Anyone have any idea how to create (or where to find) the Waterman "Butterfly" projection in ArcMap?
ESRI does not offer a way to use a waterman projection/datum because the Waterman Butterfly is converted from a sphere to a polyhedral conversion and all other projections are dependent on major axis, minor axis, semimajor axis, semiminor axis, and inverse flattening for sphere to ellipsoid conversion (technically using the following: ellipsoid, cone, or cylinder geometry). ESRI would need to do some dev work to grant users the ability to add dimensions and use other geometry for projections/datums that are non-standard. For example, currently you cannot modify inverse flattening and make your own projection in esri. In other platforms you can. ESRI does provide software developers kits for android, iOS, Java, .NET, OSX, Qt, Xamarin if you wanted to leverage developement outside of ESRI. Also, it could be done in open source like gdal ogr2ogr or quantumgis with a custom .py or c++ library. I hope this helps.