This surface shows that the
Plateau Problem can well have a
continuum of solutions: in this particular example (due to
Frank Morgan,
A smooth curve in R^3 bounding a continuum of minimal manifolds, Archives Ration. Mech. Anal. 75 (1981) 193-197.), the boundary is made of 4 coaxial circles. The
boundary is rotationally symmetric, while the
surface is not: by rotating, there are
uncountably many solutions to the Plateau problem! Of course, this particular minimal surface is not area-minimizing.
Controls: Move
mouse with left button or A key pressed to rotate.
Move
mouse with middle button or S key pressed to zoom.
The checkbox
Show half highlights the construction of this surface: in his paper, Frank Morgan starts with the least-area soap film spanning the contour shown in the model (a circle and two half circles connected by two line segments). His example is then obtained by adding a second copy of the same
soap film, rotated 180º around the line segments: by Schwarz reflection principle the two surfaces fit together to form a larger minimal surface bounded by 4 circles. The two halves are painted in different colors by checking the box
Two colors (of course,
Show half must be unchecked...)
The checkboxes
Show mesh and
Show smooth surface are self-explanatory: by default, both are checked.
The button
Change background cyclically changes the background among a few choices. The default background is flat, but less demanding in terms of graphics resources...
Webgl applet built with the library three.js (released under the MIT license). This applet is (C) 2012, Sisto Baldo and is released under GPL. Model computed with Ken Brakke's
Surface Evolver.