And if you read the MIT report, they were able to ignite a wooden “boat” with only 127 mirrors. The set up in the short story has enormous redundancy.
BUT you need a very clear day. It doesn’t take much cloud to disperse the light enough that you can’t focus it.
All you are trying to achieve and all you do achieve is being able to signal a spot miles away. Not hold your mirror on the spot perhaps a hundred yards away for an extended period.
There are two critical differences. Firstly, to achieve your aim all you need to do is flash the viewer, even for a millisecond. Not hold your spot on the person for an extended period. At a range of a hundred yards, the difference between having your spot on someone and not having your spot someone is about one quarter of a degree of arc. That’s tiny.
Secondly, the beam of light from your mirror spreads as it goes. This means that because – as you say – your viewer is miles away, the beam of light from your mirror is widely spread, such that it is quite large at the point reviewer’s location.
These two factors allow you to be extremely inaccurate while achieving your goal of simply signalling the viewer.
Further in the stadium example and your claim that redundancy will be sufficient - firstly your spot is going to be very badly focussed and area (and consequently diffusion) increases with the square. So if MIT achieved a 1 yard square spot, and your 10,000 can achieve a 10x10 yard spot (doubtful, I think) that would be 1/100th the heat per square yard.
And secondly, the holders of the mirrors are going to be substantially further away than in the MIT experiment. That involved a distance of about 50 yards. Think about the size of the stadium. To fit 10,000, all on the same side in the sun, most are going to be double that. And the light being reflected is going to diffuse with the square of distance. This drops your effect by at least a factor of 4.
300x100x4 is 120,000. 10,000 isn’t enough, and if you did have 120,000 they would have to be even further away.
I don’t think they would hold focus such that every one was within a 10x10 yard spot, i think what you’d get would be a cluster, with some spots much farther away, but a lot of spots close together at any moment.
And it doesn’t take that much concentration of sunlight to get pretty hot. Surely you started fires with a small magnifying glass as a child.
I couldn’t find the myth busters episode, but MIT seems to have lit two different boats on fire, in two different experiments. So I’m dubious about whatever myth busters did.
Close together being not very close together since there is no way to aim accurately. Once your spot is on top of an area with other spots, there is no way to tell where your spot is.
The inverse square law is a bitch. The available heat from a reflector at say 6in is 360,000 times less at 100 yards. 100 yards is 3600 inches which is 600x as far and you need to square that, which is 360,000.
MIT worked with MythBusters the second time. The point is not that it is impossible to do what the MythBusters (and MIT) did. It is that to achieve combustion, they had to alter the experimental conditions such that they were had no real world practical relevance. The only way they could get it to work was over a (relatively) short distance, on a boat that was perfectly still, and with serial aiming of mirrors, so they could tell where their spot was.
None of these conditions would be practical in the stadium example.
Well yes, if he’s looking at the stands. I’m not getting what you are trying to say here. So even assuming the ref is willing to stand in one spot long enough, I doubt even an acre of sunlight focused onto an area of 1 sq. ft. is going to instantly vaporize a referee but I think it would severely burn him in only a second. Setting his hair and clothes on fire, blinding him if he’s looking at it, and leaving him with severe burns on his skin. Continue for a few more seconds and he’d be dead. What I suggest is using a large portion of the people on one side of the stadium to reflect light onto a smaller area of mirrors across the stadium, and that smaller area is better able to focus that light on the ref. Let’s say 43,000 people have 1 sq. ft. flat reflectors on one side. They all have to focus now on a group of 560 mirrors on the other side. And let’s also say those 560 mirrors can be held in a tight formation, maybe a circle a little less than 30 feet in diameter. That smaller group could be better at focusing they’re light on a ref standing still in the middle of the field. The primary mirrors only have to reflect sunlight somewhere on the smaller secondary group of mirrors. If that smaller group, somehow can focus the light on the ref, maybe even using parabolic reflectors that will narrow the light at the specific distance to the referee, then I think if the ref is lucky he’ll be dead before he has a chance to regret pissing off the fans in the first place.
OK, let me try again. As you look around in all directions, your total field of view takes up 41,253 square degrees. If you go outside on a normal sunny day, 0.2 square degrees of that field of view will be taken up by a 6000 K body (the Sun), while all the rest of it is cold. That leads to an equilibrium temperature of a few hundred K.
With mirrors, lenses, or the like, you can increase the angular area that’s taken up by that 6000 K body. And if you’re sloppy (low-quality mirrors, say), you can decrease the effective temperature of that body (let’s assume that your mirrors are high-quality, so that’s negligible). But what you can’t ever do is increase the temperature of that body.
If you increase the angular area of the Sun to the max, instead of taking up just 0.2 square degrees of the sky, it’ll take up all 41,253 square degrees. With the Sun filling up 41,253 square degrees, you’ll come to an equilibrium temperature of 6000 K, the same as the temperature of the surface of the Sun. Use a smaller mirror setup, and you’ll get some amount of sky in between 0.2 square degrees and 41,253 square degrees, and hence a temperature somewhere between a few hundred K and 6000 K.
Now, let’s go back to our referee on the field. Before the game, he goes out and looks at Section A of the stands, and measures how much angular area that section of the stands takes up. If everyone in Section A aims a mirror at the ref, they can make the entire Section A look like the surface of the Sun. That’s not 41,253 square degrees, but it’s still a lot more than 0.2 square degrees, so the ref’s location is going to get pretty hot.
But if it’s only Section A that’s aiming at the ref, then that’s the limit. It doesn’t matter if sections G, H, I, J, and K, on the other side of the stadium, are all aiming their mirrors at Section A, for A to redirect that light to the ref. It can’t get any worse than all of Section A being sun-hot.
I don’t think that will work. 43,000 people, reflecting light onto 560 mirrors, works out to about 60 each.
Suppose I’m one off those 560 people. 60 different people, in 60 different locations, are are reflecting sunlight toward a mirror that I am holding. Those 60 incoming beams are coming from 60 (slightly) different directions; they will reflect off my flat mirror and spread out in 60 different directions. With just a single mirror, I can’t focus those 60 incoming beams of light onto a single target.
Of course a single flat mirror won’t focus light at all, but multiple flat mirrors in a tight parabolic array will, but there could be difficulties with a practical shape to the secondary array for a focal point however many cubits half a soccer field away is. I also mentioned the secondary array could have parabolic reflectors based on the idea that a smaller group could be trained and coordinated to get their mirrors in the right position. The hypothetical is already based on the coordination of 50,001 people, 50,000 holding mirrors and 1 standing still while they attempt to burn him to death. Relative to that I don’t think I’m suggesting anything outlandish as a way to improve the focus of a manually aligned mirror array.
Because it’s a diffuse “point”, not a perfectly tight point, the one guy doesn’t have to stay in exactly the right spot. 
You are. It’s physically possible for a person to stand still, even while being burned alive. It’s not physically possible to concentrate light in the way that you’re describing. It violates the Second Law of Thermodynamics.
Not really. From the perspective of an (unfortunate) ant looking up at a 6" magnifying glass on a bright day - the sunlight isn’t any more focused or brighter than looking at any individual part of the sun normally - there just happens to be a lot more “sun” now - over almost the entire field of sky. This represents pretty much the absolute maximum that can be achieved for the ant- it doesn’t matter what sort of crazy mirror arrangement or parabolic system you have feeding in to the magnifying glass- you can’t get hotter, brighter, or more focused than “I see sun over the entire field of sky”.
Of course all this discussion is politely circling around the question of etendue.
An infuriating result of the laws of the universe, expressible in a whole host of ways, all of which conspire to deprive you of a free lunch.
For the purposes of the discussion here, it isn’t possible to ever attain a geometry of mirrors where the apparent brightness over any solid angle exceeds that of the source object - in this case - the sun. It doesn’t matter if you have a relay of mirrors apparently bouncing from one set of collectors to another. You can’t get there from here. The apparent brightness can only go one way, down. Never up. All you can do to increase the energy flow is make the apparent radiating area bigger. But there isn’t a way of doing this with a smaller area. The light will always escape off to where you didn’t want it to go.
You are trying to have it all ways. You think people can focus tightly enough to burn him, but you also say their aim will be diffuse enough that he can’t run away.
Yup. I think the amount of sunlight that hits the part of the arena full of soldiers is enormous. I think it is massively redundant to kill a man. So I don’t think they need precise aim.
(I also think it’s weird to worry about whether you can get as hot as the sun by concentrating sunlight. Of course not. But why would you even want to? What in the set-up makes you think you need to do that.)
Math is hard
Math is easy, a beautiful, and fun.
Took me a while to make my way through your post. I have been thrown off by your descriptions which in the end come down to the fact that there’s only one sun in the sky, multiple mirrors doesn’t turn it into multiple suns. What I have been talking about is not somehow concentrating more light than exists, but a way to manually focus it with a lot or mirrors. So in that regard you seem to be saying that it doesn’t matter how many mirrors actually focus the light on that poor ref as long as one of them is focusing all the available light on him. Is that what you’re attempting to explain?
Multiple mirrors can make the equivalent of multiple suns, in terms of angular area of the sky.
If you have a circular mirror that takes up the same angular 0.5-degree diameter as the sun (from the point of view of the ref), the ref essential sees a binary solar system with two identical sols - one from the sky and one from the same-sized mirror. But you can’t have a single 0.5-degree-sized parabolic mirror somehow make a “double intensity, super amplified sun” (reflecting both the light from the sun and bouncing the light from another mirror on the opposite side of the stadium, focused on the same target) as your hypothetical seems to imply.
You can increase your parabolic mirror size or # of mirrors to make the equivalent of a trinary or quaternary sol system, or keep going - filling in every available square foot of the stadium with mirrors until the ref’s entire view is one big giant sol across the entire sky. But if you only have one 0.5-degree angular diameter mirror focused on the ref, it doesn’t matter how many other mirrors you’re “feeding” it, you’re not going to exceed “one sol” of effect on the ref.
That’s not my point. I’m not saying reflecting off the other side of the stadium increases the intensity of light. I’m talking about how to manually focus sunlight on something referee sized, or smaller like referee head sized, with one parabolic reflector mirror that is formed from multiple smaller mirrors. Going across the stadium is simply a way to get the sunlight reflected from a large number of poorly positioned mirrors onto one accurately positioned mirror.
By doing so, you’re not achieving anything more than using the same accurately-positioned mirror setup to reflect the light directly from the sun to the ref, without the middle-man. Adding the additional mirror bounce across the stadium accomplishes zero.