Probably my favorite are the Myriahedral projections[1], basically the Dymaxion map, but with ridiculous number of faces. In his paper he also unfolds a fractal[2].
Are there any online maps that aren't Web Mercator? Google Maps, Here.com, OpenStreetMap, Bing Maps, MapQuest are all Mercator-only, with no easy option to switch.
One of the main reasons it's so popular is that is preserves shape, angle and north is always up. This means you can create a tiling map server and use the same tiles at all scales.
Is there some nice explanation of how the projections between various EPSG codes work? I converted one projection into another but in QGIS or in python pyproj, but except for the very basics I have no idea what is going on.
>To remedy the horizontal stretching, the Mercator Projection stretches the globe out in the vertical direction an equal amount to the stretching in the vertical direction.