How the octagon method works
Seen from the side, a cylinder as long as it is wide is a square. Cut its four corners off at 45° and you have an octagon whose flats just touch the ball inside it. Take off the octagon's eight edges and you're most of the way to a sphere, with only a little left to round by eye.
Two ratios set the octagon: the octagon line is 0.293 × the diameter in from each end of the ball's section, and the groove at each end goes down to 0.414 × the diameter. That neck is the tenon that holds the ball between centers until you cut it free. With these numbers, all eight sides of the octagon come out equal.
Stock size is the square to start from: the ball plus 1/8", so there's wood to turn round. Corner depth is how much each 45° cut takes off, measured at the corner.
A worked example
For a 4" ball, start with 4 1/8" square stock and turn a cylinder exactly 4" across. Mark a 4" section, then pencil lines 1 3/16" (1.172") in from each end and one in the middle. Groove each end down to 1 21/32" (1.656"), cut the corners from the lines to the grooves at 45°, then round it off.
Check your progress with a template cut to the ball's radius from card or plywood, held against the spinning wood to find the high spots. A 4" ball roughs at about 1,500 RPM; the Lathe Speed Calculator works out any size.