"Derived" polyhedra and polytopes

Question

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?

Answer 1

What you call the derived polygon is a construction that has appeared many times in literature. I believe the first occurrence was in a 1878 paper by Darboux, "Sur un problème de géométrie élémentaire", but it has also been a topic of many problems and articles in the American Mathematical Monthly, which is where I've learned about it. :-)

Not only is the limiting shape a polygon with opposite sides parallel, but it is in fact the affine image of a regular $n$-gon (phrased this way you don't have to restrict to even number of sides). The proof boils down to the fact that every polygon is a linear combination of regular polygons, and these are eigenvectors of the associated circulant matrix. One, then observes that the limiting shape is determined by looking at the two largest eigenvalues. This is nicely explained in a monthly article, "A Polygon Problem", by E. R. Berlekamp, E. N. Gilbert and F. W. Sinden.

Another interesting perspective is to think of the limiting shape as equally distributed points on an ellipse. This, in fact, makes an analogy with the continuous version of the problem. If you have a curve in $\mathbb R^2$ and want to consider a second order parabolic evolution equation which is invariant under affine transformations, you will be studying a flow called the affine normal flow. In "Contraction of convex hypersurfaces by their affine normal", Ben Andrews proves that the affine normal flow contracts every convex hypersurface to a point, and if you scale the limiting shape accordingly, the shape converges to an ellipsoid. In fact the midpoint iteration from before is just the discrete analog of this evolution for 2D curves!

Now, coming to your question about the 3 dimensional polytopes. Your discrete evolution is clearly invariant under affine transformations and should approximate a second order evolution equation, which would then coincide with the affine normal flow. Since in dimensions higher than two, the mesh becomes more refined in every iteration, the limiting surface should in fact converge to the ellipsoid, instead of just a discrete approximation of it as in the 2D case. Unfortunately I do not know enough about numerical analysis or differential geometry to tackle this problem, and just gave a heuristic answer. If you add those tags perhaps an expert will explain how to prove a result like this.

Answer 2

Just a bit more "data" in the form of images. I started with a "random polyhedron" built from the convex hull of a small number of points. Then I iterated the process of replacing each face by its centroid, i.e., $\alpha=\beta=\gamma=\frac{1}{3}$ in my original description, and again taking the convex hull. At least visually, the data support Gjergji's suggestion that the process converges to ellipsoids (of different axes lengths).
   
So, to make an explicit conjecture out of these observations:

Conjecture. Given any convex polyhedron $P$ in $\mathbb{R}^3$, let $c(P)$ be the convex hull of the centroids of the faces of $P$. Then $c^k(P)$ converges to an ellipsoid as $k \to \infty$.

Secondarily, it makes sense to conjecture the same holds for any convex polytope $P$ in $\mathbb{R}^d$.