TITLE: Tetrahedron insphere iteration
QUESTION [24 upvotes]: I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
     

Starting with any triangle $T$, one forms $T'$ by connecting the three points
of tangency of the circle inscribed inside $T$.
Does the analogous process for tetrahedra approach a regular tetrahedron?
And the same question may be asked for simplices in $\mathbb{R}^d$.
A reference would be appreciated—Thanks!

REPLY [11 votes]: There is likely a theorem here that, for almost every start tetrahedron (left column below),
very quickly the iterates form an approximate $2$-cycle. Below shows three random
tetrahedra and five iterates each (rescaled):


Below I plot the distance between the incenter and the circumcenter of $25$ random tetrahedra, as the process is iterated and rescaled at each step. This strongly supports
fedja's conjecture. If there are exceptions, they are not common.


Here is a sample of what the inscribed and circumscribed spheres look like (with red & green centers converging):