The aim of this project was to produce a world map using a map projection similar to what you would find in a modern atlas. Little did I realise that there were so many types of projections! You can find a list of map projections here on Wiki.
After examining various projections I settled on the Mollweide projection. It looks good, shows the whole world, and it doesn't distort things too much. But first I had to produce the gridlines. Through the cartographer's guild I found FlexProjector (thanks waldronate for finding this program). It's kind of clunky, but it servers its purpose. (Just remember to set the scale to exactly 0.5, otherwise you wont get a nice 2:1 ratio grid).
After starting some maps using a grid produced with FlexProjector I stumbled across Flaming Pear's Flexify 2 plugin for Photoshop.
Wow, what a great plugin! I quickly realised that the best way to use it was to start with a rectangular map with a 2:1 ratio. I based my map on a Jurassic map (c. 170 million years ago), that I found at Global Paleogeography, and used mapping methods that I either made up myself or found here (mostly pasis's excellent tutorial, Rising up the mountains in Photoshop).
So here are the results: The base map I whipped up in a couple of hours, then I had fun playing around with various projections.
The first map use the 2:1 ratio equirectangular projection. Notice how distorted things get at the poles.
The second map is the standard Mollweide projection. Things are looking better at the poles, but the land becomes distorted at the edges of the map.
The third map is a view from the north pole, and the fourth map is a view from the south pole.
The fifth map is a view from both poles! It's a little mind boggling.
The sixth map shows the two halves of the world, while the seventh shows the world as a globe.
Flexify has about 100 or so different projections. Most are too weird to be of much use, but I included two of the more normal-looking of the strange map projections. These are perfectly valid ways of mapping a spherical object onto a flat plane, but they look strange because we're not used to seeing a world portrayed in such a fashion. They could be fun to use as an alien map.