The standard world map that you tend to see in most contexts is called the Mercator Projection, and you may or may not be aware that is is not an accurate flattened representation of the Earth. There have been a few attempts over the years to project a less distorted image; perhaps the most well known in the laser cutting world is the Dymaxion Projection, but there are a few others out there.
While looking through maps of the Port of San Francisco, Hillary Predko came across the Cahill Butterfly Projection, which offers the same flattening of the globe without distortion, but maintains symmetry and a regular grid. After falling in love with it, Hillary set about laser cutting it out of plywood, then painted the ocean and gridlines to make a beautiful display piece.
As a continuation of this project, Hillary decided to laser cut one out of leather, with the intention of stuffing and sewing it together. Leather is an excellent medium for this as it is flexible and doesn't fray. If leather isn't your bag, you could substitute for a number of heavy natural fibers, but you will have to account for the fraying. Whatever you choose, just don't try using PVC!