The limit of infinite truncations?

Question

When a regular polyhedron is made to undergo repeated truncations, is there a solid that acts as a kind of limit for this iterated process? That is, say a cube is truncated N times. As N gets larger and larger, is there a shape that is tended toward but never reached? Does the cube just shrink to a point? A sphere?

Answer 1

Strictly speaking, one need to specify precisely how the truncation is carried out in each iteration before we can have any form of definite answer.

Without further information, I will assume by truncation, you are referring to a complete truncation/rectification. For concreteness, we adopt following truncation procedure.

1. For each vertex of a polytope, find all edges attached to the vertex.
2. Remove the vertex by cutting off at midpoints of the edges.
3. If the mid-points are planar, replace the vertex by a planar polygon.
4. If not, replace the vertex by the surface of convex hull span by the mid-points.

Start from a unit cube, if we repeat truncate it, following is what we will get:

This is the result for first six iterations. They are drawn in scale and in the order from left to right, then top to bottom. Following is some numerical information about them (up to $8^{th}$ iteration):

$$\begin{array}{|r|rrr|ll|} \hline \#\text{iter} & V & E & F & \verb/Area/ & \verb/Volume/\\ \hline 1 & 12 & 24 & 14 & 4.73205080756888 & 0.833333333333333 \\ 2 & 24 & 48 & 26 & 4.05433304545186 & 0.708333333333333 \\ 3 & 48 & 96 & 50 & 3.78451479182779 & 0.661458333333333 \\ 4 & 96 & 192 & 98 & 3.66401838854418 & 0.639973958333333 \\ 5 & 192 & 432 & 242 & 3.60670601543203 & 0.630696614583333 \\ 6 & 432 & 1008 & 578 & 3.57853041917916 & 0.625325520833333 \\ 7 & 1008 & 2352 & 1346 & 3.56581924420161 & 0.623093922932943 \\ 8 & 2352 & 5424 & 3074 & 3.56017305789999 & 0.622058709462482 \\ \hline \end{array}$$

As one can see,

1. The shapes doesn't seem to converge to a point.
2. Instead, it seems to converge to some shape of finite extent.
3. The "limiting" shape doesn't look like a sphere at all.

I hope this numerical observations will help someone to construct a more concrete answer.