I had a bit of trouble envisioning how to start off the center part, but Link #6 below from Crafty Stylish gives a good picture of the beginning of the vortex of covering the laundry line.
Using up odd bits of fabric from the closet and discarding the rest, it was a good way to not only have fun seaming up a quick bowl, but it helped in de-cluttering pieces of fabric previously used in Liturgical Art projects.
These links will give you all you need to know about how to make a fabric bowl if it does not come to you intuitively. Links will also lead you to some very pretty bowls made by other bloggers.
1 here (Mrs. Tittlemouse)
2 here (Radiant Home Studio)3 here (Pretty Prudent)
4 here (This Year's Dozen) explains continuous strips and a button decoration idea
5 here (Cynthiaf) her dog sports one of her bowls on his head, sweet!
6 here for Crafty Stylish
7 here for Craftsy coils at the end
8 here Wikihow shows the zig-zag stitch in detail
Besides reading and learning from the links above, my further suggestions include the following:
Now go out and do as I say, not as I do!