Here are a few pictures of the process I took took to make my sphere chess board. Unfortunately I do not have all the calculations I made to make the square pieces fit together. However, after selecting my light wood (Soft Maple) and my dark wood (Walnut) for my board, I went to work. Using those forgotten calculations, each square piece was cut with near exactness. Each piece then went to the drill press to make holes for the magnets. After putting magnets in them, each were glued together in horizontal rings, which were octagonal shaped. With all eight rings glued up, and the magnets held in with Bondo, the two halves were glued together. With the two halves glued up, it went over to the lathe where it was turned from octagon to spherical shape. The two halves were then glued together to make the sphere, which is about 9" in diameter.
The Base was a little easier. It is about 10" in diameter, also turned on a lathe. It has three main parts to it. The stand, support and arm. The stand and what I call support looks like a disk with a ball stuck in it. They are held together with a bolt with a flat thrust bearing between them. The support has a notch cut thru it with the arm glued inside of it to hide the bolt and hold the sphere. The Arm was made using mostly a router. The router was used to make the perfect arc to appear machined. Before the arm was glued in, a drill press was used to add the decorative holes to make it appear to have measuring marks. The teeth on the arm were also made using the router. (The bit used was homemade to be at an angle instead of straight.)
The Axle on the inside of the sphere was made out of a metal rod. Also on the inside there is a small spring loaded mechanism that keeps the ball from free spinning. (Apologies for there are not pictures of this part.)
Lastly, there is a spacer and knob to turn the board. Each were designed after the Queen chess piece. The spacer was modeled after the base of the queen and keeps the base of the board away from the arm to allow the pieces to rotate around the sphere without hitting the arm. The knob at the top of the sphere was modeled after the top of the Queen (I used the queen because it is the the most powerful chess piece. And it ties the whole design together). This and the spacer were both made using the lathe.
I hope that this design was helpful, if at least inspirational to whom ever reads this.