Simple aid for turning spheres without a jig
Dennis in Southern Oregon
John K Jordan started a thread a couple of weeks ago on turning a sphere using a simple method attributed to Soren Berger that involves first turning a cylinder with a length equal to its diameter, then cutting off the corners to form an 8-sided figure. Optionally, one can then turn off the new corners to form a 16-sided figure. The result then can be smoothed out by eye to produce a pretty good sphere. The first three figures illustrate the sequence. 
I personally have used this method many times to turn spherical ornaments and spheres that become the bowls of one-piece ladles and spoons, and I find it fast and convenient. (Generally, I stop at the 8-sided figure stage.) Although I have never used a sphere turning jig, I expect that this method is far faster than using a jig if you do not require high accuracy. Furthermore, it strikes me that this method would a time-efficient way of roughing out a sphere before using the jig.
The catch is that in order to make the cuts you first need to determine where to mark and cut. At each stage, you need to mark the piece at a particular distance each side of the existing corners. In the first stage, the required distance is 0.2929 times the diameter of the sphere. In the second stage, the required distance is 0.1077 times the diameter of the sphere. Unfortunately, unless you have a calculator in the shop and a decimal ruler or caliper, this can be challenging.
I would like to offer a simple, fast, no-cost, no-math alternative that anyone can use. The fourth figure is a design graph that will let you measure the desired quantities directly using calipers for spheres up to 4 inches in diameter. The figure is easily drawn, as described below, or can be dragged off onto your desktop and printed. (It will not matter if the figure does not print to the original size, although it may change the size of the largest sphere that can be accommodated.) To use the graph, first set your calipers to the diameter of the sphere. Then, hold the calipers horizontally and move down the graph until they exactly span the distance between the reference line R and the diameter line D. Note this vertical position on the graph. (As a practical matter, taking the nearest ruled line usually is close enough.) Now, for the offset A that is required for the first stage, use your calipers or a strip of paper to measure the distance from the line R to the line A along that same horizontal line and use it mark cut lines on the work piece that distance each side of each of the corners. To obtain the offset B for the second stage do exactly the same, except copy the distance between line R and line B.
To draw a graph for yourself that will accommodate spheres of up to 4 inches diameter, obtain a sheet of quadrille paper (I used ¼” ruled paper, but any will work) and mark a point near the top left of the sheet as shown in the fourth figure and draw a vertical line down to one of the rulings near the bottom of the sheet. This is the reference line R. Then draw a horizontal base line as shown. Now, measuring from the R line along the base line, mark points at 0.431 (7/16), 1.17 (1 -3/16), and 4.0 inches. The fractional measurements shown are close approximations to the actual decimal values for those who do not have decimal rulers. If you need to turn larger spheres you can easily extend the range of the graph. Carefully align a second sheet of quadrille paper with the original piece, offsetting it to the right if necessary and extend the lines R, A, B and D onto the new sheet.
