Soap Film Cones

If one magnifies a point of a soap film, one gets a tangent cone. Recall that a cone is composed of rays emanating from a point. At regular points the tangent cone is just a plane. At a singular point (a non-regular point), the tangent cone must be three half-planes meeting at 120 degrees, or a cone over the edges of a polyhedron. For such a polyhedral cone, the cone is made of flat sheets meeting along triple lines. There are only eight polyhedra that satisfy this condition. But only one gives a true minimal cone, since the other seven cones can be deformed to soap films of less area.

In the examples below, clicking on the small image will get a full-screen version the image. You can get a 3D version by clicking on the VRML tag below a picture, if you have a VRML2 viewer.

These images were generated with my Surface Evolver program. The Evolver datafile names are in parentheses. The datafiles contain commands to evolve the surfaces shown here. The VRML files were generated by the Evolver using my vrml2.cmd script.


Candidate cones:


Triple junction (ycone.fe)

triple junction frame
Frame
triple junction film
Minimal cone

(VRML)

Tetrahedron (tetraflm.fe)

tetrahedral frame
Frame
tetrahedral film
Minimal cone

(VRML)

Cube (cubecone.fe, cubefilm.fe)

cube frame
Frame
cube cone film
Cone

(VRML)
cube film
Better film

(VRML)

Triangular Prism (prismcone.fe, prismfilm.fe)

prism frame
Frame
prism cone
Cone

(VRML)
prism film
Better film

(VRML)

Pentagonal Prism (pentprsm.fe)

pentaprism frame
Frame
pentaprism cone
Cone

(VRML)
pentaprism film
Better film

(VRML)

Dodecahedron (dodecone.fe)

dodecahedron frame
Frame
dodecahedron cone
Cone

(VRML)
dodecahedron film
Better film

(VRML)

Cone Number 8 (cone8.fe)

frame no. 8
Frame
no. 8 cone
Cone

(VRML)
no. 8 film
Better film

(VRML)

Cone Number 9 (cone9.fe)

frame no. 9
Frame
no. 9 cone
Cone

(VRML)
no. 9 film
Better film

(VRML)

Cone Number 10 (cone10.fe)

frame no. 10
Frame
no. 10 cone
Cone

(VRML)
no. 10 film
Better film

(VRML)

Reference:

Jean E. Taylor, "The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces," Annals of Mathematics 103 (1976), 489-539.
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