I dug up the tiny program I wrote to find minimal surfaces with fixed boundary in 2011 for the “grandes écoles” entrance exams. Ah, the memories…
It’s available here, along with some pictures.
It’s written in OCaml, of course!
To compile it, install OCaml, OCamlBuild, and the OpenGL libraries for OCaml.
Then run ocamlbuild main.native.
Controls are rudimentary: SPC to start/stop, left-click and scroll to move around, right-click to choose one of the four predefined boundaries, s to swap between full and hollow.
You then get images like this:
I absolutely do not guarantee that the program is correct. The surfaces found do resemble the theoretical ones, but I do not have a mathematical proof that they are the right ones.