title: 3d-printing an inherently nonorientable surface
body: <p>I'm trying to print 3D models of <a href="http://mathworld.wolfram.com/NonorientableSurface.html" rel="nofollow noreferrer">nonorientable surfaces</a>:
  Klein bottle, Kuen surface, Boy surface, etc.</p>
  
  <p>From the surface's parametric representation (x,y,z as functions of u and v) I compute
  a triangular mesh, which is mostly repairable into printable form
  by tools such as MeshLab, netfabb, and 3DEdit Pro.</p>
  
  <p>However, these tools can't restore <a href="http://wiki.netfabb.com/STL_Files_and_Triangle_Meshes" rel="nofollow noreferrer">orientability</a>, which is required for 3d printing.  (The printer <em>must</em> know where the inside is, to know where to deposit material!)
  At any line of self-intersection, the two sheets of the nonorientable surface
  disagree as to which side is "outside."  In MeshLab, one sheet is black.  In netfabb, red.  Those triangles are called flipped; their normals are reversed.</p>
  
  <p>What approaches are reasonable?</p>
  
  <ul>
  <li><p>Resolve orientability by calculating the lines of self-intersection, to separate the sheets, so each sheet is its own "shell" in 3d-printing-ese.</p></li>
  <li><p>Print not the surface enclosing a solid volume, but rather the surface as a <a href="https://www.vismath.eu/en/3d-models/3d-print-klein-bottle" rel="nofollow noreferrer">lattice</a>.  (Does that just beg the question, because extrusion "into the interior" becomes vanishingly thin at the lines of self-intersection?)</p></li>
  <li><p>Print the model as is and then its "inverse" (reversed normals), giving two incomplete parts to assemble.  For example, see how this villain has lightsabered off the part of the Kuen surface that should be atop its central two peaks.  <img src="https://i.stack.imgur.com/ydomz.jpg" alt="Kuen surface, incomplete"></p></li>
  </ul>