This image at right was scanned from the article by Tomlow listed in the references, which contains additional material on polyhedral turnings. For a detailed history, see Joseph Connors, "Ars Tornandi: Baroque Architecture and the Lathe," Journal of the Warburg and Courtauld Institutes, LIII, 1990, pp. 217-36. The techniques for turning nested spheres were apparently invented in Milan by Giovanni Ambrogio Maggiore in 1582. Interestingly, turning was considered a noble hobby, and many gentlemen and kings worked lathes as a leisure activity, e.g., Lorenz Zick taught Emperor Ferdinand in Vienna. For instructions to a lathe craftsperson on how to make these and related polyhedral forms, see the book by Holtzapffel.
Nested polyhedral balls are still made in Hong Kong, but from a plastic impregnated with bone dust, rather than ivory. Below is an image of one I bought recently (very inexpensively on an on-line auction site). It has 14 holes, positioned as the vertices of a rhombic dodecahedron. This one is 3 inches in diameter and has 12 nested spheres, with the outer one carved into dragons.
Below is another interesting example of a turned ivory carving with dodecahedrally positioned openings, and nicely carved surfaces, artist unknown. Notice the occasional five-fold openings in the "weave".