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Originally Posted by Phage
...you would eventually burst into flame and your observations would end.
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How would you burst into flames? That would imply that energy is being added to the system. From what you described, you have only made it so that energy does not become depleted by imperfections in the mirror(s).
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Originally Posted by Phage
If you moved your head you might see interesting things, such as images of previous moments in the sphere that were still bouncing around, but most likely they would be smeared beyond recognition.
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Is the light bouncing directly back and forth between two points, or randomly? That would make a difference, as with the former you would have to be in the path of the light in order to see it (I'm assuming the light isn't being reflected by air particles which would ruin the "perfect mirror" setup).