TITLE: The view from inside of a mirrored tetrahedron
QUESTION [50 upvotes]: 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.

REPLY [4 votes]: I don't know why this question got bumped to the front page, but since it did, I might as well answer that I made a movie of this a long time ago, showing the view inside a (Euclidean) regular tetrahedron with reflective faces (tinted red, green, blue and white, and intrinsically luminous so we can see something!), as well as what it's like to move along a straight line "through" the mirrors (equivalently, a billiard ball path, but with the view flipped at each reflection so that we seem to go through the mirror).
In comparison, this video, made by my friend Vincent Nesme¹, shows a rendering of the 600-cell as a regular tiling of the constantly curved 3-sphere by 600 (spherical!) regular tetrahedra, and moving in a straight line (great circle) in this geometry.

I had previously made such a video by using gnomonic projection to map the lines into Euclidean space and feeding the result into POVray, but while this correctly projects the lines, it does not correctly project their thickness.  Vincent was much more patient than I was and wrote his own raytracer for this.