The answer “cubes with no empty space are filled cubes” was perfectly decent, as was the verbal argument about limits that Viliam provided. But in case those weren’t satisfying, I’ve written a more explicit version of the proof* that the ray travels zero distance between reaching the cube and reaching a sphere. It utilizes the solution to the geometric sum, which is proved using limits:
https://en.wikipedia.org/wiki/Geometric_series
I think you can just argue by symmetry that the ray must retro-reflect.
By the symmetry you can argue that the ray hasn’t reach the sphere.
Lumifer suggested something very similar as you, but this retro-reflection is equally possible as the retro-shadowing. Which do you prefer and why? Can’t be both.
The answer “cubes with no empty space are filled cubes” was perfectly decent, as was the verbal argument about limits that Viliam provided. But in case those weren’t satisfying, I’ve written a more explicit version of the proof* that the ray travels zero distance between reaching the cube and reaching a sphere. It utilizes the solution to the geometric sum, which is proved using limits: https://en.wikipedia.org/wiki/Geometric_series
The proof (derivation?) is here: http://imgur.com/vSfABHk
I think you can just argue by symmetry that the ray must retro-reflect.
*At least, it looks like a proof to a physicist. It may not meet the standards of a proof to a mathematician.
By the symmetry you can argue that the ray hasn’t reach the sphere.
Lumifer suggested something very similar as you, but this retro-reflection is equally possible as the retro-shadowing. Which do you prefer and why? Can’t be both.
Can you explain?