Let’s look at this from a thermodynamics perspective. Energy incident on the surface will equal to the energy output of the surface. There will be energy absorbed, energy radiated, energy transmitted, and energy reflected.
Energy transmitted: this is the “clearness” of the surface.
Energy absorbed: this is the color of the surface. Actually, it’s the inverse of the color of the surface.
Energy radiated: this is not any energy generated from the surface, it’s the energy that’s been absorbed that has been released in a different form. From our perspective, it’s photons in the infrared, so we can pretty much forget about them, and pretend that energy absorbed by the surface goes into a black hole or something.
Energy reflected: this is the meat of the problem. For this, we need the BRDF, the bidirectional reflectance distribution function.[sup]1[/sup] This is a mapping, for any given point on the surface, and given incident direction of an energy source, and the direction from that point to an observer, what is the percentage chance of that energy being transmitted to the observer. Technically, we should also take into account the frequency of the energy and the polarization, but for most surfaces we can throw the former into the absorbtion and ignore the latter.
So, for this perfectly shiny object, all energy incident on the surface is perfectly reflected. No light is absorbed, and all incoming light coming in from any given angle goes out in exactly one other angle, with the angle of incidence – the angle between the incoming ray and the surface’s normal (the “up” direction at that point) – equal to the angle of reflectance between the surface’s normal and the outgoing ray. For this perfect room, the walls are exactly the same color at all points from all directions, including the direction that the observer is in, because of some strange one-way surface’s properties.
So, for this setup, if you trace a ray from the observer to a point on the sphere, and then trace the ray from that point to the point on the wall that would be seen by that observer, then the color that the observer sees in the direction of the point on the sphere is exactly the same color as the point on the wall viewed directly.
This means that, since every point on a wall is exactly the same color, and all six walls’ colors are identical, so that an observer looking at any point on the sphere will see the same color as if the sphere wasn’t there at all. So, looking through the one-way wall they will see this vast field of a single color, with no apparent floor, ceiling, or walls.
[sup]1[/sup]Some people call this the BDRF, "bi-directional reflectance function. Just ignore them, they’re wrong.