you get broken lines in the sides of the cube (the columns of the cathedral seem to be broken). It is possible in theory to devise a conformal projection that would show the sphere as a cube. However it is very difficult to find information

What you see here is the result of a long and complicated search, but with an enormous
amount of help from Daniel M. German I've able to construct the conformal projection of the
cube. If your first reaction is to mutter: "The what of the cube?" or just
"So?", and if you are interested, please follow these explanations... You
may have seen before panoramas projected in a cube: in flickr for example, there is a