I didn’t say it was impossible, just that it makes more sense as orbiting plates (and that was Dyson’s original proposal).

I was going to try to figure out what material you’d have to even build it out of, to prove you wrong that it actually would be impossible, but it turns out building a solid shell actually isn’t completely nutty from a pure structural perspective.

If we have:

• r = radius from sun’s center to the shell
• t = thickness of the shell
• ρ = density of the shell material
• M = mass of the sun
• G = gravitational constant

Then the compressive stress (σ) the material needs to withstand would be:

σ = (GMρt)/(r²)

Because:

• Each small section of the shell experiences a force F = GMm/r²
• The mass of each section (m) is its volume times density: m = Atρ where A is area
• Dividing force by area gives us the stress: σ = F/A = GMtρ/r²

So long story short, you could build it out of plain structural steel (compressive yield strength around 250-500 megapascals) with a safety factor of 5, and you’d have:

• G = 6.67e-11 m³/kg·s²
• M = 2e30 kg (sun’s mass)
• ρ = 5000 kg/m³ (steel density)
• t = 1 m (reasonable shell thickness)
• σ = 50e6 Pa (allowable stress with safety factor)

Means:

• r = √((6.67e-11 * 2e30 * 5000 * 1)/(50e6))
• r ≈ 3.66e10 meters

So you could put it at 36.6 million km from the sun, about halfway to Mercury’s orbit, or closer if you used a stronger material, or any further distance away if for some reason you wanted to (maybe to increase your margin for error in dealing with overheating issues). However… at that distance, you would need 8.4e25 kg of iron, or 14 Earth masses. If you tried to get closer to the sun to minimize your materials cost, then you’d quickly run into increasing structural strength needed. The pure size is really going to limit what elements you’re going to be able to make it out of. I guess you could say that we’re doing it out of some carbon nanostructure material or something that doesn’t exist yet, or transmuting elements to build it, in which case there’s no point in doing the math because we can’t say anything about how strong or heavy it might be… if you tried to guess based on current carbon nanotubes, you could say the compressive strength is around 100-150 GPa, so you can put it 10 million km from the sun, and you need one-third of an Earth’s mass worth of carbon, which I guess is doable if we’re assuming the technology and resource extraction has reached that point.

Or, you could just do big orbiting plates with present-day technology, and make them as thin as you want, and not have to do it as a single massive project, and be able to correct their orbits individually instead of having a single leviathan that will cause a planet-scale catastrophe if it ever falls off its balanced position, and probably some other advantages I haven’t thought of.

TL;DR Use plates.