Suppose you were standing inside a regular tetrahedron $T$ whose
internal face surfaces were perfect mirrors.
Let's assume $T$'s height is $3{\times}$ yours, so that your
eye is roughly at the centroid,
and that you look perpendicular to a face:
My question is:
Q. What do you see? Either qualitatively; or if anyone can find an image, that would be revealing.
Asking the same question for the view inside a mirrored cube is
easier to visualize: the opposing parallel square-face mirrors would produce
a "house of mirrors" effect (in three perpendicular directions).
(Image from this link.)
Addendum 1 (21Nov2014).
To respond to Yoav Kallus' (good) question, here is a quote from
Handbook of Dynamical Systems, Volume 1, Part 1.
(ed. B. Hasselblatt, A. Katok), 2010, p.194:
Addendum 2 (24Nov2014).
Now that we have Ryan Budney's amazing POV-ray images,
I would appreciate someone making an attempt to describe
his $32$-reflection image qualitatively.
I find it so complex that this may be an instance
where a thousand words might be superior to one picture.
Best Answer
Here are a couple pictures. If you'd like to do more, I created this with a simple povray script. Feel free to e-mail me and I'll send it to you.
With four reflections.
With 8 levels of reflections.
One with 24 reflections.
And one with 32 reflections, and each mirror having a slightly different tint, a red sphere, and a slightly wider viewing angle.
Here is the view from inside a mirrored tetrahedron, near one of the vertices, looking towards the opposite face. Near the centre of each face is a number indicating which vertex the triangle is opposite. The sequence shows the triangles becoming more reflective and less opaque. The field of view is 120 degrees so there is a little bit of lens distortion at the fringes of the image.
And here is the link to the scripts for the tetrahedra with the numbering and without the spheres.