Cut away the corners of a cube simultaneously: vertex truncation. Cut away its edges simultaneously: edge truncation. In woodworking, this is called beveling (using a saw) or chamfering (using a plane).
Geometrically, this can be done with vertex motion.
Imagine three points at each vertex of the cube. Let the points travel along the edges at the same speed, from time
. As the points begin to travel they form small triangular faces and the old square faces turn into octagons. When the points reach the midpoints of the edges at
, the octagons become squares; the figure is the cuboctahedron. The vertex motion for the cube ends.
Think of the triangular faces as planes cutting off the corners; they could continue to cut away the solid after
. The end result would be the dual of the cube, the octahedron, but it would be very small.
Truncating the vertices of the octahedron also ends with the cuboctahedron, so add the reverse of that truncation to the one for the cube. That gives a smooth transition from the cube to the octahedron with the second half large enough to see well.
In the case of edge truncation for the cube, again three points move from each vertex, but this time they move toward the centers of the faces meeting at the vertex. Each edge gives rise to a new hexagonal face and the old square faces shrink. As with vertex truncations, the edge truncations of duals are the same when
, so again the two truncations are joined back-to-back.