Corner Cutting the Cube

> Starting with a 1 x 1 x 1 unit cube, cut all the corners off by slicing
> through the midway points of the vertices. Now repeat the process on all
> corners again, always cutting from the midway points of the vertices.
> Continue cutting corners ad infinitum.
>
> What is the limiting volume as the number of corners cut goes to infinity?
> What is the shape?

This is is a variant of Chaikin's corner-cutting algorithm. It is similar in spirit to a number of subdivision schemes for producing smooth surfaces from an original set of control vertices, including the Doo-Sabin algorithm and the Catmull-Clark(?) algorithm, which pick distances and weights so as to produce B-spline surfaces, and the Loop scheme, which is the grandfather of much of today's work on subdivision surfaces in graphics.

It's an interesting variant because it's singular in some sense: normally corner cutting algorithms cut off a little section 'w' at either end of each edge in the original model (w = 1/4 in the original chaikin algorithm), leaving some of the original edge. This variant results in the original edge vanishing.

Notes

The initial cube faces have a limit point at their centre
The surface is always composed of triangles and quadrilaterals
The triangles correspond to the original vertices of the cube; there are eight of them
The initial triangles (in P1) have a limit point at their centre: [2/3 2/3 2/3]
This has a radius of 2/sqrt(3) = 1.155, so the limit surface cannot be a sphere
In fact, interestingly, any face of any of the surfaces has a limit point at its centre

P0 to P5, and an approximation to the limit surface

So the result seems to be a pin cushion-like surface with G1 discontinuities corresponding to the original vertices.

Gratuitous animated gifs: