This method doesn't quite work for me because I need to create negative volume along an existing edge.
Basically changing from a cross section along each edge that looks like the black lines in the image below, to one that looks like the purple lines.

There is added complexity because the tiles themselves are angled in relation to each other and each joint has three edges coming together. Also not all of the tiles are hexagons, some pentagons are required to tile a sphere.
As I mentioned to another commenter below I was able to solve this with projections and lofts but it was pretty tedious.
