OK, here is the World Map Creation System.
1) Go to Travel agents and get the times for their international flights. If the times between two cities are similar in both directions then we know that no "Jet Stream" like air currents are effecting the flight.
2) Get thin flexible wire/tube that has a bit of rigidity, like plastic fishtank hose (one for each flight detail you get).
3) Using a common scale, say 1 cm = 1 hour, cut the lengths of hose to the length of the flight times (label them with the start end cities so you remember what each is).
4) Connect all the hoses together joining the ends with the same cities together for all the lengths of hose.
What you have here is a representation of the flight times, of the various routes. But not only that, you have represented them as spatial relationships. As the actual flight times represent spatial relationships between locations on the Earth, you have just used the this data to show the spatial relationships between the various cities around the world and so reflects the actual geometrical relationship between the cities.
In other words, if the Earth is flat, then you will get a disk, if the Earht is round, then you will get a sphere.
As this is not perfect, there will be some errors (it won't be a perfectly flat plane or a perfect sphere). However, what is important is the connectivity between them. The connectivity between a Spherical and a flat Earth are so different as to be unmistakeable (the sphere will enclose a volume and the disk will not).
This is a really cheap way to make a map. It does not take a lot of effort (and some people find this fun too), and the data is available (and BTW Tom, it has been posted on the forums. Learn to use the search function or just lurk more) and easily gathered with minimal effort (a few phone calls, a trip to your local travel agents, even just go to their web sites, etc) and can be reliable as there are millions of people checking the reliability every day (every single person that has ever got on an international flight).
So here is reliable, independently verified data, and a method of turning that into an actual map for eithr a FE or RE in a way that confirms either one.
Who will take up this challenge (and yes I would like REers to have a go at this too)?