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.


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: