I have a broad experience with Mathematica, being a user since 1990. During this time, I have written three books using Mathematica as the basis, the last having been published in 2016. During this time I have imagined the possibilities of creating permanent, tangible figures to match many of the beautiful surfaces described and illustrated in my books. Wood is a beautiful medium in which to work, and it lends almost a 4th dimension to the figure with its grain. Small imperfections, even in select hardwoods, offer a pleasing foil to the deterministic and rigid lines of the figure dictated by the algebraic or trigonometric equations.
My approach has not been to design a figure to be made from a single piece of wood, as many lathe turnings are. Rather, I build the figure with uniform-depth layers of wood. This has two important effects. First, the layered construction serves to eliminate the possibility of cracking and splitting which occasionally besets a single piece of wood, even when thoroughly dried. Second, the use of layers allows one to add a pattern to the figure, in a fairly simple manner, by using two contrasting woods.
Mathematica enables one to make precise 3-D illustrations of surfaces, and to make cross-sections which are applicable to wood construction. I use Mathematica in designing each of my figures, and take the paper plots to the shop for the production phase. An example of such a paper plan is for the ellipsoid. Then cutting, gluing, shaping, and final finish are all guided by the paper plan. The contrasting woods used in the figure, as well as the exact pattern of the contrasting woods is governed by the customer's request. The available patterns allow one to customize the figure for a desired effect. Two wood types are realized with any pattern, except of course pattern {0,0,0} which is a uniform color (a single wood type).