The challenge: to construct a 3D printed Schwarz P surface piece for the Taping Shape* exhibit at the Rueben H. Fleet Science Center, which could be joined to others to create a finite part of a Schwarz P surface. I’m not the first to do this, Ken Brakke has already used his Surface Evolver program to create a beautiful and truly superior Schwarz P surface found on Shapeways.
With limited time before the exhibit, could we create a reasonable approximation of the Schwarz P surface using Cinema4D? We (Dave Pfaff and I) started by finding the minimal surface for a 4-gon with corners at the vertices of a regular octahedron. We then extended the resulting surface by 180 degree rotations about the straight boundary lines. This created a surface, but it was not quite right. We needed to cheat a bit and make the 4-gon surface closer to a quarter circle in the middle. (The actual Schwarz P surface is not circular there, but is close.)
After using the Close Polygon tool on the 4-gon, we used the Subdivide command for the 4-gon, then moved vertices closer to the circle. We subdivided again, moved vertices closer to the circle again and repeated the process. We then rotated 12 copies of the 4-gon unit around various edges to get the figure to the left.
We then arranged 6 of these units in space, and added in a cube. We used the Boole command to cut out a cubical Schwarz P unit. I then extruded the surface, and added magnet holes as described previously in this post:
I made two sizes of models: 6cmx6cmx6cm and 10cmx10cmx10cm. We printed the models on the uPrint SE printer. They printed just wonderfully. The one small flaw in the design is that there is a slightly raised line in the place where we moved vertices to the circular arc. However, the model has many strengths: aside from the line it is quite smooth, and you can almost (but not quite) see the 4-gons. Given the time restriction before the exhibition, we decided to leave the model as is.
You can find the files for the model, and instructions on how to place the magnets, here on Thingiverse.
*The Taping Shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587). (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)