Cube, a smooth algebraic surface of degree six. This is the set of real points for which ``` x^6 + y^6 + z^6 -1 = 0. ```

Is everywhere smooth! Not an actual cube; you get a cube if you replace the 6 in the equation by infinity, but then it's impossible to compute...

I have provided these files:

• `Cube_fixed_Blocky.stl` -- has the normal vectors fixed.
• `Cube_fixed_Med_Smooth.stl` -- the blocky version, normal vectors fixed
• `cube_raw_blocky.stl` -- raw triangulation coming from Bertini_real. Since the program works in arbitrary dimensions, I make no effort to control normals from it -- they don't exist for 4- and higher-dimensional surfaces, but instead a tangent space which is not immediately useful for 3d printing. The raw versions are not directly suitable for 3d printing.
• `cube_raw_med_smooth.stl` --ran through `sampler` with fairly loose tolerances. Incorrect normals.
• `input` -- the Bertini_real input file used to compute it.

This surface was sampled before I implemented cyclenumber > 1 sampling, so the surface is undersampled near critical points and singularities. It's also deliberately low-res.

Computed with a Numerical Algebraic Geometry program I wrote, called Bertini_real and printed as part of my long-term project to reproduce Herwig Hauser's gallery of algebraic surface ray-traces in my own gallery of 3d prints. The ACM ToMS algorithm number is 976; the major published paper is DOI 10.1145/3056528 with several others preceding. Bertini_real implements the implicit function theorem for algebraic surfaces and curves in any (reasonable) number of variables.

These surfaces are generally challenging to print. Rotate, and use careful support. I use Simplify3D for the manual support placement feature. These surfaces are also very tiny in scale (arbitrary units and math and all) so require significant upsizing.

See also, my Thingiverse collection of algebraic surfaces.