Image to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material.

In such structure some light can enter, and an observer inside it (e.g. located at the center of the sphere) could be able to see the reflections of himself on the internal side of the shell.

Let us suppose, additionally, that the space surrounding that structure is virtually empty, in such a way that there is not much overlap between the internal and the external reflections on the surface of the mirror.

What would an observer see inside such spherical mirror?

I tried to approach this problem with elementary Euclidean tools, but I struggle to find any reasonable answer.

Sorry for the naivety, and thanks for your suggestions!

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    $\begingroup$ Here's a nice little experiment you can do. Grab a reflective spoon or some other concave object, and hold your finger at a reasonable distance from the inside. You should see your finger reflected upside down (if not maybe you have to flip the spoon around, or hold your finger at a bigger distance). Now slowly move your finger closer to the spoon and observe what happens when you approach its focal point. $\endgroup$ – Peter Jul 28 '18 at 20:46
  • $\begingroup$ @Peter Thanks Peter, nice suggestion. But everything would change, I guess, if you are surrounded by such spoon. I would say that this problem is quantitatively related to the black body problem, which is way more complex than the experiment you mentioned, although interesting and pertinent. $\endgroup$ – user559615 Jul 28 '18 at 20:52
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    $\begingroup$ Doesn't vsauce have a video answering this exact question? $\endgroup$ – D. Brogan Jul 28 '18 at 21:00
  • $\begingroup$ @D.Brogan I didn't know vsauce, but I've just watched the video you mentioned, thanks! Very entertaining, but I am looking for a more quantitative analysis or, at least, for an idea how to mathematically approach the problem. $\endgroup$ – user559615 Jul 28 '18 at 21:16
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    $\begingroup$ You can find simulations on the web. youtube.com/watch?v=zdSC6GwNhN8 $\endgroup$ – Yves Daoust Jul 28 '18 at 21:48

The BEST Possible answer to your question: Vsauce-Inside a spherical mirror


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