Note: This page is in progress. I just wanted to get it up because I felt like that would be the kick in the shin I needed to remember to finish it.
I got this idea from Sherwood Smith, who spoke about how useful it was for her to have her worlds mapped out on a beach ball in an interview with Jo Walton at Tor.com. I'm going to provide you with the information about what I've done with the idea, like it's a craft projece, because that's pretty much what it is.
But, first, the why: Essentially, as Smith said, a world is round, so what better to map a world out on than something equally round? Inflatable beach balls have the additional advantage of being collapsible for easy travel and storage. If all you have is one kingdom that you're writing about, or even one land mass, you can probably handle mapping it out flat paper. If you have a whole world with many countries and land areas that affect each other in diverse ways, maybe a beach ball world map is for you. Another advantage to this is that it's a little easier for me to visualize the distance between places on a globe than on a flat map when I'm working on a whole world. I can take the round map and make it flat later, with better ideas of the distances and times necessary for travel.
My supplies:
-One inflatable beach ball (20" or more probably is a good idea)
-One can of White Spray Paint
-Old newspaper
-A bunch of Sharpie markers
I wanted to start out with an all-white surface, and there were no all-white beach balls available locally, so I got the spray paint. I'm also better with spray paint and pens than I am with paintbrushes. I took the beach ball out to my backyard, spread out some newspaper to protect the precious pavement beneath. I anchored the ball in place with more newspaper and rocks because it was windy out. Then I sprayed the first coat. An hour later, I sprayed the second. Then I flipped the ball over and repeated the process. (I needed two coats because my beach ball had an especially stubborn section of warning label text. Blech.) Also, I flipped my ball too soon, so some newspaper got stuck to it. I'd suggest letting it dry a bit longer than I did, impatient as I am.
With a nice, clean surface to work, I divided the ball into twenty-four sections longitudinally. Since my beach-ball already had six panels, I divided each of the six panels into four. (It was easier math for me than to take the whole circumerence of the ball and divide it by twenty-four or any other crazy thing) I penciled in time zone lines. The time zone lines are great help for then making sense of distances between places. Even if the world is bigger than Earth, like Wyld is, I know about how much bigger it is, so I can do the math to figure out the distances in ratio to Earth. I put my "Prime Meridian" in red, my "International Date Line" in orange, and did the rest of the longitudinal lines in yellow
I got lucky with latitude. The arc from pole to pole measured 21", which placed my equator at 10.5", and each fifteen degree mark was separated by 1.75" around the circumference of the ball. I wanted 15 degree marks because it seemed like a nice even way to divide up the 180 degree half of the circle. All these lines I did in yellow.