The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For example, here is what happens for a random polygon of $8$ vertices and $\alpha=\frac{1}{4}$:
          

Wikipedia: "the derived polygons ... approach a shape with opposite sides parallel and equal in length, and all have the same centroid."

Now imagine generalizing this to polyhedra, let's say, convex polyhedra in $\mathbb{R}^3$ with triangular faces. Start with a polyhedron $P$ of $F$ faces and $V$ vertices. For a given triple of positive reals $(\alpha,\beta,\gamma)$ that sum to $1$, locate a point $p$ in each face by weighting the face's three vertices by $(\alpha,\beta,\gamma)$. (So if $\alpha=\beta=\gamma=\frac{1}{3}$, $p$ is the centroid of the face.) Now replace the original polyhedron $P$ by the convex hull of the new points. Here is an example with $(\frac{1}{4},\frac{1}{2},\frac{1}{4})$:
          
You can see what happens. At first $V=8$ and $F=10$, but next $V=10$ (because every face generates a point), and by Euler's formula, $F=16$ ($F=2 V-4$). Etc.: the number of vertices and faces grows. Here are enlargements of the first and 10th polyhedra, the latter of which has $1540$ vertices:
 
It seems that, despite the combinatorial growth in $\mathbb{R}^3$ absent in $\mathbb{R}^2$, it is likely the shape is converging to a limit that may bear similarities to what can be established in 2D. My question is:

Q. Has this process been studied before? If so, does the shape approach a limit largely independent of $(\alpha,\beta,\gamma)$, analogous to the situation in 2D? If so, what characteristics does this limit shape bear to the initial polyhedron?

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For example, here is what happens for a random polygon of $8$ vertices and $\alpha=\frac{1}{4}$ (successive images rescaled):
          

MathWorld: "the derived polygons ... approach a shape with opposite sides parallel and equal in length, and all have the same centroid."

Now imagine generalizing this to polyhedra, let's say, convex polyhedra in $\mathbb{R}^3$ with triangular faces. Start with a polyhedron $P$ of $F$ faces and $V$ vertices. For a given triple of positive reals $(\alpha,\beta,\gamma)$ that sum to $1$, locate a point $p$ in each face by weighting the face's three vertices by $(\alpha,\beta,\gamma)$. (So if $\alpha=\beta=\gamma=\frac{1}{3}$, $p$ is the centroid of the face.) Now replace the original polyhedron $P$ by the convex hull of the new points. Here is an example with $(\frac{1}{4},\frac{1}{2},\frac{1}{4})$ (again, each image rescaled):
          
You can see what happens. At first $V=8$ and $F=10$, but next $V=10$ (because every face generates a point), and by Euler's formula, $F=16$ ($F=2 V-4$). Etc.: the number of vertices and faces grows. Here are enlargements of the first and 10th polyhedra, the latter of which has $1540$ vertices:
 
It seems that, despite the combinatorial growth in $\mathbb{R}^3$ absent in $\mathbb{R}^2$, it is likely the shape is converging to a limit that may bear similarities to what can be established in 2D. My question is:

Q. Has this process been studied before? If so, does the shape approach a limit largely independent of $(\alpha,\beta,\gamma)$, analogous to the situation in 2D? If so, what characteristics does this limit shape bear to the initial polyhedron?