000,000,000 bricks (540 trillion). Wait, how much do we have already? As of January 2015, LEGO reported that it had made about 760 billion bricks-- and are most likely a little over 800 billion now-- in the ballpark of around 820 billion. So... we're 1 / 658th of the way there? Ouch.
Well, that's not ... totally... insurmountable. It's pretty reasonable to assume that we have sufficient plastic on Earth to make enough bricks. They're making about 60 billion elements per year right now, which means it would take about 9,004 years at the current rate of production. But global output of plastic (not just ABS, unfortunately), is around 300 million tons. Assuming it's all 2x4 bricks, that's 117,308,000,000,000 bricks per year as a planet (117 trillion). I'll bet if we invested gobs of resources in injection molding, we could probably make it happen in under 10 years.
Alright. So... this whole putting it in space thing. That's where we cross the boundary from ludicrous into gob-smackingly outrageously insane. Taking things into space is horrifically expensive. It's around $10,000 per pound to take things into space right now, which means about $2.76 trillion for our model. And that's just the LEGO, let alone the workers, infrastructure, and everything else. This is going to take years to build, and those space-workers are going to need food, water, air, clothing, living space, heat, and huge amounts more.
As Larry Pieniazek points out in his answer, we need to build an entire infrastructure of building things in space, which we may not even have the technology to do yet. Plus, it'll take many, many years. Maybe 50-100 years just to build the infrastructure, let alone our lovely model.
So, we have to do better. My solution? It's in Oregon.
This is Crater Lake:
It's one of the deepest lakes in the world, and it could totally fit the minifig-scale Death Star horizontally:
That's right, we're going to build it underwater.
This is great, because LEGO is practically weightless underwater. The density of ABS plastic is in the ballpark of 1.05-1.30, versus water which is about 1. That means the plastic a lot lighter, and won't be under nearly as much load. Also, while the pressure at a depth of nearly 3km would crush a brick filled with air, it'll hardly do anything to the ABS with no air pockets. It should be fine.
We could build it in the ocean, but that would be a problem, because the constant currents and tides would rip it to shreds. But a lake? Much less of an issue.
The only problem is that lakes aren't as deep. Crater lake? It's a pathetic 593 meters deep. But since it's way up on a high elevation, we should be ok. We've got plenty of room to dig it deeper, right where we need it.
So, we're going to have to dig. A lot. And since the dig site is submerged, it'll be more expensive, because we'll have to get underwater equipment. It might actually be feasible to drain that area of the lake before digging to make digging cheaper, but I'm guessing that effort would actually be more expensive, since we'd have to build retaining walls to keep the water out, which are pretty pricey. We could also drain the whole lake, but since we're going to want to put the water back in when we're done, that also would add to the cost. Hence, my vote is to just dig underwater.
We're still going to dig in a bowl shape, though, so that the bottom layers of the LEGO Death Star have something to distribute the weight against more evenly. It may be a lot lighter underwater, but it's still a LOT of weight all told.
Ok. It's time to build it. It's difficult to assemble with robots, but at that depth, we're not going to have much choice. Divers won't be able to take it, so we'll need some sort of machines to do our building.
But it's cheaper (and easier) to do it by hand, so we'll divide the work. Large sections will be built by hand, and then added on to the main structure by robotic arms in the depths. Each section also has to be submerged during building to avoid air bubbles (which would otherwise crush the bricks), and then get transported down to the build site by submersibles that can attach them to the larger structure.
This will take a long time, and a lot of effort. Building sets (in normal air!) takes me around 4.2 seconds per piece when assembling normal sets. That might sound a little slow (and I'm a reasonably quick LEGO builder), but most of that time is spent hunting for pieces and following instructions. We'll probably have a lot of sections that builders can memorize, and have pieces very well laid out for our builders. But let's go with 4.2 seconds per piece anyway, because I'm a fast builder, and they're also building with their hands underwater.
Alright... that's... 71,899,859 years of labor for 1 person, working around the clock. Wow. If we're optimistic, and want the construction done in about 10 years, we need a workforce of about 14 million people working 12 hour days. At $10 / hour, that's about $6.3 trillion. Wow. But it's WAY better than spending $20 trillion getting them into space (not to mention their food)!
So. My completely hand-wavey-guess is this is going to cost us in the $15-$30 trillion ballpark, and will take us about 20-30 years. And when it's done, we'll have a model that nobody can see, given that it's totally submerged. But mission accomplished!