To smooth a manifold:
1. Select menu ‘Modify->Split Edges | Make Faces | Truncate Corners...’
2. The ‘Select Manifold’ window appears.
3. Select the Primal or Dual manifold radio button and click ‘Apply’.
Above: 4V Icosahedral Dual
Above: 4V Icosahedral Dual after Smoothing
The smooth algorithm is a simple one. First, the edges for each face are divided in two (1); the divided edges are then joined up to make new faces (2); in a projected manifold, this creates small prisms at the corners of the orginal faces. Finally, these prisms are deleted to create a new face at the original vertex (3).
Smooth can create some very complex and interesting geometries and these are best revealed using the Isolate Face tool on a smoothed manifold. Below is a sequence showing six successive smooth operations applied to a 3V Icosahedron with 'Project to Envelope' OFF. Note the appearance of a Celtic knot design in stages four and six...
Above: 3V Icosahedon. Project to envelope: OFF
Above: Smooth Pass One.
Above: Smooth Pass Two.
Above: Smooth Pass Three.
Above: Smooth Pass four Interesting Celtic knotwork appears in the geometry. (As found in the Pictish School).1
Above: Smooth Pass Five.
Above: Smooth Pass Six. Here, the Celtic knotwork has re-appeared but on a more intricate level. (As found in the Pictish School).2 Pass Six took a long time to complete.
1. See “Celtic Art, The Methods of Construction” by George Bain (published by Constable, London).