Minimal Surfaces in the 3-Sphere with Symmetries

Nicholas Schmitt

Description


Here, minimal surfaces in the 3-sphere with different symmetries are shown, e.g. surfaces with the symmetries of Platonic solids and tesselations of the 3-sphere. One image shows the stereographic projection of the minimal cube highlighting its octahedral symmetry. This minimal surface of genus 5, with octahedral symmetry. is built by forming tubes along a wireframe cube. The symmetry group, of order 96, is generated by the octahedral symmetry group S4 of order 24, together with two reflections in geodesic 2-spheres. The lines on the surface are curvature lines, and disks are cut out at each of the 16 umbilics, at which three curvature lines meet.

The surfaces in the respective WebGL (model00 - model06), with genus g, are
• tetrahedron g=3
• octahedron g=7
• cube g=5
• octahedral join g=11
• icosahedron g=19
• dodecahedron g=11
• icosahedron join g=29

Other symmetric minimal surfaces are build by putting tubes on regular tessellations of S^3. The surfaces in the repsectives WebGL (model07 - model10), with genus g, are
• 5-cell g=6
• 16-cell g=17
• 24-cell g=73
• 600-cell g=601

In addition, one can see a Torus with eight Delaunay ends (model11).

Supplement materials


References


  1. [1] Hermmann Karcher, Ulrich Pinkall, and Ivan Sterling. New minimal surfaces in S^3. J. Differential Geom., 1988. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=961512.

Dr. Nicholas Schmitt   +

University: TU Berlin
Website: http://page.math.tu-berlin.de/~schmitt/