Morgan Minimal Surface
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.
Move mouse with left button or A key pressed
Move mouse with middle button or S key pressed
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