The same dome used for Chapel 1 can be built over only four supports with a structure made of four columns and four arches:

The four columns are made of cuboctahedra, and are placed at the vertices of a square, at the same position as four of the columns of the previous construction. The arches are built on the outer edge of that square, in such a way that their tops are almost exactly where the tops of the other four columns were:

A square antiprism must be added to each column in order for the top square to have the right orientation. The structure of the arches, and their join to the columns, are better viewed in the following picture (of which there is a higher resolution version):

A small fitting problem arises now at the top central piece. We solve it by joining the four sections supported by the columns, which in this case are slightly more raised, with just two rods. The sections supported by the arches are again joined by four rods forming a square pyramid: